Mathematica Vol. 6, No. 2, 2014

Page 1


Acta Universitatis Sapientiae The scientific journal of the Sapientia University publishes original papers and deep surveys in several areas of sciences written in English. Information about the appropriate series can be found at the Internet address http://www.acta.sapientia.ro.

´ Zolt´an A. BIRO ´ ˝ Agnes PETHO

Editor-in-Chief ´ L´aszl´o DAVID Main Editorial Board ´ Zolt´an KASA

Andr´as KELEMEN Em˝od VERESS

Acta Universitatis Sapientiae, Mathematica Executive Editor ´ R´ obert SZASZ (Sapientia University, Romania) Editorial Board S´ebastien FERENCZI (Institut de Math´ematiques de Luminy, France) ˝ K´ alm´ an GYORY (University of Debrecen, Hungary) ´ (Sapientia University, Romania) Zolt´ an MAKO ˇ´IK (University of Ostrava, Czech Republic) Ladislav MIS ´ J´anos TOTH (Selye University, Slovakia) Adrian PETRUS ¸ EL (Babe¸s-Bolyai University, Romania) ´ Alexandru HORVATH (Petru Maior University of Tg. Mure¸s, Romania) ´ Arp´ad BARICZ (Babe¸s-Bolyai University, Romania) ´ ´ (Babe¸s-Bolyai University, Romania) Csaba SZANT O ´ (Babe¸s-Bolyai University, Romania) Szil´ard ANDRAS Assistant Editor ´ (Sapientia University, Romania) P´ al KUPAN Contact address and subscription: Acta Universitatis Sapientiae, Mathematica RO 400112 Cluj-Napoca Str. Matei Corvin nr. 4. Email: acta-math@acta.sapientia.ro Each volume contains two issues.

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Acta Universitatis Sapientiae

Mathematica Volume 6, Number 2, 2014

Sapientia Hungarian University of Transylvania Scientia Publishing House



Contents

R. S. Batahan, A. A. Bathanya On generalized Laguerre matrix polynomials . . . . . . . . . . . . . . . . . . 121 B. A. Bhayo, L. Yin Logarithmic mean inequality for generalized trigonometric and hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 A. A. Bouchentouf, H. Sakhi Stabilizing priority fluid queueing network model . . . . . . . . . . . . . 146 S. S. Dragomir Some inequalities of Furuta’s type for functions of operators defined by power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 S. Mustonen, P. Haukkanen, J. Merikoski Some polynomials associated with regular polygons . . . . . . . . . . . . 178 M. Z. Sarikaya, S. Erden On the weighted integral inequalities for convex function . . . . . . 194 N. Basu, A. Bhattacharyya Evolution of =-functional and ω-entropy functional for the conformal Ricci flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Contents of volume 6, 2014 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

119



Acta Univ. Sapientiae, Mathematica, 6, 2 (2014) 121–134 DOI: 10.1515/ausm-2015-0001

On generalized Laguerre matrix polynomials Raed S. Batahan

A. A. Bathanya

Department of Mathematics, Faculty of Science, Hadhramout University, 50511, Mukalla, Yemen email: rbatahan@hotmail.com

Department of Mathematics, Faculty of Education, Shabwa, Aden University, Yemen email: abathanya@yahoo.com

Abstract. The main object of the present paper is to introduce and study the generalized Laguerre matrix polynomials for a matrix that satisfies an appropriate spectral property. We prove that these matrix polynomials are characterized by the generalized hypergeometric matrix function. An explicit representation, integral expression and some recurrence relations in particular the three terms recurrence relation are obtained here. Moreover, these matrix polynomials appear as solution of a differential equation.

1

Introduction

Laguerre, Hermite, Gegenbauer and Chebyshev matrix polynomials sequences have appeared in connection with the study of matrix differential equations [8, 7, 20, 4]. In [13], the Laguerre and Hermite matrix polynomials were introduced as examples of right orthogonal matrix polynomial sequences for appropriate right matrix moment functionals of integral type. The Laguerre matrix polynomials were introduced and studied in [11, 14, 16, 17]. In [22], it is shown that these matrix polynomials are orthogonal with respect to a non-diagonal Sobolev-Laguerre matrix polynomials matrix moment functional. Recently, the 2010 Mathematics Subject Classification: 15A15, 33C45, 42C05 Key words and phrases: Laguerre matrix polynomials, three terms recurrence relation, generalized hypergeometric matrix function and Gamma matrix function

121


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R. S. Batahan, A. A. Bathanya

numerical inversion of Laplace transforms using Laguerre matrix polynomials has been given in [18]. A generalized form of the Gegenbauer matrix polynomials is presented in [2]. Moreover, two generalizations of the Hermite matrix polynomials have been given in [1, 19]. The main aim of this paper is to consider a new generalization of the Laguerre matrix polynomials. The structure of this paper is the following. After a section introducing the notation and preliminary results, we characterize, in Section 3, the definition of the generalized Laguerre matrix polynomials and an explicit representation and integral expression are given. Finally, Section 4 deals with some recurrence relations in particular the three terms recurrence relation for these matrix polynomials. Furthermore, we prove that the generalized Laguerre matrix polynomials satisfy a matrix differential equation.

2

Preliminaries

Throughout this paper, for a matrix A in CN×N , its spectrum σ(A) denotes the set of all eigenvalues of A. We say that a matrix A is a positive stable if Re(µ)> 0 for every eigenvalue µ ∈ σ(A). If f(z) and g(z) are holomorphic functions of the complex variable z, which are defined in an open set Ω of the complex plane and A is a matrix in CN×N with σ(A) ⊂ Ω, then from the properties of the matrix functional calculus [5, p. 558], it follows that f(A)g(A) = g(A)f(A). The reciprocal gamma function denoted by Γ −1 (z) = 1/Γ (z) is an entire function of the complex variable z. Then, for any matrix A in CN×N , the image of Γ −1 (z) acting on A, denoted by Γ −1 (A) is a well-defined matrix. Furthermore, if A + nI is invertible for every integer n ≥ 0,

(1)

where I is the identity matrix in CN×N , then Γ (A) is invertible, its inverse coincides with Γ −1 (A) and it follows that [6, p. 253] (A)n = A(A + I)...(A + (n − 1)I); n ≥ 1,

(2)

with (A)0 = I. For any non-negative integers m and n, from (2), one easily obtains (A)n+m = (A)n (A + nI)m , and (A)mn = mmn

m Y 1 A + (s − 1)I . m n s=1

(3)

(4)


On generalized Laguerre matrix polynomials

123

Let P and Q be commuting matrices in CN×N such that for all integer n ≥ 0 one satisfies the condition P + nI,

Q + nI,

and P + Q + nI

are invertible.

(5)

Then by [10, Theorem 2] one gets B(P, Q) = Γ (P)Γ (Q)Γ −1 (P + Q),

(6)

where the gamma matrix function, Γ (A), and the beta matrix function, B(P, Q), are defined respectively [9] by Z∞ Γ (A) = exp(−t)tA−I dt, (7) 0

and

Z1 tP−I (1 − t)Q−I dt.

B(P, Q) =

(8)

0

In view of (7), we have [10, p. 206] (A)n = Γ (A + nI)Γ −1 (A);

n ≥ 0.

(9)

If λ is a complex number with Re(λ) > 0 and A is a matrix in CN×N with A + nI invertible for every integer n ≥ 1, then the n-th Laguerre matrix (A,λ) polynomials Ln (x) is defined by [8, p. 58] (A,λ) Ln (x)

n X (−1)k λk (A + I)n [(A + I)k ]−1 xk , = k!(n − k)!

(10)

k=0

and the generating function of these matrix polynomials is given [8] by G(x, t, λ, A) = (1 − t)−(A+I) exp

−λxt 1−t

=

X

(A,λ)

Ln

(x)tn .

(11)

n≥0

According to [8], Laguerre matrix polynomials satisfy the three-term recurrence relation (A,λ) (A,λ) (n + 1)Ln+1 (x) + λxI − (A + (2n + 1)I) Ln (x) + (12) (A,λ)

(A + nI)Ln−1 (x) = θ; (A,λ)

(A,λ)

with L−1 (x) = θ and L0

n ≥ 0,

(x) = I where θ is the zero matrix in CN×N .


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R. S. Batahan, A. A. Bathanya

Definition 1 [2] Let p and q be two non-negative integers. The generalized hypergeometric matrix function is defined in the form: (13)

p Fq (A1 , . . . , Ap ; B1 . . . , Bq ; z)

=

X

(A1 )n . . . (Ap )n [(B1 )n ]−1 . . . [(Bq )n ]−1

n≥0

zn n!

,

where Ai and Bj are matrices in CN×N such that the matrices Bj ; 1 ≤ j ≤ q satisfy the condition (1). With p = 1 and q = 0 in (13), one gets the following relation due to [11, p. 213] X 1 (A)n zn , |z| < 1. (14) (1 − z)−A = n! n≥0

The following lemma provides results about double matrix series. The proof are analogous to the corresponding for the scalar case c.f [15, p. 56] and [21, p. 101]. Lemma 1 [2, 3, 19] If A(k, n) and B(k, n) are matrices in CN×N for n ≥ 0 and k ≥ 0, then it follows that: XX

A(k, n) =

n≥0 k≥0

X bn/mc X

XX

A(k, n) =

B(k, n) =

n≥0 k≥0

A(k, n − k),

(15)

A(k, n + mk),

(16)

n≥0 k=0

n≥0 k=0

and

n XX

XX n≥0 k≥0

X bn/mc X

B(k, n − mk)

; n > m,

(17)

n≥0 k=0

where bac is the standard floor function which maps a real number a to its next smallest integer. It is obviously desirable, by (2), to have the following: 1 I = (n − mk)! =

(−1)mk (−nI)mk n! m (−1)mk mk Y p − n − 1 m I ; n! m k p=1

(18) 0 ≤ mk ≤ n.


On generalized Laguerre matrix polynomials

3

125

Definition of generalized Laguerre matrix polynomials

Let A be a matrix in CN×N satisfying the spectral condition (1) and let λ be a complex number with Re(λ) > 0. For a positive integer m, we can define the generalized Laguerre matrix polynomials [GLMPs] by ! ∞ X −λxm tm (A,λ) −(A+I) = (19) Ln,m (x)tn . F(x, t, λ, A) = (1 − t) exp (1 − t)m n=0

By (14) one gets ∞ X ∞ X (−1)k λk n=0 k=0

k!n!

x

mk

(A + I + mkI)n t

n+mk

=

∞ X

(A,λ)

Ln,m (x)tn ,

n=0

which by using (17) and (3) and equating the coefficients of tn , yields an explicit representation for the GLMPs in the form: X

bn/mc (A,λ) Ln,m (x)

=

k=0

(−1)k λk (A + I)n [(A + I)mk ]−1 xmk . k!(n − mk)!

(20)

It should be observed that when m = n, the explicit representation (20) becomes (A + I)n (A,λ) Ln,n (x) = − λxn I. n! If m > n, then from (20) one gets (A,λ)

Ln,m (x) =

(A + I)n . n!

Moreover, it is evident that (A,λ)

Ln,m (0) =

(A + I)n n!

(A,λ)

(A,1)

1

and Ln,m (x) = Ln,m (λ m x).

Note that the expression (20) coincides with (10) for the case m = 1. In view of (4) and (18), we can rewrite the formula (20) in the form (A,λ) Ln,m (x)

bn/mc m (A + I)n X (−1)(m+1)k λk mk Y p − n − 1 x I k = n! k! m k=0 p=1 " m #−1 Y 1 × (A + sI k . m s=1

(21)


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R. S. Batahan, A. A. Bathanya

Therefore, in view of (13), the hypergeometric matrix representation of GLMPs is given in the form: (A + I)n n! ! −n (−n + m − 1) A + I A + mI I, · · · , I; ,··· , ; (−1)m+1 λxm . m m m m

(A,λ)

Ln,m (x) = m Fm

(22)

We give a generating matrix function of GLMPs. This result is contained in the following. Theorem 1 Let A be a matrix in CN×N satisfying (1) and let λ be a complex number with Re(λ) > 0. Then ! xt m X A + mI A+I −1 (A,λ) n t ,··· , ; −λ . (23) [(A + I)n ] Ln,m (x)t = e 0 Fm −; m m m n≥0

Proof. By virtue of (20) and applying (16), we have ! ! X tn X X (−1)k λk (A,λ) [(A + I)n ]−1 Ln,m (x)tn = [(A + I)mk ]−1 xmk tmk , n! k! n≥0

n≥0

k≥0

which, by using (4) and (13), reduces to (23).

It is clear that t

e 0 Fm

xyt m A+I A + mI −; ,··· , ; −λ m m m e

(1−y)t ty

e

0 Fm

! =

! xyt m A+I A + mI −; ,··· , ; −λ . m m m

Thus, by using (23) and applying (15), it follows that X

[(A +

(A,λ) I)n ]−1 Ln,m (xy)tn

=

n≥0

n XX (1 − y)n−k yk n≥0 k=0

(n − k)!

(A,λ)

[(A + I)k ]−1 Lk,m (x)tn .

By equating the coefficients of tn , in the last series, one gets (A,λ)

Ln,m (xy) = (A + I)n

n X (1 − y)n−k yk k=0

(n − k)!

(A,λ)

[(A + I)k ]−1 Lk,m (x).


On generalized Laguerre matrix polynomials

127

Let B be a matrix in CN×N satisfying (1). From (3), (4), (14) and (16) and taking into account (20) we have X

(A,λ)

(B)n [(A + I)n ]−1 Ln,m (x)tn

n≥0

= (1 − t)−B

X (−λ)n n≥0

n!

(B)mn [(A + I)mn ]−1

xt mn . 1−t

(24)

By using (4) and (13), the equation (24) gives the following generating function of GLMPs: X

(A,λ)

(B)n [(A + I)n ]−1 Ln,m (x)tn = (1 − t)−B

n≥0

! xt m B + (m − 1)I A + I A + mI B ,··· , ; ,··· , ; −λ . m m m m 1−t

m Fm

(25)

Clearly, (25) reduces to (19) when B = A + I. We now proceed to give an integral expression of GLMPs. For this purpose, we state the following result. Theorem 2 Let A and B be positive stable matrices in CN×N such that AB = BA. Then (A+B,λ)

Ln,m

(x) = Γ (A + B + (n + 1)I)Γ −1 (B)Γ −1 (A + (n + 1)I) Z1 (A,λ) × tA (1 − t)B−I Ln,m (xt)dt.

(26)

0

Proof. According to (8) and (20), we can write Z1 Ψ=

0 bn/mc

=

(A,λ)

tA (1 − t)B−I Ln,m (xt)dt X k=0

(−1)k λk (A + I)n [(A + I)mk ]−1 xmk B(A + (mk + 1)I, B), k!(n − mk)!

(27)

and since the summation in the right-hand side of the above equality is finite, then the series and the integral can be permuted. Hence by (6) and (9) it


128

R. S. Batahan, A. A. Bathanya

follows that X (−1)k λk xmk Γ −1 A + B + (mk + 1)I k!(n − mk)! k=0 = Γ A + (n + 1)I Γ (B)Γ −1 A + B + (n + 1)I bn/mc

Ψ = Γ A + (n + 1)I Γ (B)

(28)

X (−1)k λk xmk (A + B + I)n [(A + B + I)mk ]−1 . k!(n − mk)!

bn/mc

k=0

From (20), (27) and (28), the expression (26) holds.

We conclude this section giving an integral form of GLMPs. Theorem 3 For GLMPs the following holds Z∞

(A,λ)

xA Ln,m (x)e−x dx =

0 m F0

Γ (A + (n + 1)I) n!

! −n −n + m − 1 m+1 m . ,··· , ; −; (−1) λm m m

(29)

Proof. From (7), (9) and (20), it follows that Z∞ 0

X

bn/mc (A,λ) xA Ln,m (x)e−x dx

=

k=0

(−1)k λk (A + I)n [(A + I)mk ]−1 k!(n − mk)!

Γ (A + (mk + 1)I) X

bn/mc

= Γ (A + (n + 1)I)

k=0

(−1)k λk . k!(n − mk)!

Using (18) and taking into account (13) we arrive at (29).

4

Recurrence relations

In addition to the three terms recurrence relation, some differential recurrence relations of GLMPs are obtained here.


On generalized Laguerre matrix polynomials

129

Theorem 4 The generalized Laguerre matrix polynomials satisfy the following relations: 2m X 2m (A,λ) (−1)r (n + 1 − r)Ln+1−r,m (x) r r=0 2m−1 X 2m − 1 (A,λ) = (A + I) (−1)r Ln−r,m (x) r r=0 m X m (A,λ) − mλxm (−1)r Ln−r−m+1,m (x) r r=0 m−1 X m − 1 (A,λ) m − mλx (−1)r Ln−r−m,m (x), r

(30)

r=0

and

m X m

r

r=0

(A,λ)

(A,λ)

(−1)r DLn−r,m (x) = −λmxm−1 Ln−m,m (x).

(31)

Proof. Differentiating (19) with respect to t yields X (A,λ) X (A,λ) (1 − t)2m nLn,m (x)tn−1 = (A + I)(1 − t)2m−1 Ln,m (x)tn n≥1 m m−1

−λmx t

(1 − t)

m

X

n≥0 (A,λ) Ln,m (x)tn

m m

− λx t (1 − t)m−1

X n≥0

n≥0

With the help of the binomial theorem, it follows that 2m XX 2m n≥0 r=0

r

(A,λ)

(−1)r (n + 1)Ln+1,m (x)tn+r

X 2m−1 X 2m − 1

(A,λ)

(−1)r Ln,m (x)tn+r r n≥0 r=0 " m X X m (A,λ) m − λx m (−1)r Ln−m+1,m (x)tn+r r n≥m−1 r=0 # m−1 X X m − 1 r (A,λ) n+r + (−1) Ln−m,m (x)t . r

= (A + I)

n≥m r=0

Hence, by equating the coefficients of tn , equation (30) holds.

(A,λ)

Ln,m (x)tn .


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R. S. Batahan, A. A. Bathanya

Now, by differentiating (19) with respect to x one gets X

(A,λ) DLn,m (x)tn

n≥0

! −λmxm−1 tm −λxm tm −(A+I) = (1 − t) exp . (1 − t)m (1 − t)m

Thus, it follows that m X (A,λ) XX m (A,λ) Ln−m,m (x)tn , (−1)r DLn−r,m (x)tn = −λmxm−1 r n≥m

n≥m r=0

which, by equating the coefficients of tn , gives (31). It is worthy to mention that (30) reduces to (12) for m = 1. Also, for the case m = 1, the expression (31) gives the result for the Laguerre matrix polynomials in the form (A,λ)

DLn

(A,λ)

(A,λ)

(x) = DLn−1 (x) − λLn−1 (x).

Differentiating (19) with respect to x again we obtains X

(A,λ)

DLn,m (x)tn

n≥0

−λxm tm = −λmxm−1 tm (1 − t)−(A+(m+1)I) exp (1 − t)m X (A+mI,λ) = −λmxm−1 Ln−m,m (x)tn .

!

n≥m

Hence, by equating the coefficients of tn , we readily obtain (A,λ)

(A+mI,λ)

DLn,m (x) = −λmxm−1 Ln−m,m (x).

(32)

It may be noted that the formula (32) reduces to the result of [12, p. 16] for Laguerre matrix polynomials, when m = 1, in the form (A,λ)

DLn

(A+I,λ)

(x) = −λLn−1

(x).

Using the fact that −λxm tm −λxm tm m −(A+(m+1)I) = (1 − t) (1 − t) exp , (1 − t)−(A+I) exp (1 − t)m (1 − t)m and (19), one gets X n≥0

(A,λ) Ln,m (x)tn

m XX m (A+mI,λ) = (−1)r Ln−r,m (x)tn . r n≥r r=0


On generalized Laguerre matrix polynomials

131

Hence, we obtain that (A,λ) Ln,m (x)

m X m (A+mI,λ) = (−1)r Ln−r,m (x). r r=0

Let B be a matrix in CN×N satisfying (1) with AB = BA. Note that ! ! −λxm tm −λxm tm −(A+I) −(A−B) −(B+I) (1 − t) exp = (1 − t) (1 − t) exp . (1 − t)m (1 − t)m Using (14), (15) and (19) , it follows that X

n XX (A − B)k

(A,λ)

Ln,m (x)tn =

n≥0

k!

n≥0 k=0

Identifying the coefficients of

tn ,

(A,λ)

Ln,m (x) =

(B,λ)

Ln−k,m (x)tn .

in the last series, gives n X (A − B)k

k!

k=0

(B,λ)

Ln−k,m (x).

(33)

By reversing the order of summation in (33), we obtain that (A,λ)

Ln,m (x) =

n X (A − B)n−k k=0

(n − k)!

(B,λ)

Lk,m (x).

