DOI: 10.15415/mjis.2012.11003
On Inequalities Involving Moments of Discrete Uniform Distributions S. R. Sharma Department of Mathematics, Chitkara University, Solan-174 103, Himachal Pradesh R . Sharma Department of Mathematics, H.P. University, Summer Hill, Shimla-171 005 Abstract Some inequalities for the moments of discrete uniform distributions are obtained. The inequalities for the ratio and difference of moments are given. The special cases give the inequalities for the standard power means. Keywords and Phrases: Moments, Power means, Discrete distribution, Geometric mean. 2000 Mathematics Subject Classification. 60 E 15, 26 D 15.
1. INTRODUCTION
L
et x1, x2, …, xn denote n real numbers such that a ≤ xi ≤ b, i = 1,2,…,n. The rthorder moment µr/ of these numbers is defined as
µr/ =
1 n r ∑ xi . n i =1
(1.1)
The power mean of order r namely Mr is defined as
1
Mr = (µr/ ) r for r ≠ 0
(1.2)
and
1
Mr = lim (µr/ ) r for r = 0. r →0
(1.3)
It may be noted that M -1, M0 and M1 respectively define harmonic mean, geometric mean and Arithmetic mean. It is well known that the power mean Mr is an increasing function of r. For 0 < a ≤ xi ≤ b, i = 1, 2,..., n, we have, [1],
r
µr/ ≥ (µs/ ) s ,
Mathematical Journal of Interdisciplinary Sciences Vol. 1, No. 1, July 2012 pp. 35–46
(1.4)
where r is a positive real number and s is any real number such that r > s. If r is negative real number with r > s, the reverse inequality holds. For s = 0, the inequality (1.4) gives
©2012 by Chitkara University. All Rights Reserved.
Sharma, S. R. Sharma, R.
36
r
µr/ ≥( M o )
(1.5)
See also [2]. Some bounds for the difference and ratio of moments have also been investigated in literature; see [3-6]. Our main results give the refinements of the inequalities (1.4) and (1.5) when the minimum and maximum values of x1 namely, a and b, and the value of n is prescribed, (Theorem 2.1 and 2.2, below). The bounds for the difference and ratio of moments are obtained (Theorem 2.3-2.6, below). We also discuss the cases when the inequalities reduce to equalities. As the special cases, we get various bounds connecting lower order moments and the standard means of the n real numbers, (Inequalities 3.1 - 3.33, below), also see [7-9]. 2. MAIN RESULTS Theorem 2.1. For 0 < a ≤ xi ≤ b, i = 1, 2,...,n, we have
a r + b r n µr/ ≥ + n - 2 n
r -s s
r
/ a s + bs s µs , n
(2.1)
where n ≥ 3, r is a positive real number and is any non-zero real number such that r > s For s < r < 0 the reverse inequality holds. The inequality (2.1) becomes equality when 1
nµ ′ - a s - b s s and xn = b. x1 = a, x2 = x3 = = xn-1 = s n-2
The inequality (2.1) provides a refinement of the inequality (1.4). Proof. The rth order moment of n real numbers xi, with x1 = a and xn = b can be written as
µr/ =
a r + b r n - 2 x2 r + ... + x r n-1 + . n n n-2
(2.2)
Apply (1.4) to n - 2 real numbers x2, x3,..., xn-1, we find that r
x2 r + ... + x r n-1 x2 s + ... + x s n-1 s , ≥ n-2 n-2
(2.3)
where r is a positive real number and s is any non zero real number such that r > s. Combine (2.2) and (2.3), we get r
Also,
a r + b r n - 2 x2 s + ... + x s n-1 s µr/ ≥ + . n n n-2
Mathematical Journal of Interdisciplinary Sciences, Volume 1, Number 1, July 2012
(2.4)
µs′ =
Therefore
1 s a + x2s + + b s , n
(2.5)
x2s + x3s + + xns-1 = nµs′ - a s - b s
(2.6)
On Inequalities Involving Moments of Discrete Uniform Distributions
Substituting the value from (2.6) in (2.4), we immediately get (2.1). For s < r < 0, inequality (2.3) reverses its order [2]. It follows therefore that inequality (2.1) will also reverse its order for s < r < 0. Theorem 2.2. For 0 < a ≤ xi ≤ b, i = 1, 2, , n, we have µr/ ≥
a r + b r n - 2 M on + n n ab
where n ≥ 3 and r is a non-zero real number. The inequality (2.7) becomes equality when M n x1 = a, x2 = x2 = = xn-1 = o ab
1
r n- 2
37
,
(2.7)
n-2
and xn = b.
