42015 mjis gulzar

DOI: 10.15415/mjis.2016.42015

A Ring-shaped Region for the Zeros of a Polynomial M. H. GULZAR1, A. W. MANZOOR2 Department of Mathematics, University of Kashmir, Srinagar Email: gulzarmh@gmail.com Received:July 19, 2015| Revised: December 12, 2015| Accepted: February 11, 2016 Published online: March 30, 2016 The Author(s) 2016. This article is published with open access at www.chitkara.edu.in/publications

Abstract: Let P ( z ) =

n

∑ a j z j be a polynomial of degree n .Then according j =0

to Cauchy’s classical result all the zeros of P(z) lie in z ≤ 1 + A , where

A = max0≤ j≤n−1

aj an

. The purpose of this paper is to present a ring-shaped

region containing all or a specific number of zeros of P(z).

Mathematics Subject Classification: 30C10, 30C15.

Keywords: Polynomial, Region, Zeros.

1. INTRODUCTION The following result known as Cauchy’s Theorem [2] is well known in the theory of distribution of zeros of polynomials: n Theorem A. All the zeros of the polynomial P ( z ) = ∑ a j z j of degree n lie j =0

in the circle z ≤ 1 + M , where M = max0≤ j≤n−1

aj an

.

Several generalizations and improvements of this result are available in the literature. Mohammad [3] used Schwarz Lemma and proved the following result: n

Theorem

B.

All

the

zeros

of

the

polynomial

P( z) = ∑ a j z j

j =0 M z≤ an ≤ M , where of degree n lie in if an M = max z =1 an z n−1 + ...... + a0 = max z =1 a0 z n−1 + ...... + an−1 .

In this paper, we find a ring-shaped region containing all the zeros of P(z). We also obtain a result of similar nature for the number of zeros of the polynomial.

In fact we prove the following results:

Mathematical Journal of Interdisciplinary Sciences Vol. 4, No. 2, March 2016 pp. 177–182

Gulzar, MH Manzoor, AW

Theorem 1.All the zeros of the polynomial P ( z ) = the ring-shaped region r1 ≤ z ≤ r2 , where 4

r1 =

2

2

2

[ R a1 ( M − a0 ) + 4 a1 R 2M

2

1 3 2 M ]

n

∑ ajz j

of degree n lie in

j =0

− a1 R 2 ( M − a0 )

2

,

2 M12

r2 =

2

[ R 4 an−1 ( M1 − an )2 + 4 an

1 R 2 M 3 ]2

,

− an−1 R 2 ( M1 − an )

M = max z = R an z n + ...... + a1z , M1 = max z = R a0 z n + ...... + an−1z ,

R being any positive number.

Theorem 2. The number of zeros of the polynomial P ( z ) = n in the ring-shaped region r1 ≤ z ≤

n

∑ a j z j of degree

j =0 R ,1 < c ≤ R , does not exceed c

1 M log(1 + ), log c a0 where M is as in Theorem 1.

2. LEMMAS For the proofs of the above theorems we make use of the following results: Lemma1. Let f(z) be analytic in z ≤1 ,f(0)=a, where a <1 , f ′(0) = b and f ( z ) ≤1 for z = 1 . Then, for z ≤1 , 2

f (z) ≤

(1 − a ) z + b z + a (1 − a ) 2

a (1 − a ) z + b z + (1 − a )

The example

b z − z2 1+ a f ( z) = b 1− z − az 2 1+ a a+

shows that the estimate is sharp.

Lemma 1 is due to Govil, Rahman and Schmeisser [1].

178

.

Lemma 2. Let f(z) be analytic for z ≤ R ,f(0)=0, f ′(0) = b and f ( z ) ≤ M for z = R .

Then, for z ≤ R ,

M z M z + R2 b . f (z) ≤ 2 .

M+ b z R Lemma 2 is a simple deduction from Lemma 1 .

Lemma 3.If f(z) is analytic for z ≤ R , f (0) ≠ 0

and

f ( z ) ≤ M for

R z = R , then the number of zeros of f(z)in z ≤ ,1 < c < R is less than or c 1 M . equal to log log c f (0 ) Lemma 3 is a simple deduction from Jensen’s Theorem (see [4]).

Proof of Theorem 1. Let Q(z)= an z n + an−1z n−1 + ...... + a1z .

Then Q(z) is analytic for z ≤ R ,Q(0)=0, Q ′(0) = a1 and for z ≤ R

Q( z ) = an z n + an−1z n−1 + ...... + a1z ≤ M Hence, by Lemma 2, for z ≤ R ,

Q( z ) ≤

M z M z + a1 R 2 . .

M + a1 z R2

Therefore, for z ≤ R ,

P( z ) = Q( z ) + a0 ≥ a0 − Q( z ) M z M z + a1 R 2 ≥ a0 − 2 .