(34)

And finally, we prove the following result. Theorem 5 The GLMPs is a solution of the following differential equation " m ! Y 1 1 Θ (Θ − 1)I + (A + sI) + (−1)m λmxm m m s=1 !# (35) m Y p−n−1 1 (A,λ) Θ+ I Ln,m (x) = θ, × m m p=1

d where Θ = x dx . 1 Θxmk = kxmk . According to (22) we can write Proof. It is clear that m n n−m+1 A+I A + mI m+1 m W = m Fm − I, . . . , − I; ,..., ; (−1) λx m m m m " m #−1 bn/mc m X Y p − n − 1 Y 1 xmk = g (A + sI (−1)(m+1)k λk . m m k! k k k=0 p=1

s=1


132

R. S. Batahan, A. A. Bathanya

It follows after replacing k by k + 1 and using (3) that 1 Y Θ m m

1 1 (Θ − 1)I + (A + sI) W m m s=1 " m #−1 bn/mc m X Y p − n − 1 Y 1 = (A + sI m m k+1 k k=0 p=1

s=1

(−1)(m+1)(k+1) λk+1 m+1

= (−1)

m

λx

xm(k+1) k!

bn/mc m X Y k=0 p=1

(−1)(m+1)k λk

p−n−1 m

"

k+1

m Y 1 g (A + sI m k

#−1

s=1

xmk

k! m Y 1 p−n−1 m+1 m = (−1) λx Θ+ W. m m p=1

Therefore, W is a solution of the following differential equation " # m m Y 1 1 p−n−1 1 Y 1 m m Θ (Θ−1)I+ (A+sI) +(−1) λ x Θ+ W = θ. m m m m m s=1

p=1

(A,λ)

Since W = n![(A + I)n ]−1 Ln,m (x), then (35) follows immediately.

It is worth noticing that taking m = 1 in (35) gives the following [8] " # d2 d (A,λ) xI 2 + (A + (1 − λx)I) + λnI Ln (x) = θ. dx dx

Acknowledgments The authors wish to express their gratitude to the unknown referee for several helpful suggestions.


On generalized Laguerre matrix polynomials

133

References [1] R. S. Batahan, A new extension of Hermite matrix polynomials and its applications, Linear Algebra Appl., 419 (2006), 82–92. [2] R. S. Batahan, Generalized Gegenbauer matrix polynomials, series expansion and some properties, In: Linear Algebra Research Advances, Editor G. D. Ling, Nova Science Publishers, (2007), 291–305. [3] E. Defez, L. J´odar, Some applications of the Hermite matrix polynomials series expansions, J. Comput. Appl. Math., 99 (1998), 105–117. [4] E. Defez, L. J´odar, Chebyshev matrix polynomials and second order matrix differential equations, Util. Math., 61 (2002), 107–123. [5] N. Dunford, J. Schwartz, Linear Operators, Vol. I, Interscience, New York, (1957). [6] E. Hille, Lectures on Ordinary Differential Equations, Addison-Wesley, New York, (1969). [7] L. J´odar, R. Company, Hermite matrix polynomials and second order matrix differential equations, J. Approx. Theory Appl., 12 (2) (1996), 20–30. [8] L. J´odar, R. Company, E. Navarro, Laguerre matrix polynomials and system of second-order differential equations, Appl. Numer. Math., 15 (1994), 53–63. [9] L. J´odar, J. C. Cort´es, Some properties of gamma and beta matrix function, Appl. Math. Lett., 11 (1) (1998), 89–93. [10] L. J´odar, J. C. Cort´es, On the hypergeometric matrix function, J. Comput. Appl. Math., 99 (1998), 205–217. [11] L. J´odar, E. Defez, On Hermite matrix polynomials and Hermite matrix function, J. Approx. Theory Appl., 14 (1) (1998), 36–48. [12] L. J´odar, E. Def ez, A Connection between Laguerre’s and Hermite’s matrix polynomials, Appl. Math. Lett., 11 (1) (1998), 13–17. [13] L. J´odar, E. Defez, E. Ponsoda, Orthogonal matrix polynomials with respect to linear matrix moment functionals: Theory and applications, J. Approx. Theory Appl., 12 (1) (1996), 96–115.


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[14] L. J´odar, J. Sastre, The growth of Laguerre matrix polynomials on bounded intervals, Appl. Math. Lett., 13 (8) (2000), 21–26. [15] E. D. Rainville, Special Functions, The Macmillan Company, New York, (1960). [16] J. Sastre, E. Defez, On the asymptotics of Laguerre matrix polynomial for large x and n, Appl. Math. Lett., 19 (2006), 721–727. [17] J. Sastre, E. Defez, L. J´odar, Laguerre matrix polynomial series expansion: Theory and computer applications, Math. Comput. Modelling, 44 (2006), 1025–1043. [18] J. Sastre, E. Defez and L. J´odar, Application of Laguerre matrix polynomials to the numerical inversion of Laplace transforms of matrix functions, Appl. Math. Lett., 24 (9) (2011), 1527–1532. [19] K. A. M. Sayyed, M. S. Metwally, R. S. Batahan. On Gegeralized Hermite matrix polynomials , Electron. J. Linear Algebra, 10 (2003), 272–279. [20] K. A. M. Sayyed, M. S. Metwally, R. S. Batahan. Gegenbauer matrix polynomials and second order matrix differential equations, Divulg. Mat., 12 (2) (2004) , 101–115. [21] H. M. Srivastava, H. L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), Wiley, New York, Chichester, Brisbane, and Toronto, (1984). [22] Z. Zhu, Z. Li, A note on Sobolev orthogonality for Laguerre matrix polynomials, Anal. Theory Appl., 23 (1) (2007), 26–34.

Received: 10 November 2014


Acta Univ. Sapientiae, Mathematica, 6, 2 (2014) 135–145 DOI: 10.1515/ausm-2015-0002

Logarithmic mean inequality for generalized trigonometric and hyperbolic functions Barkat Ali Bhayo

Li Yin

Department of Mathematical Information Technology, University of Jyv¨ askyl¨ a, 40014 Jyv¨ askyl¨ a, Finland email: bhayo.barkat@gmail.com

Department of Mathematics, Binzhou University, Binzhou City, Shandong Province, 256603, China email: yinli 79@163.com

Abstract. In this paper we study the convexity and concavity properties of generalized trigonometric and hyperbolic functions in case of Logarithmic mean.

1

Introduction

Recently, the study of the generalized trigonometric and generalized hyperbolic functions has got huge attention of numerous authors, and has appeared the huge number of papers involving the equalities and inequalities and basis properties of these function, e.g. see [7, 8, 9, 6, 10, 13, 14, 18, 23] and the references therein. These generalized trigonometric and generalized hyperbolic functions p-functions depending on the parameter p > 1 were introduced by Lindqvist [19] in 1995. These functions coincides with the usual functions for p = 2. Thereafter Takesheu took one further step and generalized these function for two parameters p, q > 1, so-called (p, q)-functions. In [8], some convexity and concavity properties of p-functions were studied. Thereafter those results were extended in [5] for two parameters in the sense of Power mean inequality. In this paper we study the convexity and concavity property of p-function with 2010 Mathematics Subject Classification: 33B10; 26D15; 26D99 Key words and phrases: logarithmic mean, generalized trigonometric and hyperbolic functions, inequalities, generalized convexity

135


136

Generalized trigonometric and hyperbolic functions

respect Logarithmic mean. Before we formulate our main result we will define generalized trigonometric and hyperbolic functions customarily. The eigenfunction sinp of the so-called one-dimensional p-Laplacian problem [12] 0 −∆p u = − |u 0 |p−2 u 0 = λ|u|p−2 u, u(0) = u(1) = 0, p > 1, πp 2 ,

is the inverse function of F : (0, 1) → 0,

Zx

F(x) = arcsinp (x) =

defined as 1

(1 − tp )− p dt,

0

where 2 πp = 2arcsinp (1) = p

Z1 (1 − s)

−1/p 1/p−1

s

0

2 1 1 2π , ds = B 1 − , = p p p p sin πp

here B(., .) denotes the classical beta function. The function arcsinp is called the generalized inverse sine function, and coincides with usual inverse sine function for p = 2. Similarly, the other generalized inverse trigonometric and hyperbolic functions arccosp : (0, 1) → (0, πp /2) , arctanp: (0, 1) → (0, bp ), arcsinhp: (0, 1) → (0, cp ), arctanhp: (0, 1) → (0, ∞), where 1 1 1 1 1+p 1 1 1 bp = ψ −ψ = 2− p F , ;1 + ; , 2p 2p 2p p p p 2 1 1 p 1 1 1 cp = , F 1, ; 1 + , 2 p p 2 are defined as follows Z (1−xp ) p1 arccosp (x) =

1 p −p

(1 − t ) 0

Zx

arcsinhp (x) =

1

(1 + tp )− p dt,

Zx dt,

(1 + tp )−1 dt,

arctanp (x) = 0

Zx

(1 − tp )−1 dt,

arctanhp (x) =

0

0

where F(a, b; c; z) is Gaussian hypergeometric function [1]. The generalized cosine function is defined by d sinp (x) = cosp (x), dx

x ∈ [0, πp /2] .


B. A. Bhayo, L. Yin

137

It follows from the definition that cosp (x) = (1 − (sinp (x))p )1/p , and | cosp (x)|p + | sinp (x)|p = 1,

x ∈ R.

(1)

Clearly we get d cosp (x) = − cosp (x)2−p sinp (x)p−1 . dx The generalized tangent function tanp is defined by tanp (x) =

sinp (x) , cosp (x)

and applying (1) we get d tanp (x) = 1 + tanp (x)p . dx For x ∈ (0, ∞), the inverse of generalized hyperbolic sine function sinhp (x) is defined by Zx arcsinhp (x) = (1 + tp )−1/p dt, 0

and generalized hyperbolic cosine and tangent functions are defined by coshp (x) =

d sinhp (x), dx

tanhp (x) =

sinhp (x) , coshp (x)

respectively. It follows from the definitions that | coshp (x)|p − | sinhp (x)|p = 1.

(2)

From above definition and (2) we get the following derivative formulas, d coshp (x) = coshp (x)2−p sinhp (x)p−1 , dx

d tanhp (x) = 1 − | tanhp (x)|p . dx

Note that these generalized trigonometric and hyperbolic functions coincide with usual functions for p = 2. For two distinct positive real numbers x and y, the Arithmetic mean, Geometric mean, Logarithmic mean, Harmonic mean and the Power mean of order p ∈ R are respectively defined by A(x, y) =

x+y , 2

G(x, y) =

√ xy,


138

Generalized trigonometric and hyperbolic functions L(x, y) =

x−y , log(x) − log(y)

H(x, y) = and

x 6= y,

1 , A(1/x, 1/y)

 1/t  x t + yt , Mt =  √ 2 x y,

t 6= 0, t = 0.

Let f : I → (0, ∞) be continuous, where I is a sub-interval of (0, ∞). Let M and N be the means defined above, the we call that the function f is MNconvex (concave) if f(M(x, y)) ≤ (≥)N(f(x), f(y)) for

all x, y ∈ I .

Recently, Generalized convexity/concavity with respect to general mean values has been studied by Anderson et al. in [2]. We recall one of their results as follows Lemma 1 [2, Theorem 2.4] Let I be an open sub-interval of (0, ∞) and let f : I → (0, ∞) be differentiable. Then f is HH-convex (concave) on I if and only if x2 f 0 (x)/f(x)2 is increasing (decreasing). In [4], Baricz studied that if the functions f is differentiable, then it is (a, b)-convex (concave) on I if and only if x1−a f 0 (x)/f(x)1−b is increasing (decreasing). It is important to mention that (1, 1)-convexity means the AA-convexity, (1, 0)-convexity means the AG-convexity, and (0, 0)-convexity means GG-convexity. Motivated by the results given in [2, 4], we contribute to the topic by giving the following result. Theorem 1 Let f : I → (0, ∞) be a continuous and I ⊆ (0, ∞), then 1. L(f(x), f(y)) ≥ (≤)f(L(x, y)), 2. L(f(x), f(y)) ≥ (≤)f(A(x, y)), if f is increasing and log-convex (concave). Theorem 2 For x, y ∈ (0, πp /2), the following inequalities 1. L(sinp (x), sinp (y)) ≤ sinp (L(x, y)),

p > 1,


B. A. Bhayo, L. Yin 2. L(cosp (x), cosp (y)) ≤ cosp (L(x, y)),

139

p ≥ 2.

Theorem 3 For p > 1, we have 1. L(1/ sinp (x), 1/ sinp (y)) ≥ 1/ sinp (A(x, y)),

x, y ∈ (0, πp /2),

2. L(1/ cosp (x), 1/ cosp (y)) ≥ 1/ cosp (L(x, y)),

x, y ∈ (0, πp /2),

3. L(tanhp (x), tanhp (y)) ≤ tanhp (A(x, y)),

x, y ∈ (0, ∞),

4. L(arcsinhp (x), arcsinhp (y)) ≤ arcsinhp (A(x, y)), 5. L(arctanp (x), arctanp (y)) ≤ arctanp (A(x, y)),

2

x, y ∈ (0, 1), x, y ∈ (0, 1).

Preliminaries and Proofs

We give the following lemmas which will be used in the proof of our main result. Lemma 2 [22] Let f, g : [a, b] → R be integrable functions, both increasing or both decreasing. Furthermore, let p : [a, b] → R be a positive, integrable function. Then Zb

Zb p(x)g(x)dx ≤

p(x)f(x)dx a

Zb

a

Zb p(x)dx

a

p(x)f(x)g(x)dx.

(3)

a

If one of the functions f or g is non-increasing and the other non-decreasing, then the inequality in (3) is reversed. Lemma 3 [17] If f(x) is continuous and convex function on [a, b], and ϕ(x) is continuous on [a, b], then f

1 b−a

Zb

ϕ(x)dx

a

1 b−a

Zb f (ϕ(x)) dx.

(4)

a

If function f(x) is continuous and concave on [a, b], then the inequality in (4) reverses. Lemma 4 [3] For two distinct positive real numbers a, b, we have L < A. Lemma 5 For p > 1, the function sinp (x) is HH-concave on (0, πp /2).


140

Generalized trigonometric and hyperbolic functions

Proof. Let f(x) = f1 (x)f2 (x), x ∈ (0, πp /2), where f1 (x) = 1/ sin(x) and f2 (x) = x2 cosp (x)/ sinp (x). Clearly, f1 is decreasing, so it is enough to prove that f2 is decreasing, then the proof follows from Lemma 1. We get f20 (x) = =

sinp (x)(cosp (x) − x cosp (x)2−p sinp (x)p−1 ) − x cosp (x)2 sinp (x)2 cosp (x)2 cosp (x)2 ((1 − x tanp (x)p−1 ) tanp (x) − x) = f (x) , 3 sinp (x)2 sinp (x)2

where f3 (x) = tanp (x) − x tanp (x)p − 1. Again, one has f30 (x) = p tanp (x)p−1 (1 + tanp (x)p )x < 0. Thus, f3 is decreasing and g(x) < g(0) = 0. This implies that f20 < 0, hence f2 is strictly decreasing, the product of two decreasing functions is decreasing. This implies the proof. Proof of Theorem 1. We get Rf(x) Rx 0 f(y) 1dt y f (u)du L(f(x), f(y)) = Rf(x) . (5) = Rx f 0 (u) 1 du dt y f(u) t f(y) It is assumed that the function f(x) is increasing and log f is convex, this 0 (x) is increasing. Letting p(x) = 1, f(x) = f(u) and g(x) = implies that ff(x) 0 f (u)/f(u) in Lemma 2, we get Zx Zx Zx 0 Zx f (u) 0 1du f (u)du ≥ du f(u)du. y y y f(u) y This is equivalent to L(f(x), f(y)) =

Rx

0 y f (u)du Rx f 0 (u) y f(u) du

Rx ≥

y f(u)du Rx . y 1du

By Lemmas 3 and 4, and keeping in mind that log-convexity of f implies the convexity of f, we get ! Rx x+y y udu =f ≥ f (L(x, y)) . L(f(x), f(y)) ≥ f x−y 2 The proof of converse follows similarly. If we repeat the lines of proof of part (1), and use the concavity of the function, and Lemmas 3 & 4 then we arrive at the proof of part (2).


B. A. Bhayo, L. Yin

141

Proof of Theorem 2. It is easy to see that the function sinp (x) is increasing and log-concave. So the proof of part (1) follows easily from Theorem 1. We also offer another proof as follows: It can be observed easily that Rx Rx y cosp (u)du y cosp udu L (sinp (x), sinp (y)) = Rsin (x) = Rx cosp u , p 1 dt y sinp (u) du sinp (y) t

and sinp (L (x, y)) = sinp

x−y log yx

Rx

!

y 1du Rx 1 y u du

= sinp

! .

Clearly, cosp (u) and sinp (1/u), utilizing Chebyshev inequality, we have Zx Zx Zx Zx 1 cosp (u)du sinp (1/u)du ≤ 1du cosp usinp du. u y y y y So, we get Zx

Zx cosp udu

y

Zx sinp (1/u)du <

y

Zx 1du

y

y

cosp (u) du. sinp (u)

Where we apply simple inequality sinp u < sinp1(u) . In order to prove inequality (1), we only prove ! Rx Rx y 1du y 1du Rx ≤ sinp Rx . sin (1/u)du p y y sinp (1/u)du 1

Consider a partition T of the interval [y, x] into n equal length sub-interval by means of points y = x0 < x1 < · · · < xn = x and ∆xi = x−y n . Picking an arbitrary point ξi ∈ [xi−1 , xi ] and using Lemma 1.2, we have   n n P i=1

sinp ξ1i

 n   ≤ sinp  n P  1 i=1

ξi

 lim

n→∞

x−y n x−y P n

i=1

sinp ξ1i

 ≤ sinp  

lim

n→∞

x−y n x−y P n

i=1

1 ξi

  


142

Generalized trigonometric and hyperbolic functions

Rx Rx

y 1du

y sinp (1/u)du

Rx ≤ sinp

Rx

!

y 1du

y sinp (1/u)du

.

This completes the proof. For (2), clearly cosp (x) is decreasing and tanp (x)p−1 is increasing. One has (cosp (x))00 = cosp (x) tanp (x)p−2 (1 − p + (2 − p) tanp (x)p ) < 0, this implies that cosp (x) is concave on (0, πp /2). Using Tchebyshef inequality, we have Zx Zx Zx Zx p−1 1du cosp (u) tanp (u) du ≤ cosp (u)du tanp (u)p−1 du, y

y

y

y

which is equivalent to Rx Rx p−1 du y cosp (u)du y cosp (u) tanp (u) Rx Rx ≤ . p−1 du y tanp (u) y 1du

(6)

Substituting t = cosp (u) in (6), we get Rcosp (x) Rx Rx p−1 du cosp (y) 1dt y cosp (u) tanp (u) y cosp (u)du Rx Rx L(cosp (x), cosp (y)) = Rcos (x) = ≤ . p 1 tanp (u)p−1 du 1du dt y y cosp (y) t Using Lemma 3 and concavity of cosp (x), we obtain ! Rx x+y y udu L(cosp (x), cosp y) ≤ cosp = cosp ≤ cosp (L(x, y)) . x−y 2 Proof of Theorem 3. Let g1 (x) = 1/ cosp (x), x ∈ (0, πp /2) and g2 (x) = tanhp (x), x > 0. We get (log(g1 (x))) 00 = (p − 1) tanp (x)p−2 (1 + tanp (x)p ) > 0, and (log(g2 (x))) 00 =

1 − tanhp (x)p ((1 − p) tanhp (x)p − 1) < 0. tanhp (x)2

This implies that g1 and g2 are log-convex, clearly both functions are increasing, and log-convexity implies the convexity, so g1 and g2 are convex functions. Now the proof follows easily from Theorem 1. The rest of proof follows similarly.


B. A. Bhayo, L. Yin

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Corollary 1 For p > 1, we have 1. L(tanp (x), tanp (y)) ≥ tanp (L(x, y)), x, y ∈ (sp , πp /2), where sp is the unique root of the equation tanp (x) = 1/(p − 1)1/p , 2. L(arctanhp (x), arctanhp (y)) ≥ arctanhp (L(x, y)), x, y ∈ (rp , 1), where rp is the unique root of the equation xp−1 arctanhp (y) = 1/p. Proof. Write f1 (x) = tanp (x). We get

f10 (x) f(x)

0

=

1 + tanpp (x) tanp (x)

0 =

1 + tanpp (x) p (p − 1) tan (x) − 1 >0 p tan2p (x)

π on sp , 2p . This implies that f1 is log-convex, clearly f1 is increasing, and the proof follows easily from Theorem 1. The proof of part (2) follows similarly.

Acknowledgements The second author was supported by NSF of Shandong Province under grant numbers ZR2012AQ028, and by the Science Foundation of Binzhou University under grant BZXYL1303.