The inequality (2.1) provides a refinement of the inequality (1.5). Proof. Apply (1.5) to n - 2 real numbers x2 , x3 , , xn-1, , we find that r
x2r + x3r + ... + xnr-1 n-1 2 n-1 2 n-1 2 ≥ x2 .x3 ...xn-1 . n-2
(2.8)
Combine (2.2) and (2.8), we get r
1 a r + b r n - 2 n-1 2 n-1 2 µ ≥ + x2 .x3 ....xnn--12 . n n / r
(2.9)
Also, 1
M0 = (a ⋅ x2 ⋅ ...⋅ b)n ,
therefore
x2 x3 xn-1 =
M 0n . ab
(2.10) (2.11)
Substituting the value from (2.11) in (2.9) we immediately get (2.7). Theorem 2.3. For 0 < a ≤ xi ≤ b, i = 1,2,…,n, we have
.
µr/ - (µ
r / s s
)
r 2 a r + b r a s + b s s . ≥ - 2 n 2
(2.12)
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Sharma, S. R. Sharma, R.
where r is a positive real number and s is a nonzero real number such that r > s. For s < r < 0, the reverse inequality holds. For n = 2 inequality (2.13) becomes equality. For n ≥ 3, inequality (2.13) is sharp; equality holds when 1 a s + bs s and xn = b. x1 = a, x2 = x3 = …= xn-1 =
2
Proof. It follows from the inequality (2.1) that 38
r / s s
µ - (µ / r
)
a r + b r n ≥ + n - 2 n
r -s s
r
r
s / a s + bs s µs - (µs/ ) . n
Consider a function a r + b r n f (µs′ ) = + n - 2 n
r -s s
r
r
s / a s + bs s µs - (µs/ ) . n
(2.13)
The function f(µs′ ) is continuous in the interval [as, bs] and its derivative df r n = / d µs s n - 2
r -s s
/ a s + bs µs n
r -s s
-
r / r -s s (µs ) . s
(2.14)
vanishes at
µs/ =
a s + bs 2
(2.15)
The value of the second order derivative
at µs′ =
r -2 s r -s r -2 s d2f r r - s n s / a s + b s s / s - (µs ) , = µs n dµs/ 2 s s n - 2
a s + bs is 2 d2 f r (r - s ) 2 a s + b s = d µs/ 2 s 2 n - 2 2
It follows from (2.17) that
(2.16)
r -2 s s
d2f > 0 for r > 0 dµs/ 2
Mathematical Journal of Interdisciplinary Sciences, Volume 1, Number 1, July 2012
(2.17)
and
d2f < 0 for r < 0. dµs/ 2
Therefore, for r > 0, the function f (µs′ ) achieves its minimum when the value of µs′ is given by (2.15). Hence
On Inequalities Involving Moments of Discrete Uniform Distributions
r
2 a r + br a s + b s s . f (µs′ ) ≥ - n 2 2
.
This proves (2.12). Likewise, for r < 0, the function f (µs′ ) achieves its maximum when the value of µs′ is given by (2.15) and we conclude that
r / s s
µ - (µ / r
)
r 2 a r + br a s + b s s . ≤ - n 2 2
Theorem 2.4. For 0 < a ≤ xi ≤ b, i = 1, 2,…, n, we have 2 a r + br µr/ - M 0r ≥ n 2
(
r ab
)
(2.18)
where r is a non zero real number. For n = 2, the inequality (2.19) becomes equality. For n ≥ 3, inequality (2.19) is sharp, equality holds when x1 = a, x2 = x3 = … = xn-1 = ab and xn = b . Proof. It follows from the inequality (2.7) that r
a r + br n - 2 M 0n n-2 - M 0r . µ -M ≥ + n n ab / r
r 0
Consider a function r
a r + br n - 2 M 0n n-2 g(M0) = - M 0r . + n n ab
(2.19)
The derivative
2r n-2 M dg r -1 0 = rM 0 - 1 ab dM 0
(2.20)
Mathematical Journal of Interdisciplinary Sciences, Volume 1, Number 1, July 2012
39
Sharma, S. R. Sharma, R.
vanishes at M 0 = ab
(2.21)
The value of the second derivative 2r n-2 M d2g n ( r 1 + 2 ) r -2 0 = rM 0 - (r - 1) . n - 2 ab dM 0
40
(2.22)
at M 0 = ab is r -2 d2g 2r 2 2 ( ) . = ab dM 0 2 n - 2
Clearly,
d 2 g > 0 for M = ab . 0 dM 0 2
Therefore, for r ≠ 0, the function g(M0) attains its minimum at M 0 = ab and we have 2 a r + br g (µs′ ) ≥ n 2
(
r ab .
