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Volume 2008, Article ID 325845, 6 pagesdoi:10.1155/2008/325845 Research Article Recurring Mean Inequality of Random Variables Mingjin Wang Department of Applied Mathematics, Jiangsu Poly

Volume 2008, Article ID 325845, 6 pages

doi:10.1155/2008/325845

Research Article

Recurring Mean Inequality of Random Variables

Mingjin Wang

Department of Applied Mathematics, Jiangsu Polytechnic University, Changzhou, Jiangsu 213164, China

Correspondence should be addressed to Mingjin Wang, wang197913@126.com

Received 16 August 2007; Revised 25 February 2008; Accepted 9 May 2008

Recommended by Jewgeni Dshalalow

A multidimensional recurring mean inequality is shown Furthermore, we prove some new inequalities, which can be considered to be the extensions of those established inequalities, including, for example, the Polya-Szeg ¨o and Kantorovich inequalities

Copyright q 2008 Mingjin Wang This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1 Introduction

The theory of means and their inequalities is fundamental to many fields including mathematics, statistics, physics, and economics.This is certainly true in the area of probability and statistics There are large amounts of work available in the literature For example, some useful results have been given by Shaked and Tong 1, Shaked and Shanthikumar

2, Shaked et al 3, and Tong 4, 5 Motivated by different concerns, there are numerous ways to introduce mean values In probability and statistics, the most commonly used mean

is expectation In 6, the author proves the mean inequality of two random variables The purpose of the present paper is to establish a recurring mean inequality, which generalizes the mean inequality of two random variables ton random variables This result can, in turn, be

extended to establish other new inequalities, which include generalizations of the Polya-Szeg ¨o and Kantorovich inequalities7

We begin by introducing some preliminary concepts and known results which can also

be found in6

Definition 1.1 The supremum and infimum of the random variable ξ are defined as inf x {x : Pξ ≤ x  1} and sup x {x : Pξ ≥ x  1}, respectively, and denoted by sup ξ and inf ξ.

Definition 1.2 If ξ is bounded, the arithmetic mean of the random variable ξ, Aξ, is given by

Aξ  supξ  inf ξ

In addition, if infξ ≥ 0, one defines the geometric mean of the random variable ξ, Gξ, to be

Definition 1.3 If ξ1, , ξ n are bounded random variables, the independent arithmetic mean of the product of random variables ξ1, , ξ n , Aξ1, , ξ n is given by

Aξ1, , ξ n



n

i1 supξ in i1 infξ

Definition 1.4 If ξ1, , ξ nare bounded random variables with infξ i ≥ 0, i  1, , n, one defines the independent geometric mean of the product of random variables ξ1, , ξ nto be

Gξ1, , ξ n







 n

i1

Remark 1.5 If ξ1, , ξ nare independent, then

Aξ1, , ξ n

 A n

i1

ξ i

,

Gξ1, , ξ n

 G n

i1

ξ i

.

1.5

The mean inequality of two random variables6

Theorem 1.6 Let ξ and η be bounded random variables If inf ξ > 0 and inf η > 0, then

Eξ2·Eη2

E2ξη ≤

A2ξ, η

Equality holds if and only if

P η ξ  B a η  A b



 1,

Gη2

Eξ2 Gξ2

Eη2

1.7

for A  sup ξ, B  sup η, a  inf ξ, b  inf η.

2 Main results

Our main results are given by the following theorem

Theorem 2.1 Suppose that ξ1, , ξ n , ξ n1 are bounded random variables, inf ξ i > 0, i  1, , n  1 Let {Un} be a sequence of real numbers If