References [1] M. Abramowitz, I. Stegun, eds., Handbook of mathematical functions with formulas, graphs and mathematical tables, National Bureau of Standards, Dover, New York, 1965. [2] G. D. Anderson, M. K. Vamanamurthy, M. Vuorinen, Genenalized convexity and inequalities, J. Math. Anal. Appl. 335 (2007), 1294–1308. [3] H. Alzer, S.-L Qiu, Inequalities for means in two variables, Arch. Math. 80 (2003), 201–205. ´ Baricz, Geometrically concave univariate distributions, J. Math. Anal. [4] A. Appl. 363 (1) (2010), 182–196. ´ Baricz, B. A. Bhayo, R. Kl´en, Convexity properties of gener[5] A. alized trigonometric and hyperbolic functions, Aequat. Math. DOI 10.1007/s00010-013-0222-x.


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´ Baricz, B. A. Bhayo, M. Vuorinen, Tur´an type inequalities for gener[6] A. alized inverse trigonometric functions, available online at http://arxiv. org/abs/1209.1696. [7] B. A. Bhayo, Power mean inequality of generalized trigonometric functions, Mat. Vesnik, (to appear) http://mv.mi.sanu.ac.rs/Papers/ MV2013_033.pdf. [8] B. A. Bhayo, M. Vuorinen, On generalized trigonometric functions with two parameters, J. Approx. Theory, 164 (10) (2012),1415–1426. [9] B. A. Bhayo, M. Vuorinen, Inequalities for eigenfunctions of the pLaplacian, Issues of Analysis 2 (20), No 1, (2013), http://arxiv.org/ abs/1101.3911 [10] P. J. Bushell, D. E. Edmunds, Remarks on generalised trigonometric functions, Rocky Mountain J. Math., 42 (1) (2012), 25–57. [11] B. C. Carlson, Some inequalities for hypergeometric functions, Proc. Amer. Math. Soc., 17 (1), (1966), 32–39. [12] P. Dr´abek, R. Man´asevich, On the closed solution to some p−Laplacian nonhomogeneous eigenvalue problems, Diff. and Int. Eqns., 12 (1999), 723–740. [13] D. E. Edmunds, P. Gurka, J. Lang, Properties of generalized trigonometric functions, J. Approx. Theory, 164 (2012) 47–56, doi:10.1016/j.jat.2011.09.004. [14] W.-D. Jiang, M.-K. Wang, Y.-M. Chu, Y.-P. Jiang, F. Qi,Convexity of the generalized sine function and the generalized hyperbolic sine function, J. Approx. Theory, 174 (2013), 1–9. [15] D. B. Karp, E. G. Prilepkina, Parameter convexity and concavity of generalized trigonometric functions, arXiv:1402.3357[math.CA] [16] R. Kl´en, M. Visuri, M. Vuorinen, On Jordan type inequalities for hyperbolic functions, J. Ineq. Appl., vol. 2010, pp. 14. [17] J.-C. Kuang, Applied inequalities (Second edition), Shan Dong Science and Technology Press. Jinan, 2002.


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[18] R. Kl´en, M. Vuorinen, X.-H. Zhang, Inequalities for the generalized trigonometric and hyperbolic functions, J. Math. Anal. Appl., 409 (1) (2014), 521-29. [19] P. Lindqvist, Some remarkable sine and cosine functions, Ricerche di Matematica, Vol. XLIV (1995), 269–290. [20] D. S. Mitrinovi´c, Analytic Inequalities, Springer, New York, USA, 1970. [21] E. Neuman, J. S´andor, Optimal inequalities for hyperbolic and trigonometric functions, Bull. Math. Anal. Appl, 3(3), (2011), 177–181. http: //www.emis.de/journals/BMAA/repository/docs/BMAA3_3_20.pdf. [22] F. Qi, Z. Huang, Inequalities of the complete elliptic integrals, Tamkang J. Math, 29 (3) (1998), 165–169. [23] S. Takeuchi, Generalized Jacobian elliptic functions and their application to bifurcation problems associated with p-Laplacian, J. Math. Anal. Appl. 385 (2012) 24–35.

Received: 25 November 2014


Acta Univ. Sapientiae, Mathematica, 6, 2 (2014) 146–161 DOI: 10.1515/ausm-2015-0003

Stabilizing priority fluid queueing network model Amina Angelika Bouchentouf

Hanane Sakhi Department of Mathematics, Sciences and Technology University of Oran: Mohamed Boudiaf, USTOMB, B.P. 1505 EL-M’NAOUAR- Oran, Algeria email: sakhi.hanane@yahoo.fr

Mathematics Laboratory, Djillali Liabes University of Sidi Bel Abbes, B.P. 89, Sidi Bel Abbes, Algeria email: bouchentouf− amina@yahoo.fr

Abstract. The aim of this paper is to establish the stability of fluid queueing network models under priority service discipline. To this end, we introduce a priority fluid multiclass queueing network model, composed of N stations, N ≥ 3 and 2N classes (2 classes at each station); where in the system, each station may serve more than one job class with differentiated service priority, and each job may require service sequentially by more than one service station. In this paper the fluid model approach is employed in the study of the stability.

1

Introduction

Stochastic processing networks arise as models in manufacturing, telecommunications, computer systems and service industry. Common characteristics of these networks are that they have entities, such as jobs, customers or packets, that move along routes, wait in buffers, receive processing from various resources, and are subject to the effects of stochastic variability through such quantities as arrival times, processing times, and routing protocols. Networks arising in modern applications are often highly complex and heterogeneous. Typically, their analysis and control present challenging mathematical problems. One approach to these challenges is to consider approximate models. 2010 Mathematics Subject Classification: 60K25, 68M20, 90B22. Key words and phrases: stability, fluid models, multiclass queueing networks, fluid approximation

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Stabilizing priority fluid queueing network model

147

In the last 15 years, significant progress has been made on using approximate models to understand the stability and performance of a class of stochastic processing networks called open multiclass HL queueing networks. HL stands for a non-idling service discipline that is head-of-the-line, i.e., jobs are drawn from a buffer in the order in which they arrived. Examples of such disciplines are FIFO and static priorities. First order (functional laws of large numbers) approximations called fluid models have been used to study the stability of these networks, and second order (functional central limit theorem) approximations which are diffusion models, have been used to analyze the performance of heavily congested networks. The development of the fluid approach was inspired by the studies of some counter-examples in Kumar and Seidman [11], Rybko and Stolyar [14] and Bramson [1], etc., where the multiclass queueing networks are not stable even when the traffic intensity of each station in the network is less than one. An elegant result of the fluid model approach was proposed first in Rybko and Stolyar [14] and then generalized and refined by Dai [6], Chen [2], Dai and Meyn [8], Stolyar [15] and Bramson [1]. It states that a queueing network is stable if its corresponding fluid network model is stable. Partial converse to this result is also given in Meyn [12], Dai [7] and Puhalskii and Rybko [13]. Heng Quing Ye [10] used Kumar-Rybko-Seidman-Stolyar network for establishing the stability of fluid queueing network. In this paper, we concentrate with the capacity of some large classes of fluid multiclass queueing networks under priority service discipline. Specifically, we establish a stability condition of some heterogenous priority fluid networks with N stations and 2N job classes, where in the system, each station may serve more than one job class with differentiated service priority, and each job may require service sequentially by more than one service station. So, in our case, the network performance is improved even when more workloads are admitted for service. To stabilize our networks a number of stations should be added, these later act as regulators for the systems, adding these stations is not random, it depends essentially on higher and lower priority job classes (many-to-one mapping) and on the number of stations in the network. The fluid model approach is employed to proof the stability. The outline of the paper is as follows: At first (Section 2) we describe priority fluid multiclass queueing models, and present a powerful result on the stability of such systems given by Chen and Zhang [5], after that (Section 3) we introduce modified networks and present their stability conditions (Theorems 2 and 3), and finally we conclude this paper with a short conclusion.


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2 N-stations priority fluid multiclass network models We describe N-stations priority fluid queueing network models as (J , K, λ, m, C, P, π). Specifically, the fluid network consists of J stations (buffers) (J = N) indexed by j ∈ J = 1, N, serving K, K = 2N fluid (customer) classes indexed by k ∈ K = 1, 2N. A fluid class is served exclusively at one station, but one station may serve more than one fluid classes. σ(.) denotes a many-to-one mapping from K to J , with σ(k) indicating the station at which a class k fluid is served. A class k fluid may flow exogenously into the network at rates λ1 and λN+1 , (≥ 0), then it is served at station σ(k), with mean service time mk = 1/µk , k = 1, 2N and after being served, a fraction P pkl of fluid turns into a class l fluid, l ∈ K, and the remaining fraction, 1 − Kl=1 pkl flows out of the network. Let C(j) be the set of classes that reside in station j, alternatively, we denote by a J × K matrix C = (cij )J×K , known as the constituent matrix, where cjk = 1 if σ(k) = j, and cjk = 0 otherwise. Let Qk (t) indicates the number of class k customers in the network at time t, (Q(0) = Qk (0)) and λ = (λk ) two K-dimensional nonnegative vectors. P = (pkl )K×K a stochastic matrix with spectral radius strictly less that one, µ = (µk ) a K-dimensional positive vector. The vectors Q(0) are referred to as initial fluid level vector, λ to the exogenous inflow rate vector, µ to the processing rate vector, matrix P is referred to as flow transfer matrix. When station σ(k) devotes its full capacity to serving class k fluid (assuming that it is available to be served), it generates an outflow of class k fluid at rate µk > 0, k ∈ K. Among classes, fluid follows a priority service discipline, which is again described by a one-to-one mapping π from {1, ..., K} onto itself. Specifically, a class k has priority over a class l if π(k) < π(l) and σ(l) = σ(k), then class k job can not be served at station σ(k) unless there is no class l job. So, our multiclass fluid network consists of N stations and 2N job classes. Assume that the arrival process of class k, k = 1, 2N, customers arrive to the system following a Poisson process with arrival rates λ1 ≥ 0 and λN+1 ≥ 0, the service time for each class k customer is exponentially distributed with mean service time mk > 0. We also assume that all the inter-arrival times and service times are independent. To describe the dynamics of the fluid network, we introduce the K-dimensional fluid level process Q = {Q(t), t ≥ 0}, whose kth component Qk (t) denotes the fluid level of class k at time t; the K-dimensional time allocation process T = {T (t), t ≥ 0}, whose kth component T k (t) denotes the total amount of


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time that station σ(k) has devoted to serving class k fluid during the time interval [0, t], and the K-dimensional unused capacity process Y = {Y(t), t ≥ 0}, whose kth component Y k (t) denotes the (cumulative) unused capacity of station σ(k) during the time interval [0, t] after serving all classes at station σ(k) which have a priority no less than class k. We denote by D the K-dimensional diagonal matrix whose kth element is µk , and e is a K-dimensional vector with all components being one. Let Hk = {l : σ(l) = σ(k), π(l) ≤ π(k)} be the set of indices for all classes that are served at the same station as class k and have a priority no less than that of class k. Note that k ∈ Hk by definition. Then the dynamics of the fluid network model can be described as follows. Q(t) = Q(0) + λt − (I − P 0 )DT (t) ≥ 0,

(1)

T (·) are nondecreasing with T (0) = 0, X Y k (t) = t − T l (t) are nondecreasing, k ∈ K,

(2) (3)

l∈Hk

Z∞

Qk (t)dY k (t) = 0, k ∈ K.

(4)

0

Let Qk (t) = Qk (0) + λk t +

K X

plk µl Tl (t) − µk Tk (t) ≥ 0, k = 1, . . . , K,

(5)

l=1

be the kth coordinate of the flow balance relation (1). The equation (1) is the equivalent relation between the time allocation process T (·) and the unused capacity process Y(.). The relation (4) means that at any time t, there could be some positive remaining capacity (rate) for serving those classes at station σ(k) having a strictly lower priority than class k, only when the fluid levels of all classes in Hk (having a priority no less than k) are zero. A pair (Q, T ) (or equivalently (Q, Y)) is said to be a fluid solution if they jointly satisfy (1)-(4). For convenience, we also call Q a fluid solution if there is a T such that the pair (Q, T ) is a fluid solution. The fluid network (J , K, λ, m, C, P, π) is said to be stable if there is a time τ ≥ 0 such that Q(τ + ·) ≡ 0 for any fluid solution Q with kQ(0)k = 1; and it is said to weakly


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stable if Q(·) = 0 for any fluid solution Q with Q(0) = 0. The processes Q, Y, and T are Lipschitz continuous, and hence are differentiable almost everywhere on [0, ∞), this well-known property will be used in this paper. It is well-known that the queue length process Q(t) is a continuous time Markov chain under the Poisson arrival and exponential service assumptions. We say that the network (J , K, λ, m, C, P, π) is stable if the Markov chain Q(t) is positive recurrent. It is well-know that the Markov chain Q(t) is positive recurrent only if the traffic intensity for each station is less than one, i.e., ρj < 1 (ρj is the jth component of ρ; a traffic intensity for station j) for all j ∈ J , or in short, ρ < e, where e is a J-dimensional vector with all components being ones. The expected stationary total queue length Q is defined as " Q = lim E t→∞

X

# Qk (t) .

k∈K

The queue length Q(t) is a finite if and only if the queue length process Q is positive recurrent. Chen and Zhang [5] gave a very important result on the stability of priority fluid queueing systems, authors established the sufficient condition for the stability based on the existence of a linear Lyapunov function, this later (sufficient condition) gave the the necessary and sufficient condition for the stability. Their result is presented in the Theorem 1, in order to state it we need some additional assumptions: Let

h(k) =

+ arg max{π(l) : l ∈ H+ k } if Hk 6= ∅, 0 otherwise,

(6)

with H+ k = Hk \{k}, in words; if k is not the highest priority class at station σ(k), then h(k) is the index for the class which has the next higher priority than class k at station σ(k), otherwise h(k) = 0. θ = λ − (I − P 0 )µ0H ,

(7)

where µ0H = De0H , (e0H = (e01 , ..., e0K ) 0 ) is a K-dimensional vector with e0k = 1 if 0 H+ k = ∅ and ek = 0 otherwise. R = (I − P 0 )D(I − B),

(8)


Stabilizing priority fluid queueing network model

151

where B = (blk ) is the K × K matrix with blk = 1 if k = h(l), and blk = 0 otherwise, (l, k = 1, ...K). And let ρ = CD−1 (I − P 0 )−1 λ

(9)

be the traffic intensity of the queueing network. Theorem 1 [5] Consider a fluid network (λ, µ, P, C) under priority service discipline π. Let vector θ and matrix R be as defined in (7),(8) respectively. Assume that ρ < e. Then the fluid network is stable if there exist a K-dimensional vector h ≥ 0 such that for any given partition a and b of K satisfying if class l ∈ a, then each class k with σ(k) = σ(l) and π(k) > π(l) is also in a,

(10)

ha0 (θa + Rab xb ) < 0

(11)

we have for xb ∈ Sb := {u ≥ 0 : θb + Rb u = 0 and u ≤ e} when b 6= ∅, and xb = 0 when b = ∅. The inequality (11) is omitted to hold by default when Sb = ∅. Set a includes all classes which have zero unused capacity rate and set b includes all classes which have a positive unused capacity rate at time t.

3

Main result

In this paper, we present two theorems, we provide the proof of the first theorem, while the proof of the second one is omitted since it is similar to the former one.

3.1

Stabilizing N-stations priority fluid queueing network with some additional stations

Our multiclass queueing network consists of N stations and 2N job classes. Assume that the arrival process of class k; k = 1, 2N customers arrive to the system following a Poisson process with arrival rates λ1 and λN+1 (≥ 0) to station 1 and N + 1 respectively, the service time for each class k customer is exponentially distributed with mean service time mk > 0. We also assume that all the inter-arrival times and service times are independent. Suppose that each even class at station i = 1, N has higher priority.


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We modify our network such that if it is composed of an even number of stations we add N additional ones otherwise we add (N − 1), the explanation of this choice will be given in the rest of the paper, the modified network is illustrated in Figures 1 and 2; the additional stations are named station N + 1,...,station 2N, (N: even) (resp. station N + 1,...,station 2N − 1 (N: odd)).

Figure 1: 2N-stations priority fluid queueing network

Figure 2: 2N-1-stations priority fluid queueing network Theorem 2 Suppose ρ < e, equation (11) not satisfied. If λk1 > (1 − mk10 /mk100 )/mk1

(12)

then the queue length process Q(·) is positive recurrent. λk1 (resp. mk1 ) is the exogenous arrival rate (resp. the mean service time) of higher priority fluid class of additional stations i = N + 1, 2N, (N: even) (resp. i = N + 1, 2N − 1, (N: odd)), such that k1 = 3N + 1, 4N, (N : even) (resp. k1 = 3N, 4N − 2, (N : odd)). mk10 is the mean service time of lower


Stabilizing priority fluid queueing network model

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priority fluid class of additional stations i = N + 1, 2N, (N: even) (resp. i = N + 1, 2N − 1, (N: odd)). mk100 is the mean service time of higher priority fluid class of the original network. With

mk1 −N , k1 = 3N + 1, 4N, (N: even), 0 m k1 = mk1 −(N−1) , k1 = 3N, 4N − 2, (N: odd).

mk100 =

 N mk10 −(2N−j1 ) , k10 = 2N + 1, 5N   2 , j1 = 1, 2        5N+2 0   m(k1 −k10 )+j1 , k1 = 2 , k1 − N, j1 = 2, N

(N: even),

  N−1  mk10 −(2N−j1 ) , k10 = 2N + 1, 5N−1  2 , j1 = 1, 2        m(k1 −k10 )+j1 , k10 = 5N+1 2 , k1 − (N − 1), j1 = 4, N + 1

(N: odd).

Where for each k10 it corresponds k1 and j1 , (j1 is an even number). Via this theorem, we show that when the arrival rates of some job classes is reduced, the performance of the queue will worse. Proof. In Chen and Yao [3] and Dai [7], it was shown that to prove the stability of a queueing model, it is sufficient to study the stability of its corresponding fluid queueing model, our prove is based on this result. To understand better the phenomenon, let us examine the dynamics of the original network with no initial job. When the higher priority job classes are being served, the lower priority ones are in standby, waiting for service, (class 1 jobs can not move to class 2 and 2 cannot move to 3,... for further services, and vice versa). So, these classes will never be served at the same time and in effect form virtual stations (Dai and Vande Vate [9]). Therefore, the total nominal traffic intensity for these classes together, i.e., the virtual stations, should not exceed one for the network to be stable. The similar argument also yields that the network is unstable when the nominal traffic intensity for the virtual stations exceed one, i.e., the condition (11) is not satisfied. Now consider the modified network. The additional classes act as regulators that regulate the traffics in the network so as to stabilize the network. When the workloads of classes k1 (k1 ; defined in the theorem) are light, much service capacity of the additional stations are left to classes k10 (k10 ; defined in the theorem) and hence these later do not hold back the traffics to avoid building up of job queues at priority classes of


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the original network. Thus, the virtual stations effect prevails and the network is still unstable when the condition (11) is not satisfied. However, when the workloads of classes k1 are heavy enough such that the condition (12) holds, the service for lower priority classes at additional stations is in effect slowed down and the traffics in the original network are held back (these classes will not mutually block their services). Finally, the virtual station effect is avoided and the modified network is thus stabilized. The dynamics of the our modified fluid network model can be described as follows. Qk1 (t) = Qk1 (0) + λk1 t − µk1 T k1 (t) ≥ 0, k1 = 1, N + 1, 3N + 1, 4N, (N : even), (resp. k1 = 1, N + 1, 3N, 4N − 2) (N : odd), Qk (t) = Qk (0) + µl T l (t) − µk T k (t) ≥ 0,

(13)

(14)

(k, l) = two successive job classes, such that the kth class is the arriving lth class T k (·) are nondecreasing with T k (0) = 0, (15) k = 1, 4N, (N : even)(resp. k = 1, 4N − 2, (N : odd))   Y k1 (t) = t − T k1 (t), are nondecreasing,  Y k100 (t) = t − T k100 (t), Y k (t) = t − T l (t) − T k (t) are nondecreasing,

(16)

(17)

(k, l) = (lower priority job class, higher priority job class ) at station i, i = 1, 2N, (N : even) (resp. i = 1, 2N − 2, (N : odd)), Z∞ Qk (t)dY k (t) = 0, k = 1, 4N(N : even) (resp. k = 1, 4N − 2(N : odd)). 0

(18) The stability study of the modified fluid network will be done in three steps. 1. First step. We prove that there exists a time τ1 ≥ 0 such that Qk1 (t) = 0,

for any t ≥ τ1 ,

with k1 = 3N + 1, 4N, (N: even), (resp. k1 = 3N, 4N − 2, (N: odd)).

(19)


Stabilizing priority fluid queueing network model

155

˙ (t) > 0, then we have by equation (18) If Q k1 Y˙ k1 (t) = 0,

(20)

T˙ k1 (t) = 1,

(21)

˙ (t) = λ − µ . Q k1 k1 k1

(22)

then by conditions (16) and (20)

then by (13) and (21), we get

(l) ˙ (0)/(µ − Note that the condition ρ < e implies λk1 < µk1 . Let τ1 = Q k1 k1

λk1 ), l = 1, N2 , (N: even) (resp. l = 1, N−1 2 , (N: odd)). Then, we have (l)

Qk1 (t) = 0 for any t ≥ τ1 .