)
This proves (2.18). Theorem 2.5. For 0 < a ≤ xi ≤ b for i = 1,2,…,n, we have
µr/
(µ ) / s
r
s
s r r r -s a r + b r 1 s a b + s a + b + (n - 2) s ≥ s s s a + b n a + b
s -r s
. (2.23)
where r is a positive real number and s is any non-zero real number such that r > s. For s < r < 0, the reverse inequality holds. For n = 2, the inequality (2.23) becomes equality. For n ≥ 3, this inequality is sharp; equality 1
a r + b r r-s and xn = b. holds when x1 = a, x2 = x3 = …= xn-1= s a + bs
Mathematical Journal of Interdisciplinary Sciences, Volume 1, Number 1, July 2012
Proof. It follows from inequality (2.1) that µr/
(µ ) / s
r
s
r -s r r a + br n s / a s + b s s 1 µ ≥ + , n - 2 s n n µ / r s ( s )
(2.24)
where r is a positive real number and s is a non zero real number such that r > s. We now find the minimum value of the right side expression in (2.24) as µs′ varies over the interval [as,bs]. Consider a function r r -s s s s a r + br n s a b + 1 . (2.25) h (µs′ ) = + µ′ n - 2 s n n ′ rs (µs ) The function h( µs′ ) is continuous in the interval [as, bs] and its derivative dh r (a s + b s ) 1 = / µs sn d µs/
r +s s
r -s nµs/ - a s - b s s a r + br - s s n 2 a b +
(2.26)
vanishes at s
a s + b s n - 2 a r + b r r -s . µs′ = + n n a s + b s
(2.27)
From (2.26), if r is a positive real number and s is any non zero real numbersuch that r > s then the sign of
dh changes from negative to positive d µs′
while µs′ passes through the value given by (2.27). The function h( µs′ ) achieves its minimum at the value of µs′ given by (2.27). We therefore have,
s r r r -s + a r + b r 1 s a b s ′ h (µs ) ≥ s a + b + (n - 2) s s a + b s n a + b
s -r s
.
(2.28)
Combining (2.24), (2.25) and (2.28), the inequality (2.24) follows immediately. For s < r < 0, it follows from (2.26) that the sign of
dh changes from d µs′
positive to negative while µs′ passes through the value given by (2.27). In this case function f( µs′ ) attains its maximum at the value of µs′ given by (2.27). We therefore have,
Mathematical Journal of Interdisciplinary Sciences, Volume 1, Number 1, July 2012
On Inequalities Involving Moments of Discrete Uniform Distributions
41
Sharma, S. R. Sharma, R.
s a r + b r r -s a + b 1 s s h (µs′ ) ≤ s a + b + n 2 ( ) a s + b s a + b s n r
r
s -r s
.
(2.29)
On the other hand we conclude from theorem 2.1 that 42
µr/
(µ ) / s
r
s
r -s r r a + b r n s / a s + b s s 1 ≤ + . µ n - 2 s n n µ / r s ( s )
(2.30)
Combining (2.25), (2.26) and (2.27), we find that for s < r < 0, µr′
r
(µs′ )s
r r -s r s s s 1 a + b r n s + a b µ ′ ≤ + . s n - 2 n n ′ rs (µs )
Theorem 2.6 For 0 < a ≤ xi ≤ b for i = 1,2,…,n, we have 2
r n a + br µ ≥ M 0r 2 ab r / r
(
)
(2.31)
where r is a non zero real number. For n = 2, the inequality (2.31) becomes equality. For n ≥ 3, the inequality (2.31) is sharp; equality holds when 1
a r + b r r and xn = b. 2
x1 = a, x2 = x3= …= xn-1=
Proof. It follows from inequality (2.7) that
r r a + b r n - 2 M 0n n-2 1 F (M 0 ) = + . 2 n ab M 0r
(2.32)
where r is a non zero real number. We now find the minimum value of the right hand side expression in (2.33). Consider a function
r r a + b r n - 2 M on n-2 1 µr/ ≥ + n ab M or M 0r n
The derivative
dF 2r = dM 0 nM or +1
n r M o n-2 a r + b r . ab 2
Mathematical Journal of Interdisciplinary Sciences, Volume 1, Number 1, July 2012
(2.33)
(2.34)
vanishes at 1 r a + b r M 0 = (ab) n 2
n- 2 nr
.