M + a1 z R >0 if

a0 R 2 ( M + a1 z ) − M z ( M z + a1 R 2 ) > 0 i.e. if

2

M 2 z + a1 R 2 ( M − a0 ) z − a0 R 2 M < 0

which is true if

179

A Ring-shaped Region for the Zeros of a Polynomial

Gulzar, MH Manzoor, AW

2

4

z<

2

[ R a1 ( M − a0 ) + 4 a0 R

2

1 3 2 M ]

− a1 R 2 ( M − a0 )

2M 2

.

Thus P(z) does not vanish in 4

z<

2

2

[ R a1 ( M − a0 ) + 4 a0 R

2

1 3 2 M ]

2M 2

− a1 R 2 ( M − a0 ) .

On the other hand, let

1 R( z ) = z n P( ) = a0 z n + a1z n−1 + ...... + an−1z + an z = H ( z ) + an ,

where

H ( z ) = a0 z n + a1z n−1 + ...... + an−1z .

Then H(z) is analytic and H ( z ) ≤ M1 for z ≤ R , H(0)=0, H ′(0) = an−1 .

Hence, by Lemma 2, for z ≤ R ,

M1 z M1 z + an−1 R 2 .

H (z) ≤ 2 .

M1 + an−1 z R Therefore, for z ≤ R ,

R ( z ) = an + H ( z ) ≥ an − H ( z ) M z M z + an−1 R 2 ≥ a n − 12 . 1 M1 + an−1 z R >0 if

an R 2 ( M1 + an−1 z ) − M1 z ( M1 z + an−1 R 2 ) > 0 i.e. if

2

M12 z + an−1 R 2 ( M1 − an ) z − an R 2 M1 < 0

which is true if

180

4

z<

2

2

[ R an−1 ( M1 − an ) + 4 an R

2

1 3 2 M1 ]

− an−1 R 2 ( M1 − an )

1 3 2 M1 ]

− an−1 R 2 ( M1 − an ) .

.

2 M12

Thus P(z) does not vanish in 4

z<

2

2

[ R an−1 ( M1 − an ) + 4 an R

2

2 M12

Hence all the zeros of R(z) lie in 1

2

z≥

[ R 4 an−1 ( M1 − an )2 + 4 an R 2 M13 ]2 − an−1 R 2 ( M1 − an )

.

2 M12 n

1 z

Thus all the zeros of z P ( ) lie in 4

z≥

2

2

[ R an−1 ( M1 − an ) + 4 an R

2

1 3 2 M1 ]

− an−1 R 2 ( M1 − an )

.

2 M12

1 z

Hence all the zeros of P ( ) lie in 4

z≥

2

2

[ R an−1 ( M1 − an ) + 4 an R

2

1 3 2 M1 ]

2 M12

− an−1 R 2 ( M1 − an ) .

This implies that all the zeros of P(z) lie in

2 M12

z≤ 2

[ R 4 an−1 ( M1 − an )2 + 4

1 2 3 2 an R M1 ]

.

− an−1 R 2 ( M1 − an )

Consequently, it follows that all the zeros of P(z) lie in the ring-shaped region r1 ≤ z ≤ r2 and the proof of Theorem 1 is complete.

Proof of Theorem 2. To prove Theorem 2, we need to show only that the number

R 1 M ,1 < c < R does not exceed log(1 + ). c log c a0 Since P(z) is analytic for z ≤ R , P(0)= a0 and for z ≤ R , of zeros of P(z) in z ≤

P( z ) = an z n + an−1z n−1 + ....... + a1z + a0

181

A Ring-shaped Region for the Zeros of a Polynomial

Gulzar, MH Manzoor, AW

≤ an z n + an−1z n−1 + ....... + a1z + a0 ≤ M + a0

,

it follows , by Lemma 3, that the number of zeros of P(z) in z ≤

1

does not exceed log log c follows.

M + a0 a0

=

R ,1 < c ≤ R c

1 M log(1 + ) and Theorem 2 log c a0

ACKNOWLEDGEMENT The authors are thankful to the referee for his value-able suggestions.

REFERENCES [1] Govil N. K., Rahman Q. I and Schmeisser G.,On the derivative of a polynomial, Illinois Math. Jour. 23, 319-329(1979).

[2] Marden M,, Geometry of Polynomials, Mathematical Surveys Number 3, Amer.

Math.. Soc. Providence, RI, (1966).

[3] Mohammad Q. G., On the Zeros of a Polynomial, Amer. Math. Monthly, 72, 631-633 (1966). http://dx.doi.org/10.2307/2313853 [4] Titchmarsh E.C., Theory of Functions, Oxford University Press, London, (1949).

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