(23) (l)

Letting τ1 = max(1/µk1 − λk1 ), we have that τ1 ≥ max(τ1 ), (each l corresponds to k1 ) under the assumption kQ(0)k = 1. Now, the conclusion (23) leads to the claim (19). 2. Second step. We prove that there exists a time τ2 ≥ τ1 such that Qk100 (t) = 0, for any t ≥ τ2 , where k100 is the higher priority job class at station i, i = 1, N.  N  k10 − (2N − j1 ), k10 = 2N + 1, 5N  2 , j1 = 1, 2          (k1 − k10 ) + j1 , k10 = 5N+2 2 , k1 − N, j1 = 2, N 00 k1 =   N−1  k10 − (2N − j1 ), k10 = 2N + 1, 5N−1  2 , j1 = 1, 2        (k1 − k10 ) + j1 , k10 = 5N+1 2 , k1 − (N − 1), j1 = 4, N + 1

(24)

(N: even),

(N: odd).

˙ (t) = 0, and then T˙ (t) = λ m , k = Under the condition (19), we have Q k1 k1 k1 1 k1 3N + 1, 4N, (N: even) (resp. k1 = 3N, 4N − 2, (N: odd)), for all time t ≥ τ1 . Combined with (17), this gives rise to Y˙ k10 (t) = t − T˙ k10 (t) − T˙ k1 (t) ≥ 0,


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k10 are classes of lower priority at additional stations,

k1 − N, k1 = 3N + 1, 4N, (N: even), 0 k1 = k1 − (N − 1), k1 = 3N, 4N − 2, (N: odd). and

T˙ k10 (t) ≤ 1 − T˙ k1 (t) = 1 − λk1 mk1 , for any t ≥ τ1 .

(25)

Then, ˙ 00 (t) = µ 0 T˙ 0 (t) − µ 00 T˙ 00 (t) ≤ µ 0 (1 − λ m ) − µ 00 < 0, where for Q k1 k1 k1 k1 k1 k1 k1 k1 k1 each k10 it corresponds k100 for any t ≥ τ2 , where the last inequality is implied by the assumption that λk1 > (1 − mk10 /mk100 )/mk1 . (l)

Let τ2 =

˙ Q k 00 (τ1 )

l = 1, N (N: even), (resp. l = 1, N − 1 (N:

µk 00 −µk 0 (1−λk1 mk1 ) , 1

1

1

odd)). Then, we have (l)

Qk100 (t) = 0 for any t ≥ τ2 . Let τ2 = max

µk100

1 + Θτ1 − µk10 (1 − λk1 mk1 )

(26) !

with Θ being the Lipschitz constant for the fluid level process Q(t). Then we (l) have that τ2 ≥ max(τ2 ). Now, the conclusion (26) implies the claim (24). Before to pass to the last step, we prove separately that QN+1 (t) = 0 for any t ≥ τ2 , “the case of network with even number of stations”. ˙ ˙ ˙ If Q N+1 (t) = 0, this implies Y N+1 (t) = 0, which in turn implies that T N+1 (t) = ˙ (t) = λ −µ , with λ <µ (since ρ < e). So 1, then we have Q N+1

there exists

τ20

N+1

N+1

N+1

N+1

˙ ˙ =Q N+1 (0)/µN+1 −λN+1 , such that QN+1 (t) = 0 for any t ≥ τ2 .

Third step. We prove that there exists a time τ ≥ τ2 (≥ 0) such that Ql (t) = 0, for t ≥ τ,

(27)

l represents job classes of lower priority at station i = 1, 2N (N: even) (resp. i = 1, 2N − 1 (N: odd), which together with equations (19) and (24) implies Q(t) = 0 for t ≥ τ.


Stabilizing priority fluid queueing network model Let W i (t) = (λ1 ml1 + λN+1 ml2 )t −

X

157

T k (t), i = 1, N,

k:σ(k)=i

with l1 = 1, N and l2 = N = 1, 2N job classes at the same station in the original network. W i 0 (t)=   λ1 mk10 t − T k10 (t),    

0

k1 = 2N + 1, 5N 2 , N: even, 0

(resp. k1 = 2N + 1, 2N + 5N−1 2 , N: odd) 0 5N+2  λN+1 mk10 t − T k10 (t), k1 = 2 , 3N, N: even,    0  (resp. k1 = 5N+1 2 , 3N − 1, N: odd)

for τ ≥ τ2 . Here W(t) can be explained as the immediate workload in the system at time t. Define fi (t) = k10 W i (t), with k10 a lower priority job class in the additional stations. fi 0 (t) = k100 W i 0 (t), with k100 a higher priority job class in the original network. For each i (resp. i 0 ) it corresponds to k10 (resp. k100 ). Then, it is direct to verify that, for t ≥ τ2 , ˙ (t) > 0, for i = 1, 4N, (N: even, (resp. i = 1, 4N − 2, (N: odd)) f˙i (t) < 0 if Q i And f1 (t) ≤ fN (t) if Q1 (t) = 0, fi (t) ≤ fi−1 (t) if Qi (t) = 0, i = 2, 3N, (N: even) fj (t) ≤ fi (t) if

X

(resp. i = 2, 3N − 1, (N: odd)) Qj (t) = 0, j = 1, 3N, (N: even)

j6=i

(resp. j = 1, 3N − 1, (N: odd)) fN (t) ≤ fN (t) if Q3N (t) = 0, (N: even) (resp. Q3N−1 (t) = 0, (N: odd)) Now applying the piecewise linear Lyapunov function approach for the multiclass fluid network model described in Theorem 3.1 of Chen and Ye [4], we obtain the conclusion (27).


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3.2

A. A. Bouchentouf, H. Sakhi

Stabilizing N-stations priority fluid queueing networks with N additional stations

Our N-stations multiclass queueing network is the same is above. Suppose in this case that the higher priority is devoted to classes N, N + 2, 2N.

Figure 3: 2N-stations priority fluid queueing network We modify our network by adding N stations, (see Figure 3), compared to the original network, there are N additional stations, namely the station N + 1, . . . , station 2N, such that, 3N + 1 job class has high priority at station N + 1, and 3N + 2, 4N job classes have higher priority at stations N + 2, 2N. Now, let us introduce the second main result. Theorem 3 Suppose ρ < e holds, equation (11) not satisfied. If λ3N+1 > (1 − m2N+1 /mN )/m3N+1 , λk2 > (1 − mk20 /mk200 )/mk2 ,

(28)

k200 = N, 2N, k20 = 2N + 1, 3N, k2 = 3N + 1, 4N, where for each k2 it corresponds to k20 and k200 . Then the queue length process Q(·) is positive recurrent. In this case, when the higher priority classes are being served, the lower priority ones cannot be served, (class 1 cannot move to class 2 and 2 cannot move to 3,..., for further service, and vice versa. So, these later form a virtual stations. Therefore, these later, should not exceed one for the network to be stable. Now, let us consider the modified network. The additional classes 2N+1 and 3N + 1, 4N act as regulators that regulate the traffics. When the workloads of classes 3N + 1 and 3N + 2, 4N are light, much service capacity of stations


Stabilizing priority fluid queueing network model

159

N + 1, . . . , 2N are left to classes 2N + 1, 3N respectively and hence these later do not hold back the traffics to avoid building up of job queues at higher priority classes of the original network. Thus, the virtual stations effect prevails and the network is still unstable. However, when the workloads of classes 3N + 1, 4N are heavy enough such that the condition (28) holds, the service for lower priority classes 2N + 1, 3N is in effect slowed down and the traffics to the higher priority classes N and N + 2, 2N are held back. Finally, the virtual stations effect is avoided and the modified network is thus stabilized. Then, following the same steps given in theorem 2, it is not difficult to prove that there exists a time τ1 ≥ 0 such that Qk2 (t) = 0, k2 = 3N + 1, 4N, for any t ≥ τ1 .

(29)

after that, we prove that there exists a time τ2 ≥ τ1 such that QN (t) = Qk200 (t) = 0, k200 = N + 2, 2N for any t ≥ τ2 .

(30)

and finally, we prove that there exists a time τ ≥ τ2 (≥ 0) such that Qk40 (t) = 0 k40 =lower priority job class at stations i = 1, 2N, for t ≥ τ. (31)

4

Conclusion

Multiclass queueing networks are effective tools for modelling many industrial settings. One setting for which the model is particularly attractive is the production flow within semiconductor manufacturing facilities. In this paper we have studied the stabilization of N-stations queueing networks using its corresponding fluid network. The resulting model, fluid queueing networks with additional stations depending on the service priority and on the number of stations in the network are formally presented in Section 3. Beyond the presentation of our modified network models “fluid networks with additional stations”, the primary concern of the paper is the stability of such networks. Nevertheless, stability of the artificial fluid model implies stability of the original network (see Theorems 2 and 3).

References [1] M. Bramson, Stability of two families of queueing networks and a discussion of fluid limits, Queueing Syst., 23 (1998), 7–31.


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[2] H. Chen, Fluid approximations and stability of multiclass queueing networks: Workconserving discipline, Ann. Appl. Probab., 5 (1995), 637–655. [3] H. Chen, D. D. Yao, Fundamentals of Queueing Networks: Performance, Asymptotics and Optimization, Springer-Verlag New York, Inc. (2001). [4] H. Chen, H. Q. Ye, Piecewise linear Lyapunov function for the stability of priority multiclass queueing networks, IEEE Trans. Automat. Control, 47 (4) (2002), 564–575. [5] H. Chen, H. Zhang, Stability of multiclass queueing networks under priority service disciplines, Oper. Res., 48 (2000), 26–37. [6] J. G. Dai, On positive Harris recurrence of multiclass queueing networks: a unified approach via fluid models, Ann. Appl. Probab., 5 (1995), 49–77. [7] J. G. Dai, A fluid-limit model criterion for instability of multiclass queueing networks, Ann. Appl. Probab., 6 (1996), 751–757. [8] J. G. Dai, S. P. Meyn, Stability and convergence of moments for multiclass queueing networks via fluid models, IEEE Trans. Automat. Control, 40 (1995), 1899–1904. [9] J. G. Dai, J. H. Vande Vate, Global Stability of Two-Station Queueing Networks. Proceedings of Workshop on Stochastic Networks: Stability and Rare Events, Editors: Paul Glasserman, Karl Sigman and David Yao, Springer-Verlag, Columbia University, New York., (1996), 1–26. [10] H. Q. Ye, A paradox for admission control of multiclass queueing network with differentiated service, J. Appl. Probab., 44 (2) (2007), 321–331. [11] P. R. Kumar, T. I. Seidman, Dynamic instabilities and stabilization methods in distributed real-time scheduling of manufacturing systems, IEEE Trans. Automat. Control, 35 (1990), 289–298. [12] S. Meyn, Transience of multiclass queueing networks via fluid limit models, Ann. Appl. Probab., 5 (1995), 946–957. [13] A. Puhalskii, A. N. Rybko, Non-ergodicity of queueing networks under nonstability of their fluid models, Probl. Inf. Transm., 36 (1) (2000), 26– 48.


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[14] A. N. Rybko, A. L. Stolyar, Ergodicity of stochastic processed describing the operations of open queueing networks, Problemy Peredachi Informatsii, 28 (1992), 2–26. [15] A. L. Stolyar, On the stability of multiclass queueing network: a relaxed sufficient condition via limiting fluid processes, Markov Process. Related Fields, 1 (4) (1995), 491–512.

Received: 15 November 2013


Acta Univ. Sapientiae, Mathematica, 6, 2 (2014) 162–177 DOI: 10.1515/ausm-2015-0004

Some inequalities of Furuta’s type for functions of operators defined by power series Sever S. Dragomir Mathematics, College of Engineering & Science Victoria University, PO Box 14428 Melbourne City, MC 8001, Australia School of Computational & Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa email: sever.dragomir@vu.edu.au

Abstract. Generalizations of Kato and Furuta inequalities for power series of bounded linear operators in Hilbert spaces are given. Applications for normal operators and some functions of interest such as the exponential, hyperbolic and trigonometric functions are provided as well.

1

Introduction

In the following we denote by B (H) the Banach algebra of all bounded linear operators on a complex Hilbert space (H; h·, ·i) . If P is a positive selfadjoint operator on H, i.e. hPx, xi ≥ 0 for any x ∈ H, then the following inequality is a generalization of the Schwarz inequality in H |hPx, yi|2 ≤ hPx, xi hPy, yi , (1) for any x, y ∈ H. The following inequality concerning the norm of a positive operator is of interest as well, see [13, p. 221]. 2010 Mathematics Subject Classification: 47A63, 47A99 Key words and phrases: Bounded linear operators, operator inequalities, Kato’s inequality, functions of normal operators, Euclidian norm and numerical radius

162


Some inequalities of Furuta’s type

163

Let P be a positive selfadjoint operator on H. Then kPxk2 ≤ kPk hPx, xi

(2)

for any x ∈ H. The “square root” of a positive selfadjoint operator on H can be defined as follows, see for instance [13, p. 240]: If the operator A ∈ B (H) is selfadjoint √ and positive, then there exists a unique positive selfadjoint operator B := A ∈ B (H) such that B2 = A. If A is invertible, then so is B. If A ∈ B (H) , then the operator A√∗ A is selfadjoint and positive. Define the “absolute value” operator by |A| := A∗ A. In 1952, Kato [14] proved the following celebrated generalization of Schwarz inequality for any bounded linear operator T on H: D ED E |hTx, yi|2 ≤ |T |2α x, x |T ∗ |2(1−α) y, y (K) for any x, y ∈ H and α ∈ [0, 1] . In order to generalize this result, in 1994 Furuta [12] obtained the following result:

D E 2 D ED E

α+β−1 x, y ≤ |T |2α x, x |T ∗ |2β y, y (F)

T |T | for any x, y ∈ H and α, β ∈ [0, 1] with α + β ≥ 1. If one analyses the proof from [12], that one realizes that the condition α, β ∈ [0, 1] is taken only to fit with the result from the Heinz-Kato inequality |hTx, yi| ≤ kAα xk B1−α y (HK) for any x, y ∈ H and α ∈ [0, 1] where A and B are positive operators such that kTxk ≤ kAxk and kT ∗ yk ≤ kByk for all x, y ∈ H. Therefore, one can state the more general result: Theorem 1 (Furuta Inequality, 1994, [12]) Let T ∈ B (H) and α, β ≥ 0 with α + β ≥ 1. Then for any x, y ∈ H we have the inequality (F). If we take β = α, then we get

D E 2 D ED E

2α−1 x, y ≤ |T |2α x, x |T ∗ |2α y, y

T |T | for any x, y ∈ H and α ≥ 12 . In particular, for α = 1 we get D ED E |hT |T | x, yi|2 ≤ |T |2 x, x |T ∗ |2 y, y

(3)

(4)


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S. S. Dragomir

for any x, y ∈ H. If we take T = N a normal operator, i.e., we recall that NN∗ = N∗ N, then we get from (F) the following inequality for normal operators

D ED E E 2 D

2α 2β α+β−1 (5) x, y ≤ |N| x, x |N| y, y

N |N| for any x, y ∈ H and α, β ≥ 0 with α + β ≥ 1. This implies the inequalities

D E 2 D ED E

2α−1 x, y ≤ |N|2α x, x |N|2α y, y

N |N| for any x, y ∈ H and α ≥

(6)

1 2

and, in particular, D ED E |hN |N| x, yi|2 ≤ |N|2 x, x |N|2 y, y

(7)

for any x, y ∈ H. Making y = x in (6) produces

D E D E

2α−1 2α |N| |N| N x, x ≤ x, x

for any x ∈ H and α ≥

1 2

and, in particular, D E |hN |N| x, xi| ≤ |N|2 x, x

for any x ∈ H. If we take β = 1 − α with α ∈ [0, 1] in (5), then we get D ED E 2 2α 2(1−α) |hNx, yi| ≤ |N| x, x |N| y, y

(8)

for any x, y ∈ H. We can state the following corollary of Furuta’s inequality for the numerical radius w of an operator V ∈ B (H), namely w (V) = supkxk=1 |hVx, xi|, which satisfies the following basic inequalities 1 kVk ≤ w (V) ≤ kVk . 2 Corollary 1 Let T ∈ B (H) and α, β ≥ 0 with α + β ≥ 1. Then we have 1 w T |T |α+β−1 ≤ |T |2α + |T ∗ |2β . (9) 2


Some inequalities of Furuta’s type In particular, we also have 1 w T |T |2α−1 ≤ |T |2α + |T ∗ |2α , 2 for any α ≥

1 2

165

(10)

and, as a special case,

1 2 2 (11) |T | + |T ∗ | . 2 Proof. We have from (F) for any x ∈ H that

D E

D E1/2 D E1/2

α+β−1 |T |2α x, x |T ∗ |2β x, x x, x ≤ (12)

T |T | Dh i E 1 |T |2α + |T ∗ |2β x, x ≤ 2 where α, β ≥ 0 with α + β ≥ 1. Utilising the inequality in (12) and taking the supremum over x ∈ H, kxk = 1 we get

D E

= sup T |T |α+β−1 x, x

w T |T |α+β−1 w (T |T |) ≤

kxk=1

≤ =

Dh i E 1 |T |2α + |T ∗ |2β x, x sup 2 kxk=1 1 2α 2β |T | + |T ∗ | . 2

For various interesting generalizations, extension of Kato and Furuta inequalities, see the papers [3]-[12], [17]-[21] and [23]. Motivated by the above results, we establish in this paper some generalizations of Kato and Furuta inequalities for functions of operators that can be expresses as power series with real coefficients. Applications for some functions of interest such as the exponential, hyperbolic and trigonometric functions are provided as well.

2

Functional inequalities

P n Now, by the help of power series f (z) = ∞ n=0 an z we can naturally construct another power series which will have as coefficientsP the absolute values of the n coefficient of the original series, namely, fA (z) := ∞ n=0 |an | z . It is obvious that this new power series will have the same radius of convergence as the original series. We also notice that if all coefficients an ≥ 0, then fA = f.


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S. S. Dragomir

P P∞ n n Theorem 2 Let f (z) = ∞ n=0 an z and be g (z) = n=0 bn z be two functions defined by power series with real coefficients and both of them convergent on the open disk D (0, R) ⊂ C, R > 0. If T is a bounded linear operator on the Hilbert space H and z, u ∈ C with the property that |z|2 , |u|2 , kT k2 < R,

(13)

|hTf (z |T |) g (u |T |) x, yi|2 ED E D ≤ fA |z|2 gA |u|2 fA |T |2 x, x |T ∗ |2 gA |T ∗ |2 y, y

(14)

then we have the inequality

for any x, y ∈ H. Proof. From Furuta’s inequality (F) we have for any natural numbers n ≥ 0 and m ≥ 1 the following power inequality

D E D E1/2 D E1/2

n+m−1 2n ∗ 2m |T | |T | |T | T x, y ≤ x, x y, y , (15)

where x, y ∈ H. If we multiply this inequality with the positive quantities |an | |z|n and |bm−1 | |u|m−1 , use the triangle inequality and the Cauchy-Bunyakowsky-Schwarz discrete inequality we have successively:

k l

X X D E

n+m−1 n m−1 T |T | x, y

(16) an z bm−1 u

n=0 m=1 k X l X

n=0 m=1 k X

D E

|an | |z|n |bm−1 | |u|m−1 T |T |n+m−1 x, y

n

|an | |z|

D

|T |

2n

l E1/2 X D E1/2 |bm−1 | |u|m−1 |T ∗ |2m y, y x, x

n=0

k X

m=1

|an | |z|

!1/2 * k X 2n

n=0

×

l X

2(m−1)

|bm−1 | |u|

+1/2 |an | |T |2n x, x

n=0 !1/2 * l X

m=1

for any x, y ∈ H and k ≥ 0, l ≥ 1.

m=1

+1/2 ∗ 2m

|bm−1 | |T |

y, y


Some inequalities of Furuta’s type

167

Observe also that k X l X

D E an zn bm−1 um−1 T |T |n+m−1 x, y

(17)

n=0 m=1

* =

T

k X

! n

an z |T | n

n=0

l X

! m−1

bm−1 u

m−1

|T |

+ x, y

m=1

for any x, y ∈ H and k ≥ 0, l ≥ 1. Making use of (16) and (17) we get

* ! +

! l k

X X

m−1 n x, y

bm−1 um−1 |T | an zn |T |

T

m=1 n=0 !1/2 * k +1/2 k X X |an | |z|2n |an | |T |2n x, x ≤ n=0

×

l X

n=0 !1/2 *

|bm−1 | |u|

m=1

2(m−1)

∗2

|T |

l X

(18)

+1/2 ∗ 2(m−1)

|bm−1 | |T |

y, y

m=1

for any x, y ∈ H and k ≥ 0, l ≥ 1. P n Due to the assumption (13) in the theorem, we have that the series ∞ n=0 an z P P P ∞ ∞ ∞ 2m 2n m n ∗ m |T | , m=0 bm u |T | , n=0 |an | |T | and m=0 |bm | |T | are convergent in P P 2n 2m B (H) and the series ∞ and ∞ are convergent in R n=0 |an | |z| m=0 |bm | |u| and then, by taking the limit over k → ∞ and l → ∞ in (18), we deduce the desired result (14). Remark 1 The above inequality (14) can provide various particular instances of interest. For instance, if we take g = f in Theorem 2 then we get

D E

(19)

Tf2 (z |T |) x, y

D E1/2 D E1/2 |T ∗ |2 fA |T ∗ |2 y, y ≤ fA |z|2 fA |T |2 x, x for any x, y ∈ H. Also if we take g (z) = 1 in (14), then we get D ED E |hTf (z |T |) x, yi|2 ≤ fA |z|2 fA |T |2 x, x |T ∗ |2 y, y for any x, y ∈ H.