(2.35)
From (2.34), if r is a non zero real number then the sign of
dF changes dM 0
fromnegative to positive while Mo passes through the value given by (2.35). The function f(Mo) attains its minimum at the value of Mo given by (2.35). We therefore have 2 r n a + br . F (M 0 ) ≥ r 2 ab
(
(2.36)
)
Combining (2.32), (2.33) and (2.36), inequality (2.31) follows immediately. 3. SOME SPECIAL CASES From the application point of view, it is of interest to know the bounds for the first four moments and the bounds for standard power means (namely, Arithmetic mean, ( µ1/ = x ), Geometric mean (G) and Harmonic mean (H)). These bounds are also of fundamental interest in the theory of inequalities. By assigning particular values to r and s, in the generalized inequalities obtained in this paper, we can find inequalities connecting various power means and moments. The following inequalities can be deduced easily from the generalized inequalities given in Theorems 2.1 and 2.2: 2
µ2/ ≥
a 2 + b2 n / a +b µ1 , + n n - 2 n 3
µ2/ ≥
a 2 + b 2 n - 2 + n n
(3.1) 2
abnH , abn - (a + b) H
(3.2)
2
a 2 + b 2 n - 2 G n n-2 µ ≥ + , n n ab / 2
2
a + b n - 2 µ ≥ + n n / 1
abnH , abn - (a + b) H
(3.3)
(3.4)
1
a + b n - 2 G n n-2 µ ≥ + , n n ab / 1
(3.5)
Mathematical Journal of Interdisciplinary Sciences, Volume 1, Number 1, July 2012
On Inequalities Involving Moments of Discrete Uniform Distributions
43
Sharma, S. R. Sharma, R.
1 n-2 n G ≥ (ab) n - a+b H ab
n-2 n
,
2
44
(3.6)
3
a 3 + b3 n / a + b µ , + n - 2 1 n n
µ3/ ≥
a 3 + b3 n / a 2 + b 2 2 , µ ≥ + µ2 n n - 2 n
µ4/ ≥
a 4 + b4 n / a 2 + b 2 µ ≥ + µ2 n n - 2 n
(3.7)
3
/ 3
3
a 4 + b 4 n + n - 2 n
(3.8)
4
/ a +b µ1 , n
(3.9)
2
and
/ 4
(3.10)
4
a 4 + b 4 3 n / a 3 + b3 3 µ ≥ + µ3 . n n - 2 n / 4
(3.11)
The following lower bounds for the difference of moments and means are deduced from the generalized inequalities given in Theorems 2.3 and 2.4:
1 2 (b - a ) , 2n (b - a ) 2 2ab µ2/ - H 2 ≥ 1+ , 2 n ( a + b) µ2/ - µ1/ 2 ≥
µ2/ - G 2 ≥
µ1/ - H ≥
(b - a) 2 , n
(3.12) (3.13) (3.14)
2
µ1/ - G ≥
µ3/ - µ1/ 3 ≥
n( a + b)
,
( b - a )2 , n
n-2 n
2 ab G - H ≥ a + b
(b - a )
(
ab
2
)
n
2 2 ab n - a + b ,
3 a +b 2 (b - a ) , 4 n
Mathematical Journal of Interdisciplinary Sciences, Volume 1, Number 1, July 2012
(3.15) (3.16)
(3.17) (3.18)
3 / 2 2
/ 3
µ -µ
3 2 a 3 + b3 a 2 + b 2 2 , ≥ - n 2 2
2 a 4 + b 4 a + b - 2 n 2
4
µ4/ - µ1/ 4 ≥
,
2 a 4 + b 4 a 2 + b 2 µ - µ ≥ - 2 n 2
2
/ 4
/2 2
and
4 / 3 3
/ 4
µ -µ
(3.19)
(3.20)
(3.21)
4 2 a 4 + b 4 a 3 + b3 3 . ≥ - n 2 2
(3.22)
The following lower bounds for the ratio of moments and means are deduced from the generalized inequalities given in Theorems 2.5 and 2.6:
µ2/ (a 2 + b 2 )n , ≥ /2 2 (a + b) + (n - 2)(a 2 + b 2 ) µ1
(3.23)