(20)


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Corollary 2 With the assumptions of Theorem 2 we have the norm inequality kTf (z |T |) g (u |T |)k2 ≤ fA |z|2 gA |u|2 fA |T |2 |T ∗ |2 gA |T ∗ |2

(21)

and the numerical radius inequality w (Tf (z |T |) g (u |T |)) i1/2 1 h 2 2 2 2 ≤ fA |z| gA |u|2 fA |T | + |T ∗ | gA |T ∗ | . 2

(22)

Proof. The inequality (21) follows from (14) by taking the supremum over x, y ∈ H with kxk = kyk = 1. From (14) we also have the inequality |hTf (z |T |) g (u |T |) x, xi| E1/2 D E1/2 i1/2 D h |T ∗ |2 gA |T ∗ |2 x, x fA |T |2 x, x ≤ fA |z|2 gA |u|2 i E1/2 i1/2 Dh 1 h 2 ≤ fA |T |2 + |T ∗ |2 gA |T ∗ |2 x, x fA |z| gA |u|2 2 for any x ∈ H, which, by taking the supremum over kxk = 1 produces the desired result (22). The following result also holds: P n Theorem 3 Let f (z) = ∞ n=0 an z be a function defined by power series with real coefficients and convergent on the open disk D (0, R) ⊂ C, R > 0. If T is a bounded linear operator on the Hilbert space H with the property that kT k2 < R, then we have the inequality

D ED E 2 D E

2 2 2 2 2 (23)

T |T | f |T | x, y ≤ |T | fA |T | x, x |T ∗ | fA |T ∗ | y, y for any x, y ∈ H. Proof. From Furuta’s inequality (F) we have for any natural numbers n ≥ 1 the power inequality

D E D E1/2 D E1/2

2n−1 |T ∗ |2n y, y x, y ≤ |T |2n x, x (24)

T |T | where x, y ∈ H.


Some inequalities of Furuta’s type

169

If we multiply this inequality with the positive quantities |an−1 | , use the triangle inequality and the Cauchy-Bunyakowsky-Schwarz discrete inequality we have successively

* +

k

X

an−1 T |T |2n−1 x, y

(25)

n=1 k X n=1 k X

D E

|an−1 | T |T |2n−1 x, y

D E1/2 D E1/2 |an−1 | |T |2n x, x |T ∗ |2n y, y

n=1

* k X

+1/2 * k +1/2 X |an−1 | |T |2n x, x |an−1 | |T ∗ |2n y, y

n=1

n=1

for any x, y ∈ H and k ≥ 1. Observe also that k X

an−1 T |T |2n−1 = T |T |

n=1

|an−1 | |T |2n = |T |2

k X

n=1 k X

an−1 |T |2(n−1) ,

n=1

k X

and

k X

|an−1 | |T |2(n−1)

n=1

|an−1 | |T ∗ |2n = |T ∗ |2

n=1

k X

|an−1 | |T ∗ |2(n−1)

n=1

for any k ≥ 1. Therefore, by (25) we have the inequality * T |T | * ≤

k X

+2 an−1 |T |

n=1 k X 2

|T |

2(n−1)

(26)

x, y + k X |T ∗ |2 |an−1 | |T ∗ |2(n−1) y, y

+* |an−1 | |T |2(n−1) x, x

n=1

for any x, y ∈ H and k ≥ 1.

n=1


170

S. S. Dragomir

P 2n Due to the assumption kT k2 < R, we have that the series ∞ n=0 an |T | , P P∞ ∞ 2n ∗ 2n are convergent in B (H) and taking the and n=0 |an | |T | n=0 |an | |T | limit over k → ∞ in (26) we deduce the desired result from (23). Corollary 3 With the assumptions of Theorem 3 we have the norm inequality 2 2 2 2 2 2 T |T | f |T | ≤ |T | fA |T | |T ∗ | fA |T ∗ | and the numerical radius inequality 1 w T |T | f |T |2 ≤ |T |2 fA |T |2 + |T ∗ |2 fA |T ∗ |2 . 2 The following result for functions of normal operators holds. P n Theorem 4 Let f (z) = ∞ n=0 an z be a function defined by power series with real coefficients and convergent on the open disk D (0, R) ⊂ C, R > 0. If N is a normal operator on the Hilbert space H and α, β ≥ 0 with α + β ≥ 1 with the property that kNk2α , kNk2β < R, then we have the inequality

D E ED E 2 D

(α+β−1) x, y ≤ fA |N|2α x, x fA |N|2β y, y

f N |N|

(27)

for any x, y ∈ H. Proof. Utilising Furuta’s inequality written for Nn we have

D E 2 D ED E

n n α+β−1

2β x, y ≤ |Nn |2α x, x |(Nn )∗ | y, y

N |N | for any x, y ∈ H. Since N is normal, then |Nn |2 = (Nn )∗ Nn = N∗ ...N∗ N...N = N∗ ...NN∗ ...N = ... = (N∗ N) ... (N∗ N) = |N|2n for any natural number n, and, similarly, 2

2

|(Nn )∗ | = |(N∗ )n | = |N∗ |2n = |N|2n for any n ∈ N.

(28)


Some inequalities of Furuta’s type

171

These imply that |Nn |2α = |N|2αn , |(Nn )∗ | = |N|2βn and |Nn |α+β−1 = |N|(α+β−1)n for any α, β ≥ 0 and for any n ∈ N. Utilising the spectral representation for Borel functions of normal operators on Hilbert spaces, see for instance [1, p. 67], we have for any α, β ≥ 0 and for any n ∈ N that Z (α+β−1)n n = zn |z|(α+β−1)n dP (z) N |N| σ(N) Z h in z |z|(α+β−1) dP (z) = σ(N)

in h = N |N|(α+β−1) , where P is the spectral measure associated to the operator N and σ (N) is its spectrum. Therefore, the inequality (28) can be written as

Dh in E Dh in E1/2 Dh in E1/2

(α+β−1) |N|2β y, y x, y ≤ |N|2α x, x (29)

N |N| for any x, y ∈ H and for any n ∈ N. If we multiply the inequality (29) by |an | ≥ 0, sum over n from 0 to k ≥ 1 and utilize the Cauchy-Bunyakowsky-Schwarz discrete inequality, we have successively

* +

k

X

in h

an N |N|(α+β−1) x, y

(30)

n=0 k X n=0 k X

Dh in E

|an | N |N|(α+β−1) x, y

|an |

Dh

|N|2α

in

E1/2 Dh in E1/2 |N|2β y, y x, x

n=0

* k X

+1/2 * k +1/2 h in h in X 2α 2β |an | |N| |an | |N| x, x y, y

n=0

n=0

for any x, y ∈ H and for any k ≥ 1. Since kNk2α , kNk2β < R then N |N|(α+β−1) < R and the series ∞ X n=0

∞ h in X h in |an | |N|2α , |an | |N|2β n=0


172

S. S. Dragomir

and ∞ X

h in an N |N|(α+β−1)

n=0

are convergent in the Banach algebra B (H) . Taking the limit over k → ∞ in the inequality (30) we deduce the desired result from (27). Corollary 4 With the assumptions of Theorem 4, we have the inequality 2 (α+β−1) 2α 2β f N |N| ≤ fA |N| fA |N| .

(31)

Remark 2 If we take β = 1 − α with α ∈ [0, 1] in (27), then we get the following generalization of Kato’s inequality for normal operators (8) E ED D |hf (N) x, yi|2 ≤ fA |N|2α x, x fA |N|2(1−α) y, y

(32)

where x, y ∈ H and kNk2α , kNk2(1−α) < R.

3

Applications

As some natural examples that are useful for applications, we can point out that, if

f (z) = g (z) =

∞ X (−1)n n=1 ∞ X n=0

h (z) =

∞ X n=0

l (z) =

∞ X n=0

n!

zn = ln

1 , z ∈ D (0, 1) ; 1+z

(−1)n 2n z = cos z, z ∈ C; (2n) ! (−1)n 2n+1 z = sin z, z ∈ C; (2n + 1) ! (−1)n zn =

1 , z ∈ D (0, 1) ; 1+z

(33)


Some inequalities of Furuta’s type

173

then the corresponding functions constructed by the use of the absolute values of the coefficients are ∞ X 1 n 1 fA (z) = z = ln , z ∈ D (0, 1) ; (34) n! 1−z n=1

gA (z) =

∞ X n=0

hA (z) =

∞ X n=0

lA (z) =

∞ X

1 z2n = cosh z, z ∈ C; (2n) ! 1 z2n+1 = sinh z, z ∈ C; (2n + 1) ! zn =

n=0

1 , z ∈ D (0, 1) . 1−z

Other important examples of functions as power series representations with nonnegative coefficients are: exp (z) = 1 ln 2

∞ X 1 n z n!

z ∈ C,

(35)

n=0 ∞ X

1 z2n−1 , z ∈ D (0, 1) ; 2n − 1 n=1 ∞ X Γ n + 21 −1 √ sin (z) = z2n+1 , z ∈ D (0, 1) ; π (2n + 1) n! 1+z 1−z

=

n=0

tanh−1 (z) =

∞ X n=1

2 F1 (α, β, γ, z)

=

1 z2n−1 , 2n − 1

z ∈ D (0, 1)

∞ X Γ (n + α) Γ (n + β) Γ (γ) n=0

n!Γ (α) Γ (β) Γ (n + γ)

zn , α, β, γ > 0,

z ∈ D (0, 1) ; where Γ is the Gamma function. Example 1 Let x, y ∈ H. a) If we take f (z) = sin z and g (z) = cos z in (14), then we get |hT sin (z |T |) cos (u |T |) x, yi|2 ≤ sinh |z|2 cosh |u|2 E D ED × sinh |T |2 x, x |T ∗ |2 cosh |T ∗ |2 y, y

(36)


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S. S. Dragomir

for any z ∈ C and T ∈ B (H) . 1 1 b) If we take f (z) = ln 1+z and g (z) = ln 1−z in (14), then we get

D E 2

−1 −1

T ln (1H + z |T |) ln (1H − z |T |) x, y

!2 1 ≤ ln 1 − |z|2 −1 −1 |T ∗ |2 ln 1H − |T ∗ |2 y, y × ln 1H − |T |2 x, x

(37)

for any z ∈ C and T ∈ B (H) with |z| < 1 and kT k < 1. c) If we take f (z) = exp(z) and g (z) = exp (z) in (14), then we get |hT exp [(z + u) |T |] x, yi|2 ≤ exp |z|2 exp |u|2 ED D E × exp |T |2 x, x |T ∗ |2 exp |T ∗ |2 y, y

(38)

for any z, u ∈ C and T ∈ B (H) . d) By the inequality (20) we have

D E 2 D ED E

2 2 −1 −1 −1 ∗2 |z| |T | x, x |T | y, y sin

T sin (z |T |) x, y ≤ sin

(39)

and

D E 2

T tanh−1 (z |T |) x, y

D ED E ≤ tanh−1 |z|2 tanh−1 |T |2 x, x |T ∗ |2 y, y

(40)

for any z ∈ C and T ∈ B (H) with |z| < 1 and kT k < 1. Example 2 Let x, y ∈ H. 1 a) If we take f (z) = 1±z in (23), then we get

2 −1

T |T | 1H ± |T |2 x, y

−1 −1 2 2 ∗2 ∗2 |T | 1H − |T | ≤ |T | 1H − |T | x, x y, y

(41)


Some inequalities of Furuta’s type for any T ∈ B (H) with kT k < 1. 1 b) If we take f (z) = ln 1±z in (23), then we get

2 −1

T |T | ln 1H ± |T |2 x, y

−1 −1 2 2 ∗2 ∗2 |T | ln 1H − |T | y, y ≤ |T | ln 1H − |T | x, x for any T ∈ B (H) with kT k < 1. c) If we take f (z) = exp (z) in (23), then we get

D E 2

2

T |T | exp |T | x, y

ED D E ≤ |T |2 exp |T |2 x, x |T ∗ |2 exp |T ∗ |2 y, y

175

(42)

(43)

for any T ∈ B (H) . Example 3 Let N be a normal operator on the Hilbert space H, α, β ≥ 0 with α + β ≥ 1 and x, y ∈ H. 1 in (27), then we get a) If we take f (z) = 1±z

2 −1

1H ± N |N|(α+β−1) x, y

(44)

−1 −1 2α 2β ≤ 1H − |N| x, x 1H − |N| y, y provided kNk < 1. In particular, we have

D E 2

−1

(1H ± N) x, y

−1 −1 2α 2(1−α) x, x y, y , ≤ 1H − |N| 1H − |N|

(45)

for α ∈ [0, 1] . b) If we take f (z) = exp (z) in (27), then we get

D ED E 2 D E

(α+β−1) x, y ≤ exp |N|2α x, x exp |N|2β y, y . (46)

exp N |N| As a special case, we have D ED E |hexp (N) x, yi|2 ≤ exp |N|2α x, x exp |N|2(1−α) y, y , for α ∈ [0, 1] .

(47)


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References [1] W. Arveson, A Short Course on Spectral Theory, 2002, Springer-Verlag Inc., New York. [2] S. S. Dragomir, The hypo-Euclidean norm of an n-tuple of vectors in inner product spaces and applications, J. Inequal. Pure Appl. Math., 8 (2) (2007), Article 52, 22 pp. [3] M. Fujii, C.-S. Lin, R. Nakamoto, Alternative extensions of Heinz-KatoFuruta inequality, Sci. Math., 2 (2) (1999), 215–221. [4] M. Fujii and T. Furuta, L¨owner-Heinz, Cordes and Heinz-Kato inequalities, Math. Japon., 38 (1) (1993), 73–78. [5] M. Fujii, E. Kamei, C. Kotari and H. Yamada, Furuta’s determinant type generalizations of Heinz-Kato inequality, Math. Japon., 40 (2) (1994), 259–267 [6] M. Fujii, Y.O. Kim, Y. Seo, Further extensions of Wielandt type HeinzKato-Furuta inequalities via Furuta inequality, Arch. Inequal. Appl., 1 (2) (2003), 275–283. [7] M. Fujii, Y. O. Kim, M. Tominaga, Extensions of the Heinz-Kato-Furuta inequality by using operator monotone functions, Far East J. Math. Sci. (FJMS), 6 (3) (2002), 225–238. [8] M. Fujii, R. Nakamoto, Extensions of Heinz-Kato-Furuta inequality, Proc. Amer. Math. Soc., 128 (1) (2000), 223–228. [9] M. Fujii, R. Nakamoto, Extensions of Heinz-Kato-Furuta inequality. II., J. Inequal. Appl., 3 (3) (1999), 293–302. [10] T. Furuta, Equivalence relations among Reid, L¨owner-Heinz and HeinzKato inequalities, and extensions of these inequalities, Integral Equations Operator Theory, 29 (1) (1997), 1–9. [11] T. Furuta, Determinant type generalizations of Heinz-Kato theorem via Furuta inequality, Proc. Amer. Math. Soc., 120 (1) (1994), 223–231. [12] T. Furuta, An extension of the Heinz-Kato theorem, Proc. Amer. Math. Soc., 120 (3) (1994), 785–787.


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[13] G. Helmberg, Introduction to Spectral Theory in Hilbert Space, John Wiley & Sons, Inc. -New York, 1969. [14] T. Kato, Notes on some inequalities for linear operators, Math. Ann., 125 (1952), 208-212. [15] F. Kittaneh, Notes on some inequalities for Hilbert space operators, Publ. Res. Inst. Math. Sci., 24 (2) (1988), 283–293. [16] F. Kittaneh, Norm inequalities for fractional powers of positive operators,Lett. Math. Phys., 27 (4) (1993), 279–285. [17] C.-S. Lin, On Heinz-Kato-Furuta inequality with best bounds, J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math., 15 (1) (2008), 93–101. [18] C.-S. Lin, On chaotic order and generalized Heinz-Kato-Furuta-type inequality, Int. Math. Forum, 2 (37-40) (2007), 1849–1858. [19] C.-S. Lin, On inequalities of Heinz and Kato, and Furuta for linear operators, Math. Japon., 50 (3) (1999), 463–468. [20] C.-S. Lin, On Heinz-Kato type characterizations of the Furuta inequality. II., Math. Inequal. Appl., 2 (2) (1999), 283–287. [21] C. A. McCarthy, cp . Israel J. Math., 5(1967), 249-271. [22] G. Popescu, Unitary invariants in multivariable operator theory, Mem. Amer. Math. Soc., 200 (941) (2009), vi+91 pp. [23] M. Uchiyama, Further extension of Heinz-Kato-Furuta inequality, Proc. Amer. Math. Soc., 127 (10) (1999), 2899–2904.

Received: 13 June 2014


Acta Univ. Sapientiae, Mathematica, 6, 2 (2014) 178–193 DOI: 10.1515/ausm-2015-0005

Some polynomials associated with regular polygons Seppo Mustonen

Pentti Haukkanen

Department of Mathematics and Statistics FI-00014 University of Helsinki, Finland email: seppo.mustonen@survo.fi

School of Information Sciences FI-33014 University of Tampere, Finland email: pentti.haukkanen@uta.fi

Jorma Merikoski School of Information Sciences FI-33014 University of Tampere, Finland email: jorma.merikoski@uta.fi

Abstract. Let Gn be a regular n-gon with unit circumradius, and m = n−1 bn 2 c, µ = b 2 c. Let the edges and diagonals of Gn be en1 < · · · < enm . We compute the coefficients of the polynomial (x − e2n1 ) · · · (x − e2nµ ). They appear to form a well-known integer sequence, and we study certain related sequences, too. We also compute the coefficients of the polynomial (x − s2n1 ) · · · (x − s2nm ), where sni = cot

(2i − 1)π , 2n

i = 1, . . . , m.

We interpret sn1 as the sum of all individual edges and diagonals of Gn divided by n. We also discuss the interpretation of sn2 , . . . , snm , and present a conjecture on expressing sn1 , . . . , snm using en1 , . . . , enm . 2010 Mathematics Subject Classification: 11B83, 11C08, 15B36, 51M20 Key words and phrases: polynomial, regular polygon, eigenvalue

178


Polynomials associated with regular polygons

1

179

Introduction

Throughout, Gn is a regular n-gon with unit circumradius, and jnk n−1 m= . , µ= 2 2 Long time ago Kepler observed [2] that the squares of the edge and diagonals of G7 are the zeros of the polynomial x3 − 7x2 + 14x − 7. This raises a general question: Are the squares of (the lengths of) the edge and diagonals of Gn , excluding the diameter, the zeros of a monic polynomial of degree µ with integer coefficients? Yes, they are. This follows from Savio’s and Suruyanarayan’s [6] results, which, however, do not give the polynomial explicitly. We will do it in Section 2. A natural further question concerns the edge and diagonals themselves, instead of their squares. They are not zeros of a polynomial described above, but we will in Section 3 see that the squared sum of all individual edges and diagonals is the largest zero of a monic polynomial of degree m with integer coefficients. We will study geometric interpretation of the square roots of the other zeros in Section 4. In Section 5, we will present a conjecture on expressing these square roots as simple linear combinations of the edge and diagonals. We will in Section 6 notify that the coefficients of the first-mentioned polynomial form an OEIS [4] sequence, and also study OEIS sequences corresponding to certain related polynomials. Finally, we will complete our paper with conclusions and further questions in Section 7.

2

Squared chords

Let (the lengths of) the edge and diagonals of Gn be en1 < · · · < enm . Call them (the lengths of) the chords. Then kπ , k = 1, . . . , m. n Our problem is to find the coefficients amk and bmk of the polynomials enk = 2 sin

Am (x) = (x − e2n+2,1 ) · · · (x − e2n+2,m ) = xm + am,m−1 xm−1 + · · · + am1 x + am0 ,

(1)

where n is even, and Bm (x) = (x − e2n1 ) · · · (x − e2nm ) = xm + bm,m−1 xm−1 + · · · + bm1 x + bm0 , (2)


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where n is odd. We solve it in two theorems. Mustonen [3] found them experimentally and sketched their proofs. Let tridiagm (x, y) denote the symmetric tridiagonal m × m matrix with all main diagonal entries x and first super- and subdiagonal entries y. For m ≥ 2, define Am = tridiagm (2, 1) and Bm is as Am but the (m, m) entry equals 3. Also define A1 = (2) and B1 = (3). Denote by spec the (multi)set of eigenvalues. Lemma 1 For all m ≥ 1,

2 kπ

spec Am = 4 sin

k = 1, . . . , m = {e2n+2,1 , . . . , e2n+2,m }, n+2

k = 1, . . . , m = {e2n1 , . . . , e2nm }. spec Bm = 4 sin2

n Proof. See [1, 5, 6].

(3)

Theorem 1 In (1), m−k

amk = (−1)

m+1+k . 2k + 1

(4)

Proof. Denoting Pm (x) = xm +

m−1 X

(−1)m−k

k=0

m+1+k k x , 2k + 1

our claim is that Pm (x) = Am (x) for all m ≥ 1. Expanding det (xIm − Am ) along the last row, we have Am+1 (x) = (x − 2)Am (x) − Am−1 (x) for all m ≥ 2. Since P1 (x) = x − 2 = A1 (x)

(5)


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181

and P2 (x) = x2 − 4x + 3 = A2 (x), the claim (5) follows by showing that Pm+1 (x) = (x − 2)Pm (x) − Pm−1 (x)

(6)

for all m ≥ 2. Mustonen [3] did it by using Mathematica. We will do the computations algebraically in the appendix. The formula (4) yields amm = 1, consistently with the coefficient of xm in (1). It also allows to define a00 = 1. The polynomial ˜ m+1 (x) = (x − 4)Am (x) = xm+1 + αm+1,m xm + · · · + αm+1,1 x + αm+1,0 A has e2n+2,m+1 = 4 as the additional zero. By (4), m+k m+1+k m−k+1 αm+1,k = (−1) +4 . 2k − 1 2k + 1 (We define nk = 0 if k < 0.)

(7)

(8)

Theorem 2 In (2), +1 m+k m+1+k m+k m−k = (−1) + . (9) m − k 2k + 1 2k + 1 2k + 1

m−k 2m

bmk = (−1)

Proof. The second equation follows from trivial computation. To show the first, denote Qm (x) = xm +

m−1 X

(−1)m−k

k=0

2m + 1 m + k k x m − k 2k + 1

and claim that Qm (x) = Bm (x) for all m ≥ 1. Expanding det (xIm − Bm ), we have Bm+1 (x) = (x − 3)Am (x) − Am−1 (x) for all m ≥ 2. Since Q1 (x) = x − 3 = B1 (x)

(10)


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and Q2 (x) = x2 − 5x + 5 = B2 (x), the claim (10) follows by showing that Qm+1 (x) = (x − 3)Pm (x) − Pm−1 (x)

(11)

for all m ≥ 2. Mustonen [3] did also this by using Mathematica, and we will do the computations algebraically in the appendix. For k = m, the first expression in (9) is undefined but the second is defined. (We define nk = 0 if n < k.) It gives bmm = 1, the coefficient of xm in (2). It also allows to define b00 = 1. Corollary 1 The sum of all individual squared chords of Gn is n2 . Their product is nn . Proof. By Theorems 1 and 2 (or by [7, Eqs. (20) and (24)]), we obtain e22m,1 + · · · + e22m,m−1 = −am−1,m−2 = 2(m − 1), e22m+1,1 + · · · + e22m+1,m = −bm,m−1 = 2m + 1, and e22m,1 · · · e22m,m−1 = (−1)m am−1,0 = m, e22m+1,1 · · · e22m+1,m = (−1)m bm0 = 2m + 1. Denoting by Σn the sum and by Πn the product of all individual squared chords of Gn , we therefore have Σ2m = 2m · 2(m − 1) + m · 4 = (2m)2 , Σ2m+1 = (2m + 1)(2m + 1) = (2m + 1)2 , and Π2m = m2m 4m = (2m)2m ,

Π2m+1 = (2m + 1)2m+1 .


Polynomials associated with regular polygons

3

183

Sum of chords

The sum of all individual chords of Gn is Sn = nsn , where sn = en1 + · · · + en,m−1 + 21 enm = en1 + · · · + en,m−1 + 1 if n is even, and sn = en1 + · · · + enm if n is odd, is the sum of different (lengths of) chords but the diameter is halved. Theorem 3 For all n ≥ 3, sn = cot

π . 2n

Proof. We have [7, Eq. (21)] n−1 X k=1

sin

kπ π = cot . n 2n

(12)

If n is even, this implies sn =

m−1 X

2 sin

k=1

m−1 2m−1 X X kπ 1 kπ kπ + ·2= +1+ = sin sin n 2 n n k=1

k=m+1

2m−1 X k=1

sin

kπ π = cot . n 2n

If n is odd, then sn =

m X k=1

2 sin

m 2m 2m X kπ kπ X kπ π kπ X = sin + sin = sin = cot . n n n n 2n k=1

k=m+1

k=1

Is sn a zero of a monic √ polynomial of degree m with integer coefficients? Yes for s4 = cot π8 = 1 + 2; it is a zero of x2 − 2x − 1. On the other hand, it is easy p √ π to see that s5 = cot 10 = 5 + 2 5 is not a zero of such a polynomial. But


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√ √ s25 = 5 + 2 5 is a zero of x2 − 10x + 5, and the other zero is 5 − 2 5 = cot2 3π 10 . √ 2 − 6x + 1, and the other 2 has this property: it is a zero of x Also s24 = 3 + 2 √ zero is 3 − 2 2 = cot2 3π 8 . Generally, denoting sni = cot

(2i − 1)π , 2n

i = 1, . . . , m,

this motivates us to study for even n the coefficients of the polynomial Um (x) = (x − s2n1 ) · · · (x − s2nm ) = xm + um,m−1 xm−1 + · · · + um1 x + um0 , (13) and for odd n those of Vm (x) = (x − s2n1 ) · · · (x − s2nm ) = xm + vm,m−1 xm−1 + · · · + vm1 x + vm0 . (14) We will see that they all are integers. The largest zero is s2n = s2n1 . Mustonen [3] found the following theorem experimentally and also presented its proof. Yaglom and Yaglom [9, Eqs. (7) and (8)] formulated (16) differently. Theorem 4 In (13), n . 2k

(15)

n . 2k + 1

(16)

k

umk = (−1)

In (14), k

vmk = (−1) Proof. We have [10]

Pm k n n−2k t k=0 (−1) 2k cot n−2k−1 . cot nt = Pm n k t k=0 (−1) 2k+1 cot

(17)

Denote

(2i − 1)π , 2n Since cot nti = 0, (17) yields ti =

m X k=0

(−1)k

i = 1, . . . , m.

n cotn−2k ti = 0. 2k

(18)


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185

First assume n even. The polynomial ˜ m (x) = U

m X

(−1)

m−k

k=0

n k x 2k

is monic and has degree m. For all i = 1, . . . , m, m 2m 2k X 2m sni = s2m−2l (−1)l 2k 2m − 2l ni k=0 l=0 m m X X l 2m 2m−2l l n = (−1) s = (−1) cotn−2l ti = 0 2l ni 2l

˜ m (s2 ) = U ni

m X

(−1)m−k

l=0

l=0

by (18). Hence ˜ m (x) = (x − s2 ) · · · (x − s2 ) = Um (x), U nm n1 and (15) follows. Second, assume n odd. The polynomial V˜m (x) =

m X k=0

(−1)

m−k

n xk 2k + 1

is monic and has degree m. For all i = 1, . . . , m, m 2m + 1 2m + 1 2k X l (−1) s2m−2l = (−1) s = 2k + 1 ni 2m − 2l + 1 ni k=0 l=0 m m X X 2m + 1 −1 l 2m+1−2l −1 l 2m + 1 (−1) s = sni (−1) s2m+1−2l sni ni 2m − 2l + 1 ni 2l l=0 l=0 m X −1 l n = sni (−1) cotn−2l ti = 0, 2l

V˜m (s2ni ) =

m X

m−k

l=0

again by (18). Hence V˜m (x) = (x − s2n1 ) · · · (x − s2nm ) = Vm (x), and (16) follows.


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Corollary 2 The number s2n is the largest zero of the polynomial xm + um,m−1 nxm−1 + · · · + um1 nm−1 x + um0 nm if n is even, and that of xm + vm,m−1 nxm−1 + · · · + vm1 nm−1 x + vm0 nm if n is odd.

4

Interpreting sn,m−k+1 , k = 1, . . . , b n−1 3 c, n odd

The zeros of Am (x) and Bm (x) describe the squared chords of G2m+2 and G2m+1 , respectively, excluding the diameter. The largest zero of Um (x), s22m,1 = s22m , and that of Vm (x), s22m+1,1 = s22m+1 , describe the squared sum of chords but halving the diameter. In other words, the sum of all individual chords of Gn is divided by n and the result is squared. What about the other zeros? kπ Let the vertices of Gn be P0 , . . . , Pn−1 , where Pk = (cos kπ n , sin n ). Then kπ enk = P0 Pk = 2 sin n , k = 1, . . . , m. Since P0 Pn−k = P0 Pk , we define en,n−k = enk , k = 1, . . . , m.

Fix n and denote ek = enk for brevity. Assume that 3k < n; i.e., k < n3 . Then the line segments P0 P2k and Pk Pn−k intersect; let Qk be their intersection point and denote xk = P0 Qk . Because 4Qk P0 Pk ∼ 4Qk P2k Pn−k , we have ek xk = . e2k − xk e3k Hence 2kπ 2 sin kπ ek e2k n sin n = = 3kπ ek + e3k sin kπ n + sin n

xk =

2kπ 2 sin kπ n sin n

sin( 2kπ n

kπ n )

+

sin( 2kπ n

+

kπ n )

=

2kπ sin kπ n sin n

sin

2kπ n

cos

kπ n

= tan

kπ . n

If n is odd, then tan

π kπ kπ [2(m − k) + 1]π = cot − = cot = sn,m−k+1 . n 2 2m + 1 2n


Polynomials associated with regular polygons

187

n−1 Thus sn,m−k+1 = P0 Qk , k = 1, . . . , b n−1 3 c. In other words, the b 3 c smallest zeros of Vm (x) are the squared line segments P0 Qk , k = 1, . . . , b n−1 3 c. Mustonen [3] found this experimentally. The largest zero is already interpreted, but the interpretation of the rest of zeros remains open. For some experimental observations, see [3]. Interpretation of the zeros of Um (x), except the largest, remains open, too.

5

Expressing sn1 , . . . , snm using en1 , . . . , enm

Mustonen’s [3] experiments make conjecture that, given n, there are numbers (i) λnk ∈ {0, ±1}, i, k = 1, . . . , m, such that (i)

(i)

(i)

0 sni = λn1 en1 + · · · + λn,m−1 en,m−1 + λnm enm ,

where

0 enm =

1 2 enm enm

i = 1, . . . , m,

if n is even, if n is odd.

In other words, cot

h π (m − 1)π mπ i (2i − 1)π (i) (i) (i) = 2 λn1 sin + · · · + λn,m−1 sin + θn λnm sin , 2n n n n

where

θn =

1 2

if n is even, 1 if n is odd. (1)

(1)

This is true by (12) when i = 1 (sn1 = sn , λn1 = · · · = λnm = 1) but remains generally open. For example, let n = 15. Denoting sk = s15,k and ek = e15,k for brevity, we have [3, p. 17] s1 s2 s3 s4 s5 s6 s7

= = = = = = =

e1 + e2 +

e3 + e3 +

e4 + e5 + e6 + e7 e6 e5 e1 − e2 + e3 − e4 + e5 − e6 + e7 −e3 + e6 e1 − e2 + e3 + e4 − e5 + e6 − e7 e1 + e2 − e3 − e4 + e5 + e6 − e7 .


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We study the zero coefficients in general. If and only if d = gcd(n, 2i−1) > 1, then Gn ”inherits” the chord (2i − 1)π 2n from Gd . Then the chords of Gd are enough to express sni , and the coefficients of the remaining chords are zero. Indeed, in our example, π π 2π 3π = cot = 2 sin + sin , s2 = s15,2 = cot 30 10 5 5 π π 5π = cot = 2 sin , s3 = s15,3 = cot 30 6 3 9π 3π π 2π s5 = s15,5 = cot = cot = 2 − sin + sin , 30 10 5 5 showing that s3 is ”inherited” from G3 , and s2 and s5 from G5 . So we conjecture additionally that if and only if n is a prime or a power (i) of 2, then each λnk ∈ {±1}. Mustonen [3] gives also other experimental results (i) and conjectures about the structure of the three-dimensional array (λnk ), and presents an efficient algorithm to compute these numbers. sni = cot

6

Connections with OEIS sequences

The (lexicographically ordered) sequence (amk ) is A053122 in OEIS. Its first six terms are a00 = 1, a10 = −2, a11 = 1, a20 = 3, a21 = −4, a22 = 1. The OEIS sequence A132460 consists of the numbers

tnk

t = 1, n = 0, 1, 2, . . . , n0 n −k n−k−1 k = (−1) ( + ), n = 2, 3, . . . , k = 1, . . . , m. k k−1

The first six terms of its subsequence corresponding to odd values of n are t10 = 1 = b00 , t30 = 1 = b11 , t31 = −3 = b10 , t50 = 1 = b22 , t51 = −5 = b21 , t52 = 5 = b20 . In general, bmk = t2m+1,m−k . Also the characteristic polynomials of certain other tridiagonal matrices have connections with OEIS sequences. We study two of them. Let tridiag(a, b, c) denote the tridiagonal matrix with main diagonal, subdiagonal and superdiagonal entries those of vectors a, b and c, respectively, and denote x(k) = x, . . . , x, k copies. For m ≥ 3, define Cm = tridiag ((2(m) ), ((−1)(m−2) , −2), (−2, (−1)(m−2) ))


Polynomials associated with regular polygons and

C2 =

2 −2 −2 2

189

,

C1 = (2).

For m ≥ 1, consider the polynomial Cm (x) = det (xIm − Cm ) = xm + cm,m−1 xm−1 + · · · + cm1 x + cm0 and define C0 (x) = 1, c00 = cmm = 1. The sequence A140882 consists of the numbers (−1)m cmk . Since C0 (x) = 1, C1 (x) = x − 2, C2 (x) = x2 − 4x, C3 (x) = x3 − 6x2 + 8x, its first ten terms are 1, 2, −1, 0, −4, 1, 0, −8, 6, −1, as listed in [4]. ˜ 1 (x) = x2 − 4x = C2 (x) and xA ˜ 2 (x) = x3 − 6x2 + 8x = C3 (x), We have xA and generally ˜ m (x) Cm+1 (x) = xA

(19)

for all m ≥ 1. This can be proved similarly to the proofs of Theorems 1 and 2. By (8), a formula for A140882 is then obtained. By (19), (7) and (3),

spec Cm = spec Am−2 ∪ {0, 4} = 4 sin2

k = 0, . . . , m − 1 2m − 2 for m ≥ 3. Finally, the sequence A136672 motivates us to study the polynomial Fm+1 (x) = (x − 2)Am (x) = xm+1 + fm+1,m xm + · · · + fm+1,1 x + fm+1,0

(20)

and its connections with the matrix Dm , defined by Dm = tridiag((2(m) ), ((−1)(m−2) , 0), ((−1)(m−1) )) if m ≥ 3, and D2 =

2 −1 0 2

,

D1 = (2).

By Theorem 1, m+k m+1+k fm+1,k = (−1)m−k+1 ( +2 ). 2k − 1 2k + 1 For m ≥ 1, consider the polynomial Dm (x) = det (xIm − Dm ) = xm + dm,m−1 xm−1 + · · · + dm1 x + dm0

(21)


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S. Mustonen, P. Haukkanen, J. Merikoski

and define D0 (x) = 1, d00 = dmm = 1. The sequence A136672 consists of the numbers (−1)m dmk . We have D0 (x) = 1, D1 (x) = x − 2, D2 (x) = x2 − 4x + 4, D3 (x) = x3 −6x2 +11x−6. So its first ten terms are 1, 2, −1, 4, −4, 1, 6, −11, 6, −1, as listed in [4]. Since F1 (x) = x − 2 = D1 (x), F2 (x) = x2 − 4x + 4 = D2 (x), and F3 (x) = x3 − 6x2 + 11x − 6 = D3 (x), it seems that Dm (x) = Fm (x)

(22)

for all m ≥ 1. This can be proved similarly to the previous proofs. By (21), a formula for A136672 follows. By (22), (20) and (3),

spec Dm = spec Am−1 ∪ {2} = 4 sin2

k = 1, . . . , m − 1 ∪ {2} 2m for m ≥ 2.

7

Conclusions and further questions

The squared chords of Gn , excluding the diameter, are the zeros of a monic polynomial of degree µ with integer coefficients. Including the diameter, the degree is m. The squared sum of all individual chords is the largest zero of a monic polynomial of degree m with integer coefficients. An equivalent fact is that the squared sum of all different (lengths of) chords but the diameter is halved, is a zero of such a polynomial. The zeros of this polynomial seem to be linear combinations of the chords with all coefficients 0 or ±1. Lemma 1, stating that e2n1 , . . . , e2nµ are the eigenvalues of a tridiagonal matrix with integer entries, follows from certain properties of the Chebychev polynomials. So squared chords have interesting connections with these topics. But what about s2n1 , . . . , s2nm ? Are also they the eigenvalues of such a tridiagonal matrix? This question remains open. The coefficients of the polynomial (x − e2n1 ) · · · (x − e2nµ ) form an OEIS sequence, and so do also those of certain related polynomials. What about the coefficients of (x − s2n1 ) · · · (x − s2nm )? Do also they form such a sequence? This question remains open, too.


Polynomials associated with regular polygons

191

Appendix: Proofs of (6) and (11) Proof of (6) (x − 2)Pm (x) − Pm−1 (x) m m−1 X X k m−k m + 1 + k m−1−k m + k = (x − 2) x − xk (−1) (−1) 2k + 1 2k + 1 k=0 k=0 m−1 m X X m+1 m−k m + 1 + k k+1 m−k m + 1 + k −x + (−1) x −2 (−1) xk 2k + 1 2k + 1 k=0 k=0 m−1 X m+k k − (−1)m−1−k x 2k + 1 k=0 m m X X m+1+k k m+k k (−1)m+1−k x (−1)m+1−k x +2 = xm+1 + 2k + 1 2k − 1 k=0 k=1 m−1 X m+k k − (−1)m+1−k x 2k + 1 k=0 2m 2m + 1 m+1 =x − +2 xm 2m − 1 2m + 1 m−1 X m+k m+1+k m+k m+1−k + (−1) +2 − xk 2k − 1 2k + 1 2k + 1 k=1 m+1 m m+1 + (−1) 2 − 1 1 m−1 X m+2+k k = xm+1 − (2m + 2)xm + (−1)m+1−k x + (−1)m+1 (m + 2) 2k + 1 k=1 m+1 X m+1−k m + 1 + 1 + k = (−1) xk = Pm+1 (x). 2k + 1 k=0


192

S. Mustonen, P. Haukkanen, J. Merikoski Proof of (11) (x − 3)Pm (x) − Pm−1 (x) 2m 2m + 1 m+1 = ··· = x − +3 xm 2m − 1 2m + 1 m−1 X m+k m+1+k m+k m+1−k + (−1) +3 − xk 2k − 1 2k + 1 2k + 1 k=1 m+1 m m+1 + (−1) 3 − 1 1 m−1 X m+1+k k m+1 m m+1−k 2m + 3 x =x − (2m + 3)x + (−1) m−k+1 2k + 1 k=1

m+1

+ (−1) (2m + 3) m X m+1 m+1−k 2(m + 1) + 1 m + 1 + k =x + (−1) xk = Qm+1 (x). m+1−k 2k + 1 k=0

References [1] N. D. Cahill, J. R. D’Errico, J. P. Spence, Complex factorization of the Fibonacci and Lucas numbers, Fibonacci Quart., 41 (2003), 13–19. [2] H. M. S. Coxeter, Regular Convex Polytopes, Cambridge U. Pr., 1974. [3] S. Mustonen, Lengths of edges and diagonals and sums of them in regular polygons as roots of algebraic equations, 2013, 44 pp. http://www.survo.fi/papers/Roots2013.pdf [4] The On-Line Encyclopedia of Integer Sequences (OEIS). http://oeis.org/ [5] D. E. Rutherford, Some continuant determinants arising in physics and chemistry, I, Proc. Royal Soc. Edinburgh, 62A (1947), 229–236. [6] D. Y. Savio, E. R. Suryanarayan, Chebychev polynomials and regular polygons, Amer. Math. Monthly, 100 (1993), 657–661. [7] E. W. Weisstein, Sine, MathWorld –A Wolfram Web Resource. http://mathworld.wolfram.com/Sine.html


Polynomials associated with regular polygons

193

[8] E. W. Weisstein, Tangent, MathWorld –A Wolfram Web Resource. http://mathworld.wolfram.com/Tangent.html [9] A. M. glom, I. M. glom, lementarny˘ i vyvod formul Vallisa, Le˘ ibnica i ˘ ilera dl qisla π, Uspehi Matem. Nauk, 8 (1953), 181–187. (A. M. Yaglom, I. M. Yaglom, An elementary derivation of the Wallis, Leibniz and Euler formulas for the number π, Uspekhi Matem. Nauk, 8 (1953), 181–187.) [10] http://functions.wolfram.com/ElementaryFunctions/Cot/27/01/ 0002/

Received: 4 August 2014


Acta Univ. Sapientiae, Mathematica, 6, 2 (2014) 194–208 DOI: 10.1515/ausm-2015-0006

On the weighted integral inequalities for convex function Mehmet Zeki Sarikaya

Samet Erden

Department of Mathematics, Faculty of Science and Arts, Duzce University, Konuralp Campus, Duzce-Turkey email: sarikayamz@gmail.com

Department of Mathematics, Faculty of Science, Bartin University, Bartin-Turkey email: erdem1627@gmail.com

Abstract. In this paper, we establish several weighted inequalities for some differantiable mappings that are connected with the celebrated Hermite-Hadamard-Fej´er type and Ostrowski type integral inequalities. The results presented here would provide extensions of those given in earlier works.