µ3/ (a 2 - ab + b 2 )n 2 , ≥ /3 2 µ1 a + b + (n - 2) a 2 - ab + b 2
(3.24)
µ3/
(3.25)
µ2/
3
2
≥
n (a 3 + b3 ) (a 2 + b 2 )3 + (n - 2)(a 3 + b3 ) 2
1
, 2
µ4/ n3 (a 4 + b 4 ) , ≥ /4 4 1 3 µ1 4 4 3 3 (a + b) + (n - 2)(a + b )
(3.26)
µ4/ n( a 4 + b 4 ) , ≥ µ2/ 2 (a 2 + b 2 ) 2 + (n - 2)(a 4 + b 4 )
(3.27)
µ4/ µ3/
4
3
≥
3
n (a 4 + b 4 )
(a 3 + b3 ) 4 + (n - 2)(a 4 + b 4 )3 2
µ1/ 1 a + b , ≥ 2 n - 2 + H n ab
1
,
(3.28)
3
(3.29)
Mathematical Journal of Interdisciplinary Sciences, Volume 1, Number 1, July 2012
On Inequalities Involving Moments of Discrete Uniform Distributions
45
Sharma, S. R. Sharma, R.
2
µ1/ a + b n , ≥ G 2 ab
(3.30)
2
G a + b n , ≥ H 2 ab
a 2 + b2 n µ2/ ≥ G 2 2ab
(3.31)
2
46
and
µ2/ 1 ≥ 3 2 H n
(3.32) 3
1 2 2 2 (a + b )(a + b) 3 . + n 2 2 2 ab
(3.33)
Acknowledgement The second author acknowledges the support of the UGC-SAP. REFERENCES [1] G.H. Hardy, J.E. Littlewood and G. Polya, Inequalities, New York, Cambridge University Press, Cambridge, U.K. (1934). [2] R. Sharma, R,G, Shandil, S. Devi and M. Dutta, Some Inequalities between Moments of probability distribution, J. Inequal. Pure and Appl. Math., 5 (2): Art. 86, (2004). [3] B.C. Rennie, On a Class of Inequalities, J. Austral. Math. Soc., 3: 442–448, (1963). [4] S. Ram, S. Devi, and R. Sharma, Some inequalities for the ratio and difference of moments, International Journal of Theoretical and Applied Sciences, 1(1): 103–110, (2009). http://dx.doi.org/10.1017/S1446788700039057 [5] R. Sharma, A. Kaura, M. Gupta and S. Ram, Some bounds on sample parameters with refinements of Samuelson and Brunk inequalities. Journal of Mathematical Inequalities, 3:99–106, (2009). http://dx.doi.org/10.7153/jmi-03-09 [6] R. Sharma, Some more inequalities for arithmetic mean, harmonic mean and variance, Journal of Mathematical Inequalities, 2: 109–114, (2008). http://dx.doi.org/10.7153/jmi-02-11 [7] E.N. Laguerre, Surune mathode pour obtenir par approximation les raciness d’ une Equation algebrique qul a tautes ses raciness [in French], Nouvelles Annales de Mathematiques (Paris), 2e Serie 19: 161–171 and 193–202, (1980) [JFM12:71]. [8] A. Lupas, Problem 246, Mathematicki vesnik (Belgrade), 8 (23): 333, (1971). [9] S.T. Jensen and G.P.H. Styan, Some comments and a bibliography on the Laguerre Samuelson inequalities with extensions and applications to statistics and matrix theory, Analytic and Geometric inequalities and their applications (Hari M. Srivastava and Themistocles M. Rassias, eds), Springer-Verlag, (1999).
SitaRam Sharma is Assistant Professor Department of Applied Sciences, Chitkara University Himachal Pradesh. Rajesh Sharma is Associate Professor, Department of Mathematics Himachal Pradesh.
Mathematical Journal of Interdisciplinary Sciences, Volume 1, Number 1, July 2012