1

Introduction

The following result is known in the literature as Ostrowski’s inequality [10]: Theorem 1 Let f : [a, b]→ R be a differentiable mapping on (a, b) whose 0 derivative f : (a, b)→ R is bounded on (a, b), i.e., kf0 k∞ = sup |f0 (t)| < ∞. t∈(a,b)

Then, the inequality:

" #

Zb a+b 2

0 (x − ) 1 1 2

≤ f

f(x) − f(t)dt + (b − a)

∞ b−a 4 (b − a)2

(1)

a

holds for all x ∈ [a, b]. The constant

1 4

is the best possible.

2010 Mathematics Subject Classification: 26D07, 26D15 Key words and phrases: Ostrowski’s inequality, Montgomery’s identities, convex function, H¨ older inequality

194


On the weighted integral inequalities

195

Inequality (1) has wide applications in numerical analysis and in the theory of some special means; estimating error bounds for some special means, some mid-point, trapezoid and Simpson rules and quadrature rules, etc. Hence inequality (1) has attracted considerable attention and interest from mathematicans and researchers. Due to this, over the years, the interested reader is also refered to ([1]-[7],[12]-[17]) for integral inequalities in several independent variables. In addition, the current approach of obtaining the bounds, for a particular quadrature rule, have depended on the use of Peano kernel. The general approach in the past has involved the assumption of bounded derivatives of degree greater than one. If f : [a, b] → R is differentiable on [a, b] with the first derivative f0 integrable on [a, b], then Montgomery identity holds: Zb Zb 1 f(x) = f(t)dt + P(x, t)f0 (t)dt, b−a a

(2)

a

where P(x, t) is the Peano kernel defined by  t−a   , a≤t<x   b−a P(x, t) :=   t−b   , x ≤ t ≤ b. b−a Definition 1 The function f : [a, b] ⊂ R → R, is said to be convex if the inequality f(λx + (1 − λ)y) ≤ λf(x) + (1 − λ)f(y) holds for all x, y ∈ [a, b] and λ ∈ [0, 1] . We say that f is concave if (−f) is convex. The following inequality is well known in the literature as the HermiteHadamard integral inequality (see, [11]): Zb a+b 1 f(a) + f(b) f ≤ f(x)dx ≤ (3) 2 b−a a 2 holds, where f : I ⊂ R → R is a convex function on the interval I of real numbers and a, b ∈ I with a < b. The most well-known inequalities related to the integral mean of a convex function are the Hermite-Hadamard inequalities or its weighted versions, the so-called Hermite-Hadamard-Fej´er inequalities (see, [18]-[22]). In [8], Fej´er gave a weighted generalization of the inequality (3) as the following:


196

M. Z. Sarikaya, S. Erden

Theorem 2 Let f : [a, b] → R, be a convex function, then the inequality Zb Zb Z a+b 1 f(a) + f(b) b f w(x)dx ≤ f(x)w(x)dx ≤ w(x)dx (4) 2 b−a a 2 a a holds, where w : [a, b] → R is nonnegative, integrable, and symmetric regarding x = a+b 2 . In [18], some inequalities of Hermite-Hadamard-Fej´er type for differentiable convex mappings were proved using the following lemma. Lemma 1 Let f : I◦ ⊂ R → R be a differentiable mapping on I◦ , a, b ∈ I◦ with a < b, and w : [a, b] → [0, ∞) be a differentiable mapping. If f0 ∈ L[a, b], then the following equality holds: Zb Zb 1 a+b 1 f(x)w(x)dx − f w(x)dx b−a a b−a 2 a (5) Z1 0 = (b − a) k(t)f (ta + (1 − t)b)dt 0

for each t ∈ [0, 1], where  Rt 1  0 w(as + (1 − s)b)ds, t ∈ [0, 2 ) k(t) =  R1 − t w(as + (1 − s)b)ds, t ∈ [ 21 , 1]. The main result in [18] is as follows: Theorem 3 Let f : I◦ ⊂ R → R be a differentiable mapping on I◦ , a, b ∈ I◦ with a < b, and w : [a, b] → [0, ∞) be a differentiable mapping and symmetric 0 to a+b 2 . If |f | is convex on [a, b] , then the following inequality holds:

Zb Zb

1

a+b 1

f(x)w(x)dx − f w(x)dx

b − a

b−a 2 a a (6) ! Zb h i |f0 (a)| + |f0 (b)| 1 2 2 (x (b ≤ w(x) − a) − − x) dx . 2 (b − a)2 a+b 2

In this article, using functions whose derivatives absolute values are convex, we obtained new inequalities of Fejer-Hermite-Hadamard type and Ostrowski type. The results presented here would provide extensions of those given in earlier works.


On the weighted integral inequalities

2

197

Main results

We will establish some new results connected with the left-hand side of (4) and Ostrowski type inequalities used the following Lemma. Now, we give the following new Lemma for our results: Lemma 2 Let f : I◦ ⊆ R → R be a differentiable mapping on I◦ , a, b ∈ I◦ with a < b and let w : [a, b] → R. If f0 , w ∈ L[a, b], then, for all x ∈ [a, b], the following equality holds:  α  α Zx Zt Zb Zb  w(s)ds f0 (t)dt −  w(s)ds f0 (t)dt a

a

x

t

 x α  b α  Z Z =  w(s)ds +  w(s)ds  f(x) a

(7)

x

t α−1  α−1 Z Zb Zb − α  w(s)ds w(t)f(t)dt − α  w(s)ds w(t)f(t)dt. Zx

a

a

x

t

Proof. By integration by parts, we have the following equalities: t α x    α

Z Zx Zt Zx Zt

 w(s)ds f0 (t)dt =  w(s)ds f(t) − α  w(s)ds w(t)f(t)dt

a a a a a a (8) x α t α−1 x Z Z Z =  w(s)ds f(x) − α  w(s)ds w(t)f(t)dt a

a

a

and Zb x

b α Z  w(s)ds f0 (t)dt t

b α−1  b α

Zb Zb Z

w(t)f(t)dt =  w(s)ds f(t)

+ α  w(s)ds

x t t x b α b α−1 b Z Z Z w(t)f(t)dt. = −  w(s)ds f(x) + α  w(s)ds x

x

t

(9)


198

M. Z. Sarikaya, S. Erden

Subtracting (8) from (9), we obtain (7) Zx a

t α  α Z Zb Zb  w(s)ds f0 (t)dt −  w(s)ds f0 (t)dt a

x

t

α  α  b  x Z Z =  w(s)ds +  w(s)ds  f(x) x

a

 α−1  α−1 Zx Zt Zb Zb −α  w(s)ds w(t)f(t)dt − α  w(s)ds w(t)f(t)dt. a

a

x

t

This completes the proof.

Corollary 1 Under the same assumptions as in Lemma 2, if we put α = 1, then the following identity holds: b  Z Zb  w(s)ds f(x) − w(t)f(t)dt a  Zx Zt

a

Zb

b  Z 0 =  w(s)ds f (t)dt −  w(s)ds f (t)dt a

0

a

x

(10)

t

Remark 1 If we take w(s) = 1 in (10), the idendity (10) reduces to the identity (2). Definition 2 Let f ∈ L1 [a, b]. The Riemann-Liouville integrals Jαa+ f and Jαb− f of order α > 0 with a ≥ 0 are defined by Jαa+ f(x) =

1 Γ (α)

Jαb− f(x) =

1 Γ (α)

and

Zx

(x − t)α−1 f(t)dt, x > a

a

Zb

(t − x)α−1 f(t)dt, x < b

x

respectively. Here, Γ (α) is the Gamma function and J0a+ f(x) = J0b− f(x) = f(x).


On the weighted integral inequalities

199

Corollary 2 Under the same assumptions as in Lemma 2, if we put w(s) = 1, then the following equality holds: [(x − a)α + (b − x)α ] f(x) − Γ (α + 1)Jαx− f(a) − Γ (α + 1)Jαx+ f(b) (11) Zx α

0

Zb

0

(t − a) f (t)dt − (b − t)α f (t)dt.

= a

x

Corollary 3 Under the same assumptions of Corollary 2 with x = a+b 2 , the idendity (11) becomes to the following identity a+b Γ (α + 1) α α f − 1−α J − f(a) + J a+b + f(b) ( 2 ) 2 2 (b − a)α ( a+b 2 )   a+b   b 2   Z   Z 1 α 0 α 0 (b (t − t) f (t)dt − a) f (t)dt − = .  21−α (b − a)α      a a+b 2

Now, by using the above lemma, we prove our main theorems: Theorem 4 Let f : I◦ ⊆ R → R be a differentiable mapping on I◦ , a, b ∈ I◦ with a < b and let w : [a, b] → R be continuous on [a, b]. If |f0 | is convex on [a, b], then the following inequality holds:

 x α  b α 

Z Z

 w(s)ds +  w(s)ds  f(x)

a x

   α−1 α−1

Zx Zt Zb Zb

−α  w(s)ds w(t)f(t)dt − α  w(s)ds w(t)f(t)dt

a a x t kwkα[a,x],∞ (b − a)(x − a)α+1 (x − a)α+2 0 ≤ − |f (a)| b−a α+1 α+2 kwkα[a,x],∞ (x − a)α+2 0 kwkα[x,b],∞ (b − x)α+2 0 + |f (b)| + |f (a)| b−a α+2 b−a α+2 kwkα[a,x],∞ (b − a)(b − x)α+1 (b − x)α+2 0 − |f (b)| + b−a α+1 α+2


200

M. Z. Sarikaya, S. Erden kwkα[a,b],∞

(b − a)(x − a)α+1 (b − x)α+2 − (x − a)α+2 ≤ + b−a α+1 α+2 α+2 α+2 α+1 (x − a) − (b − x) (b − a)(b − x) 0 |f (b)| + + α+1 α+2

0

|f (a)|

where α > 0 and kwk[a,b],∞ = sup |w(t)| . t∈[a,b]

Proof. We take absolute value of both sizes of (7), we find that

 x α  b α 

Z Z

 w(s)ds +  w(s)ds  f(x)

a x

   α−1 α−1

Zx Zt Zb Zb

−α  w(s)ds w(t)f(t)dt − α  w(s)ds w(t)f(t)dt

a a x t

    α α

Zx Zt Zb Zb

0

0

≤ 

w(s)ds

 f (t) dt + 

w(s)ds

 |f (t)|dt

a

a

x

Zx ≤

kwkα[a,x],∞

t

Zb

0

0

(t − a) f (t) dt + kwk[x,b],∞ (b − t)α |f (t)|dt α

a

Zx = kwkα[a,x],∞

x

t − a

α 0 b−t (t − a) f ( a+ b) dt b−a b−a

a

Zb +

kwkα[x,b],∞

0 b−t t − a

(b − t) f ( a+ b) dt b−a b−a

α

x 0

Since |f | is convex on [a.b], it follows that

 x α  b α 

Z Z

 w(s)ds +  w(s)ds  f(x)

a x

  α−1  α−1

Zx Zt Zb Zb

w(t)f(t)dt − α  w(s)ds w(t)f(t)dt

−α  w(s)ds

a a x t


On the weighted integral inequalities Zx ≤

kwkα[a,x],∞

(t − a)

α

201

b − t

0

t − a 0 |f (b)| dt

f (a) + b−a b−a

a

Zb

b−t 0 t−a 0 |f (a)| + |f (b)| dt b−a b−a x kwkα[a,x],∞ (b − a)(x − a)α+1 (x − a)α+2 0 (x − a)α+2 0 |f (a)| + = − |f (b)| b−a α+1 α+2 α+2

kwkα[x,b],∞ (b − x)α+2

0

(b − a)(b − x)α+1 (b − x)α+2 0 f (a) + − |f (b)| +

b−a α+2 α+1 α+2

α kwk[a,b],∞ (b − a)(x − a)α+1 (b − x)α+2 − (x − a)α+2 0 ≤ + |f (a)| b−a α+1 α+2 (b − a)(b − x)α+1 (x − a)α+2 − (b − x)α+2 0 + + |f (b)| . α+1 α+2 +

kwkα[x,b],∞

(b − t)α

Hence, the proof of theorem is completed.

Corollary 4 Under the same assumptions as in Theorem 4, if we take w(s) = 1, then the following inequality holds: |[(x − a)α + (b − x)α ] f(x) − Γ (α + 1) [Jαx− f(a) + Jαx+ f(b)]|

1 (b − a)(x − a)α+1 (b − x)α+2 − (x − a)α+2 0 ≤ + |f (a)| b−a α+1 α+2 (b − a)(b − x)α+1 (x − a)α+2 − (b − x)α+2 0 + + |f (b)| . α+1 α+2 Remark 2 If we take x =

a+b 2

in (12), we get

a+b

2α−1 Γ (α + 1) α α

f

− J f(a) + J f(b) − + a+b a+b

( 2 ) ( 2 ) 2 (b − a)α (b − a)

0

0

f (a) + f (b)

4 (α + 1) which is proved by Sarikaya and Yildirim in [19].

(12)


202

M. Z. Sarikaya, S. Erden

Corollary 5 Under the same assumptions as in Theorem 4, if we take α = 1, then the following inequality holds:

 b

Z

Zb

 w(s)ds f(x) − w(t)f(t)dt

a a

kwk[a,b],∞ (b − a)(x − a)2 (b − x)3 − (x − a)3 0 ≤ |f (a)| + b−a 2 3 2 (x − a)3 − (b − x)3 (b − a)(b − x) 0 + |f (b)| . + 2 3 Corollary 6 Under the same assumptions of Corollary 5 with x = get

 b

Z

Zb

 w(s)ds f a + b − w(t)f(t)dt

2

a a 

0

0

 (b − a)2 kwk[a,b],∞ f (a) + f (b)

 . ≤ 4 2

a+b 2 ,

we

(13)

Remark 3 If we take w(s) = 1 in (13), we have

0

0

Zb

(b − a) f (a) + f (b)

a+b 1

f   − f(t)dt

2 b−a 4 2

a

which is proved by Kırmacı in [9]. 0

Corollary 7 Under the same assumptions as in Theorem 4, if we put |f (a)| = 0 |f (b)| in (10), then the following inequality holds:

 x α  b α 

Z Z

 w(s)ds +  w(s)ds  f(x)

a x

 α−1 t α−1

x Z Z Zb Zb

    w(s)ds w(t)f(t)dt − α w(s)ds w(t)f(t)dt

−α

a a x t


On the weighted integral inequalities

0

α

f (a) kwk[a,x],∞

0

α

f (a) kwk[x,b],∞

(x − a)α+1 + α + 1 α+1

0

α i

f (a) kwk[a,b],∞ h (x − a)α+1 + (b − x)α+1 ≤ α+1 ≤

203

(b − x)α+1

Theorem 5 Let f : I◦ ⊆ R → R be a differentiable mapping on I◦ , a, b ∈ I◦ with a < b and let w : [a, b] → R be continuous on [a, b]. If |f0 |q is convex on [a, b], q > 1, then the following inequality holds:

 α  b α   α−1

Zx Z Zx Zt

 w(s)ds +  w(s)ds  f(x) − α  w(s)ds w(t)f(t)dt

a x a a

  α−1

1 Zb Zb

kwkα αp+1 p (x − a) [a,x],∞

−α  w(s)ds w(t)f(t)dt ≤ 1

αp + 1 q (b − a)

x t q1 kwkα (x − a)2 0 (b − a)2 − (b − x)2 0 [x,b],∞ q q + |f (a)| + |f (b)| 1 2 2 (b − a) q 1 q1 2 − (x − a)2 (14) (b − x)αp+1 p (b − x)2 0 (b − a) 0 |f (a)|q + |f (b)|q αp + 1 2 2  1 ! kwkα[a,b],∞  (x − a)αp+1 p (b − a)2 − (b − x)2 0 |f (a)|q ≤ 1  αp + 1 2 q (b − a) (x − a)2 0 + |f (b)kq 2 (b − 2

x)2

0

1 p

+

(b − x)αp+1 αp + 1 2

|f (a)|q +

where α > 0,

q1

+

1 q

!1

p

2

(b − a) − (x − a) 0 |f (b)|q 2

!1  q 

= 1, and kwk[a,b],∞ = sup |w(t)| . t∈[a,b]


204

M. Z. Sarikaya, S. Erden

Proof. We take absolute value of (7). Using Holder’s inequality, we find that

 x α  α  b

Z Z

 w(s)ds +  w(s)ds  f(x)

x a

α−1  α−1 t

x Zb Zb Z Z

w(t)f(t)dt − α  w(s)ds w(t)f(t)dt

−α  w(s)ds

x a a t

α

α Zx Zt Zb Zb

0

0 ≤

w(s)ds

|f (t)|dt +

w(s)ds

|f (t)|dt

a a

x t

αp  p1  x 

1 q

Zx Zt Z

0 q 

   ≤ |f (t)| dt

w(s)ds dt

a

a

a

αp  p1  b  q1 

Z Zb Zb

0 + 

w(s)ds

dt  |f (t)|q dt

x x t 1 x 1 x p q Z Z 0 α αp q ≤ kwk[a,x],∞  |t − a| dt  |f (t)| dt a

a

b  p1  b  q1 Z Z 0 + kwkα[x,b],∞  |b − t|αp dt  |f (t)|q dt x

x

0 q Since f (t) is convex on [a, b]

q

q

q

0 b−t t − a

≤ b − t

f0 (a)

+ t − a

f0 (b)

f ( a + b)

b−a b−a

b−a b−a

(15)

From (15), it follows that

 x α  b α 

Z Z

 w(s)ds +  w(s)ds  f(x)

a x

α−1  α−1 t

x Z Z Zb Zb

    w(s)ds w(t)f(t)dt

−α w(s)ds w(t)f(t)dt − α

a a x t


On the weighted integral inequalities

kwkα[a,x],∞ 1

(b − a) q

(x − a)αp+1 αp + 1

p1

205

(b − a)2 − (b − x)2 0 |f (a)|q 2

1 1 α (x − a)2

0

q q kwk[x,b],∞ (b − x)αp+1 p + +

f (b)

1 2 αp + 1 (b − a) q !1 q kwkα[a,b],∞ (b − a)2 − (x − a)2 0 (b − x)2 0 q q |f (a)| + |f (b)| ≤ 1 2 2 (b − a) q  !1 !1 q  (x − a)αp+1 p (b − a)2 − (b − x)2 0 2 (x − a) 0 |f (a)|q + |f (b)|q  αp + 1 2 2 αp+1

+

(b − x) αp + 1

!1

p

(b − 2

x)2

0

|f (a)|q +

2

2

(b − a) − (x − a) 0 |f (b)|q 2

!1  q 

which completes the proof.

Corollary 8 Under the same assumptions as in Theorem 4, if we put w(s) = 1, then the following inequality holds: |[(x − a)α + (b − x)α ] f(x) − Γ (α + 1) [Jαx− f(a) + Jαx+ f(b)]| ≤  !1  (x − a)αp+1 p  αp + 1 +

(b − x)αp+1 αp + 1

!1

p

1 1

(b − a) q !1

(b − a)2 − (b − x)2 0 (x − a)2 0 |f (a)|q + |f (b)|q 2 2

(b − a)2 − (x − a)2 0 (b − x)2 0 |f (a)|q + |f (b)|q 2 2

q

(16) !1  q 

.

Remark 4 If we take x = a+b 2 in (16), we have

a+b 2α−1 Γ (α + 1) α α

f − J − f(a) + J a+b + f(b) a+b

( 2 ) ( 2 ) 2 (b − a)α  !1 !1  0 0 0 0 q q q q q (b − a)  3|f (a)| + |f (b)| |f (a)| + 3|f (b)| q  ≤ + 1 4 4 4(αp + 1) p which is proved by Sarikaya and Yildirim in [19].


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Corollary 9 Let the conditions of Theorem 5 hold. If we take α = 1 in (14), then the following inequality holds:

 b

Z

Zb

kwk[a,b],∞

 w(s)ds f(x) − w(t)f(t)dt ≤ 1

(b − a) q a a  !1 !1 q  (x − a)p+1 p (b − a)2 − (b − x)2 0 (x − a)2 0 q q |f (a)| + |f (b)|  p+1 2 2 !1 !1  q

2 2 p+1 p 2 q (b − a) − (x − a) 0 (b − x) (b − x) 0

q |f (b)| +

f (a) +  p+1 2 2 Corollary 10 Under the same assumptions of Corollary 9 with x = get

 b 

Z Zb

(b − a)2 kwk[a,b],∞

a + b

 w(s)ds f − w(t)f(t)dt 1 1

2

22+ q (p + 1) p a a  !1 !1  0 0  3|f0 (a)|q + |f0 (b)|q q q q |f (a)| + 3|f (b)| q  + .   2 2 Remark 5 If we take w(s) = 1 in (17), we have

Zb

a+b

1

f

− f(t)dt

2 b−a

a  !1  3|f0 (a)|q + |f0 (b)|q q (b − a) + ≤ 1 1 2 22+ q (p + 1) p  which is proved by Kırmacı in [9].

0

|f

(a)|q

+ 3|f 2

0

(b)|q

a+b 2 ,

we

(17)

!1  q 


On the weighted integral inequalities

207

References [1] F. Ahmad, N. S. Barnett, S. S. Dragomir, New weighted Ostrowski and Cebysev type inequalities, Nonlinear Anal., 71 (12) (2009), 1408–1412. [2] F. Ahmad, A. Rafiq, N. A. Mir, Weighted Ostrowski type inequality for twice differentiable mappings, Global Journal of Research in Pure and Applied Math., 2 (2) (2006), 147–154. [3] N. S. Barnett, S. S. Dragomir, An Ostrowski type inequality for double integrals and applications for cubature formulae, Soochow J. Math., 27 (1) (2001), 109–114. [4] N. S. Barnett, S. S. Dragomir, C. E. M. Pearce, A Quasi-trapezoid inequality for double integrals, ANZIAM J., 44 (2003), 355–364. [5] S. S. Dragomir, P. Cerone, N. S. Barnett, J. Roumeliotis, An inequlity of the Ostrowski type for double integrals and applications for cubature formulae, Tamsui Oxf. J. Math., 16 (1) (2000), 1–16. [6] S. Hussain, M. A. Latif, M. Alomari, Generalized duble-integral Ostrowski type inequalities on time scales, Appl. Math. Letters, 24 (2011), 1461– 1467. [7] M. E. Kiris, M. Z. Sarikaya, On the new generalization of Ostrowski type inequality for double integrals, International Journal of Modern Mathematical Sciences, 9 (3) 2014, 221–229. ¨ [8] L. Fej´er, Uber die Fourierreihen, II. Math. Naturwiss Anz. Ungar. Akad. Wiss., 24 (1906), 369–390. (Hungarian). [9] U. S. Kırmacı, Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comp., 147 (2004), 137–146. ¨ [10] A. M. Ostrowski, Uber die absolutabweichung einer differentiebaren funktion von ihrem integralmitelwert, Comment. Math. Helv., 10 (1938), 226– 227. [11] J. Peˇcari´c, F. Proschan, Y. L. Tong, Convex functions, partial ordering and statistical applications, Academic Press, New York, 1991.


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[12] A. Qayyum, A weighted Ostrowski-Gr¨ uss type inequality and applications, Proceeding of the World Cong. on Engineering, 2 (2009), 1–9. [13] A. Rafiq, F. Ahmad, Another weighted Ostrowski-Gr¨ uss type inequality for twice differentiable mappings, Kragujevac Journal of Mathematics, 31 (2008), 43–51. [14] M. Z. Sarikaya, On the Ostrowski type integral inequality, Acta Math. Univ. Comenianae, Vol. LXXIX, 1 (2010), 129–134. [15] M. Z. Sarikaya, On the Ostrowski type integral inequality for double integrals, Demonstratio Mathematica, Vol. XLV, 3 (2012), 533–540. [16] M. Z. Sarikaya, H. Ogunmez, On the weighted Ostrowski type integral inequality for double integrals, The Arabian Journal for Science and Engineering (AJSE)-Mathematics, 36 (2011), 1153–1160. [17] M. Z. Sarikaya, On the generalized weighted integral inequality for double integrals, Annals of the Alexandru Ioan Cuza University-Mathematics, accepted. [18] M. Z. Sarikaya, On new Hermite Hadamard Fejer Type integral inequalities, Studia Universitatis Babe¸s-Bolyai Mathematica, 57 (3) (2012), 377– 386. [19] M. Z. Sarikaya, H. Yildirim, On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals, Submitted. [20] K-L. Tseng, G-S. Yang, K-C. Hsu, Some inequalities for differentiable mappings and applications to Fejer inequality and weighted trapozidal formula, Taiwanese J. Math, 15 (4) (2011), 1737–1747, [21] C.-L. Wang, X.-H. Wang, On an extension of Hadamard inequality for convex functions, Chin. Ann. Math., 3 (1982), 567–570. [22] S.-H. Wu, On the weighted generalization of the Hermite-Hadamard inequality and its applications, The Rocky Mountain J. of Math., 39 (5) (2009), 1741–1749.

Received: 16 May 2014


Acta Univ. Sapientiae, Mathematica, 6, 2 (2014) 209–216 DOI: 10.1515/ausm-2015-0007

Evolution of =-functional and ω-entropy functional for the conformal Ricci flow Nirabhra Basu

Arindam Bhattacharyya

Department of Mathematics, The Bhawanipur Education Society College, Kolkata-700020, West Bengal, India email: nirabhra.basu@hotmail.com

Department of Mathematics Jadavpur University, Kolkata-700032, India email: bhattachar1968@yahoo.co.in

Abstract. In this paper we define the =-functional and the ω-entropy functional for the conformal Ricci flow and see how they evolve according to time.

1

Introduction

In 1982 R. Hamilton introduced Ricci flow as a deformation of Riemannian metric [3], [4]. After him many scientists gave attention on it and in 2003–2004 G. Perelman [1], [2] used it to prove Poincar´e conjecture. Meanwhile in 2004 A. E. Fischer introduced the concept of conformal Ricci flow equation which is given by g ∂g +2 S+ = −pg ∂t n (1) R(g) = −1. Here p is a scalar non dynamical field. As conformal Ricci flow equation is analogous to the Navier-Stokes equation of fluid mechanics, the scalar field p is also called conformal pressure field.

2010 Mathematics Subject Classification: 53C44, 35K65, 58D17 Key words and phrases: Ricci flow, conformal Ricci flow, entropy functional

209


210

N. Basu, A. Bhattacharyya

The name conformal Ricci flow was introduced because of the role that conformal geometry plays in constraining the scalar curvature and because these equations are the vector field sum of a conformal flow equation and a Ricci flow equation. For the classical Ricci flow equation and the conformal Ricci flow equation, the volume and scalar curvature behave somewhat oppositely. In classical Ricci flow equation, the volume is preserved, that is vol(M, g) = 1, but for non-static flows the scalar is not preserved, whereas for conformal Ricci flow equation the scalar curvature R(g) is kept constant to −1 and for non-static flows the volume varies. Comparing the classical and conformal Ricci flow equations, we observe that the constraint equation changes from vol(M, g) = 1 for the classical Ricci flow to R(g) = −1 for the conformal Ricci flow with the concomitant change of the configuration space from M1 to M−1 . Since M1 is a codimension-1 submanifold of M whereas M−1 is a codimension C∞ (M, <) submanifold of M, M−1 is a much smaller configuration space than M1 . In the view point of geometry having a smaller configuration space is potentially better. From the lecture note of P. Topping [5], we have been introduced the concept of the =-functional and Perelman’s ω entropy functional for Ricci flow. In our paper we have defined the =-functional and ω-entropy functional regarding conformal Ricci flow and have shown how they evolve with respect to time t.

2

The =-functional for the conformal Ricci flow

Let M be a fixed closed manifold, g is a Riemannian metric and f is a function defined on M to the set of real numbers <. Then the =-functional on pair (g, f) is defined as Z =(g, f) = −1 + |∇f|2 e−f dV. (2) Now we establish how the =-functional changes according to time under conformal Ricci flow. Theorem 1 In conformal Ricci flow, the rate of change of =-functional with respect of time is given by Z d 2 ∂f =(g, f) = −2Ric(∇f, ∇f) − + p g(∇f, ∇f) − 2 (∆f − |∇f|2 ) dt n ∂t ∂f 1 ∂g + (−1 + |∇f|2 ) − + tr e−f dV, ∂t 2 ∂t


Evolution of = and ω-entropy functional for the conformal Ricci flow

211

R where =(g, f) = (−1 + |∇f|2 )e−f dV. Proof. ∂ ∂ ∂g ∂f 2 |∇f| = g(∇f, ∇f) = (∇f, ∇f) + 2g ∇ , ∇f . ∂t ∂t ∂t ∂t So using proposition 2.3.12 of [5] we can write Z d ∂g ∂f =(g, f) = (∇f, ∇f) + 2g ∇ , ∇f e−f dV dt ∂t ∂t Z ∂f 1 ∂g −f 2 e dV. + (−1 + |∇f| ) − + tr ∂t 2 ∂t Using integration by parts of equation (3), we get Z Z ∂f ∂f 2g ∇ , ∇f e−f dV = −2 (∆f − |∇f|2 )e−f dV. ∂t ∂t Now putting (5) in (4), we get Z ∂g ∂f d =(g, f) = (∇f, ∇f) − (∆f − |∇f|2 ) dt ∂t ∂t ∂f 1 ∂g 2 +(−1 + |∇f| ) − + tr e−f dV. ∂t 2 ∂t

(3)

(4)

(5)

(6)

Using (1) in (6), we get the following result for conformal Ricci flow, as Z d 2 =(g, f) = − 2Ric(∇f, ∇f) − + p g(∇f, ∇f) dt n (7) ∂f 1 ∂g ∂f 2 2 −f e dV. − 2 (∆f − |∇f| ) + (−1 + |∇f| ) − + tr ∂t ∂t 2 ∂t Hence the proof.

3

ω-entropy functional for the conformal Ricci flow

Let M be a closed manifold, g is a Riemannian metric on M and f is a smooth function defined from M to the set of real numbers <. We define ω-entropy functional as Zh i ω(g, f, τ) = τ −1 + |∇f|2 + f − n udV, (8)


212

N. Basu, A. Bhattacharyya

R where τ > 0 is a scale parameter and u is defined as u(t) = e−f(t) ; M udV = 1. We would also like to define heat operator acting on the function f : M × ∂ ∂ − ∆ and also, ♦∗ := − ∂t − ∆ − 1, conjugate to ♦. [0, τ] −→ < by ♦ := ∂t ∗ We choose u, such that ♦ u = 0. Now we prove the following theorem. Theorem 2 If g, f, τ evolve according to ∂g 2 = −2Ric − +p g ∂t n ∂τ = −1 ∂t ∂f n = −∆f + |∇f|2 + 1 + ∂t 2τ

(9) (10) (11)

and the function v defined as v = [τ(2∆f − |∇f|2 − 1) + f − n]u, then theR rate of ∗ change of ω-entropy functional for conformal Ricci flow is dω dt = − M ♦ v, where un ♦∗ v = 2u(∆f − |∇f|2 − 1) − − v − uτ[4 < Ric, Hessf > 2τ 2 + p g(∇f, ∇f) − 2g(∇|∇f|2 , ∇f) + 4g(∇(∆f), ∇f) + 2|Hessf|2 ]. + n Proof.

v v v u = ♦∗ u + u♦∗ . u u u We have defined previously that ♦∗ u = 0, so v ♦∗ v = u♦∗ u ∗ ∗ ♦ v = u♦ [τ(2∇f − |∇f|2 − 1) + f − n]. ♦∗ v = ♦∗

We shall use the conjugate of heat operator, as defined earlier as ♦∗ = ∂ ∂ + ∆ + 1 . Therefore ♦∗ v = −u ∂t + ∆ + 1 [τ(2∆f−|∇f|2 −1)+f−n] ⇒ − ∂t ∂ ∂ u−1 ♦∗ v = − ∂t + ∆ [τ(2∆f−|∇f|2 −1)]− ∂t + ∆ f−[τ(2∆f−|∇f|2 −1)+f−n] using equation (10), we have ∂ −1 ∗ 2 u ♦ v = (2∆f − |∇f| − 1) − τ + ∆ (2∆f − |∇f|2 − 1) ∂t (12) ∂f v − − ∆f − . ∂t u


Evolution of = and ω-entropy functional for the conformal Ricci flow

213

∂ ∂ ∂ Now using the equality ∂t (2∆f − |∇f|2 − 1) = 2 ∂t (∆f) − ∂t |∇f|2 and the proposition 2.5.6 of [5], we have

∂ ∂f ∂g (2∆f−|∇f|2 −1) = 2∆ +4 < Ric, Hessf > − (∇f, ∇f)−2g ∂t ∂t ∂t

∂ ∇f, ∇f . ∂t

Now using the conformal Ricci flow equation (1), we have ∂f ∂ (2∆f − |∇f|2 − 1) = 2∆ + 4 < Ric, Hessf > +2Ric(∇f, ∇f) ∂t ∂t ∂ 2 + p g(∇f, ∇f) − 2g ∇f, ∇f . + n ∂t

(13)

Using (11) in (13), we get ∂ n (2∆f − |∇f|2 − 1) = 2∆ −∆f + |∇f|2 + 1 + ∂t 2τ + 4 < Ric, Hessf > +2Ric(∇f, ∇f) 2 ∂ + + p g(∇f, ∇f) − 2g ∇f, ∇f . n ∂t

(14)

Now let us compute ∆(2∆f − |∇f|2 − 1) = 2∆2 f − ∆|∇f|2 .

(15)

Using (14) and (15) in (12) we obtain after a brief calculation u−1 ♦∗ v = (2∆f − |∇f|2 − 1) − τ − 2∆2 f + 2∆|∇f|2 + 4 < Ric, Hessf > +2Ric(∇f, ∇f) 2 ∂ + + p g(∇f, ∇f) − 2g ∇f, ∇f n ∂t ∂f v + 2∆2 f − ∆|∇f|2 ] − − ∆f − ∂t u = ∆f − |∇f|2 − 1 − τ[∆|∇f|2 + 4 < Ric, Hessf > +2Ric(∇f, ∇f) 2 ∂ ∂f v + + p g(∇f, ∇f) − 2g ∇f, ∇f − − n ∂t ∂t u


214

N. Basu, A. Bhattacharyya

= ∆f − |∇f| − 1 − τ ∆|∇f|2 + 4 < Ric, Hessf > +2Ric(∇f, ∇f) 2 ∂ n v + + p g(∇f, ∇f) − 2g ∇f, ∇f + ∆f − |∇f|2 − 1 − − n ∂t 2τ u v n 2 − − τ ∆|∇f|2 + 4 < Ric, Hessf > +2Ric(∇f, ∇f) = 2(∆f − |∇f| − 1) − 2τ u ∂ 2 + p g(∇f, ∇f) − 2g ∇f, ∇f + n ∂t 2

n u ♦ v = 2(∆f − |∇f| − 1) − − τ(2∆f − |∇f|2 − 1) + f − n] − τ[∆|∇f|2 2τ + 4 < Ric, Hessf > +2Ric(∇f, ∇f) ∂ 2 + p g(∇f, ∇f) − 2g ∇f, ∇f + n ∂t −1

2

n u ♦ v = 2(∆f − |∇f| − 1) − − f + n − τ 2∆f − |∇f|2 − 1 + ∆|∇f|2 2τ + 4 < Ric, Hessf > +2Ric(∇f, ∇f) (16) 2 ∂f + + p g(∇f, ∇f) − 2g ∇ , ∇f n ∂t −1

2

using (11), we get n −1 ∗ 2 − f + n − τ 2∆f − |∇f|2 − 1 + ∆|∇f|2 u ♦ v = 2 ∆f − |∇f| − 1 − 2τ 2 + p g(∇f, ∇f) + 4 < Ric, Hessf > +2Ric(∇f, ∇f) + (17) n n 2 − 2g ∇ −∆f + |∇f| + + 1 , ∇f . 2τ We can rewrite (17) in the following way n − f + n − τ[2∆f − |∇f|2 − 1 u−1 ♦∗ v = 2(∆f − |∇f|2 − 1) − 2τ 2 + 4 < Ric, Hessf > + + p g(∇f, ∇f) n 2

− 2g(∇|∇f| , ∇f) + 4g(∇(∆f), ∇f)] + τ[−∆|∇f|2 − 2Ric(∇f, ∇f) + 2g(∇(∆f), ∇f)]

(18)


Evolution of = and ω-entropy functional for the conformal Ricci flow

215

and using Bochner formula in (18) and simplifying, we get n − f + n − τ[2∆f − |∇f|2 − 1 2τ 2 + 4 < Ric, Hessf > +( + p)g(∇f, ∇f) − 2g(∇|∇f|2 , ∇f) n + 4g(∇(∆f), ∇f)] − 2τ|Hessf|2 n u−1 ♦∗ v = 2(∆f − |∇f|2 − 1) − − [τ(2∆f − |∇f|2 − 1) + f − n] 2τ 2 + p g(∇f, ∇f) − 2g(∇|∇f|2 , ∇f) − τ[4 < Ric, Hessf > + n u−1 ♦∗ v = 2(∆f − |∇f|2 − 1) −

+ 4g(∇(∆f), ∇f)] − 2τ|Hessf|2 i.e. v n − − τ[4 < Ric, Hessf > u−1 ♦∗ v = 2(∆f − |∇f|2 − 1) − 2τ u 2 + + g(∇f, ∇f) − 2g(∇|∇f|2 , ∇f) n

(19)

+ 4g(∇(∆f), ∇f)] − 2τ|Hessf|2 . So finally we have un − v − uτ[4 < Ric, Hessf > ♦∗ v = 2u(∆f − |∇f|2 − 1) − 2τ 2 + + p g(∇f, ∇f) − 2g(∇|∇f|2 , ∇f) n

(20)

+ 4g(∇(∆f), ∇f) + 2|Hessf|2 ]. Now using remark 8.2.7 of [5], we get dω =− dt

Z ♦∗ v. M

So the evolution of ω with respect to time can be found by this integration.

Acknowledgements We would like to thank honorable referee for valuable suggestions to improve the paper.


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References [1] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv.org/abs/math/0211159, (2002) 1–39. [2] G. Perelman, Ricci flow with surgery on three manifolds, arXiv.org/ abs/math/0303109, (2002), 1–22. [3] R. S. Hamilton, Three Manifold with positive Ricci curvature, J. Differential Geom., 17 (2) (1982), 255–306. [4] B. Chow, P. Lu, L. Ni, Hamilton’s Ricci Flow, American Mathematical Society Science Press, 2006. [5] P. Topping, Lecture on The Ricci Flow, Cambridge University Press, 2006. [6] A. E. Fischer, An introduction to conformal Ricci flow, Class. Quantum Grav., 21 (2004), S171–S218

Received: 13 May, 2013


Contents Volume 6, 2014

S. D. Purohit, R. K. Raina Some classes of analytic and multivalent functions associated with q-derivative operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5 G. C. Rana Hydromagnetic thermoslutal instability of Rivlin-Ericksen rotating fluid permeated with suspended particles and variable gravity field in porous medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 T. M. Seoudy, M. K. Aouf Some applications of differential subordination to certain subclass of p-valent meromorphic functions involving convolution . . . . . . . 46 Cs. Sz´ ant´ o On some Ringel-Hall numbers in tame cases . . . . . . . . . . . . . . . . . . . . 61 ´ Sz´ A. az A particular Galois connection between relations and set functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 I. Sz¨ oll˝ osi On the combinatorics of extensions of preinjective Kronecker modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 E. Wolf Composition followed by differentiation between weighted Bergman spaces and weighted Banach spaces of holomorphic functions . . 107

217


R. S. Batahan, A. A. Bathanya On generalized Laguerre matrix polynomials . . . . . . . . . . . . . . . . . . 121 B. A. Bhayo, L. Yin Logarithmic mean inequality for generalized trigonometric and hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 A. A. Bouchentouf, H. Sakhi Stabilizing priority fluid queueing network model . . . . . . . . . . . . . 146 S. S. Dragomir Some inequalities of Furuta’s type for functions of operators defined by power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 S. Mustonen, P. Haukkanen, J. Merikoski Some polynomials associated with regular polygons . . . . . . . . . . . . 178 M. Z. Sarikaya, S. Erden On the weighted integral inequalities for convex function . . . . . . 194 N. Basu, A. Bhattacharyya Evolution of =-functional and ω-entropy functional for the conformal Ricci flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Contents of volume 6, 2014 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

218




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