n

i1 Eξ2

i

E2n

then

n1

i1 Eξ2

i

E2n1

i1 ξ i ≤ A

2

ξ1, , ξ n1

G2ξ1, , ξ n1  Un. 2.2

Proof Let A i  sup ξ i , a i  inf ξ i , i  1, , n  1 We have

Pξ1· · · ξ n A n1 − a1· · · a n ξ n1

A1· · · A n ξ n1 − ξ1· · · ξ n a n1

≥ 0 1. 2.3 So

PA1· · · A n1  a1· · · a n1

ξ1· · · ξ n1 ≥ A1a1· · · A n a n ξ2

n1  A n1 a n1 ξ2

1· · · ξ2

n

 1, 2.4 which implies that



A1· · · A n1  a1· · · a n1

Eξ1· · · ξ n1

≥ A1a1· · · A n a n Eξ2

n1



 A n1 a n1 Eξ2

1· · · ξ2

n



. 2.5 Using the Jensen inequality7 and assumption 2.1, we get



A1· · · A n1  a1· · · a n1

Eξ1· · · ξ n1

≥ A1a1· · · A n a n Eξ2

n1



 A n1 a n1 E2

ξ1· · · ξ n

≥ A1a1· · · A n a n Eξ2

n1



 A n1 a n1 Eξ2

1· · · Eξ2

n

Un

≥ 2



A1a1· · · A n a n Eξ2

n1



A n1 a n1 Eξ2

1· · · Eξ2

n

Un

1/2

.

2.6

Hence,

G2

ξ1, , ξ n1

Eξ2

1· · · Eξ2

n1

Un

1/2

≤ Aξ1, , ξ n1

Eξ1· · · ξ n1

from which2.2 follows

Combining this result with Theorem 1.6, the following recurring inequalities are immediate

Corollary 2.2 Let ξ1, , ξ n be bounded random variables If inf ξ i > 0, i  1, , n, then

Eξ2

2

E2

ξ1ξ2

 ≤ A

2

ξ1, ξ2



G2ξ1, ξ2

 ,

Eξ2

3

E2

ξ1ξ2ξ3

 ≤ A

2

ξ1, ξ2, ξ3



G2ξ1, ξ2, ξ3

A

2

ξ1, ξ2



G2ξ1, ξ2

 ,

.

n

k1 Eξ2

k

E2n

k1 ξ k ≤ n

k2

A2ξ1, ξ k

G2ξ1, ξ k 

2.8

3 Some applications

In this section, we exhibit some of the applications of the inequalities just obtained We make use of the following known lemma which we state here without proof

Lemma 3.1 If 0 < m2≤ m1≤ M1≤ M2, then

1/2m1 M1





m1M1

≤ 1/2



m2 M2





m2M2

Theorem 3.2 the extensions of the inequality of Polya-Szeg¨o Let a ij > 0, a i  minj a ij , A i  maxj a ij , for i  1, , n and j  1, , m Then,

n

i1

m



j1

a2

ij ≤ m n−2

4n−1

n

k2



a1· · · a k  A1· · · A k2

a1· · · a k A1· · · A k

m



j1

n

i1

a ij

2

Proof This result is a consequence of inequality2.8 Let ξ1have the distribution

Pξ1 a1j

 m1, j  1, , m. 3.3

We definen − 1 functions as follows:

f i

a1j

 a ij , i  2, , n, j  1, , m. 3.4 Letξ i  f i ξ1, i  2, , n Then,

Eξ2

i  m1m

j1

a2

ij , i  1, , n,

Eξ1· · · ξ n 1

m

m



j1

n

i1

a ij ,

Aξ1, , ξ k

 1 2



a1· · · a k  A1· · · A k

, Gξ1, , ξ k

a1· · · a k A1· · · A k

3.5

Inequality2.8 then becomes

n

i1 1/mm j1 a2

ij



1/mm j1n i1 a ij2 ≤ n

k2



1/2a1· · · a k  A1· · · A k2



a1· · · a k A1· · · A k2 , 3.6 from which our result follows

Remark 3.3 For n  2, we can get the inequality of Polya-Szeg¨o 7:

m



k1

a2

k

m



k1

b2

k

≤ 1 4

AB

ab 



ab AB

2

m



k1

a k b k

2

wherea k , b k > 0, k  1, , m, a  min a k , A  max a k , b  min b k, andB  max b k

Theorem 3.4 the extensions of Kantorovich’s inequality Let A be an m × m positive Hermitian

matrix Denote by λ1 and λ m the maximum and minimum eigenvalues of A, respectively For real

β1, , β n and β  β1 · · ·  β n , and any vector 0 / x ∈ R m ,the following inequality is satisfied:

n

i1 x∗A β i x



x∗A β/2 x2 ≤



x∗xn−2

4n−1

n

k2



l1· · · l k  L1· · · L k2

l1· · · l k L1· · · L k , 3.8

where

l i

⎧

⎨

⎩

λ β i /2

m , β i ≥ 0,

λ β i /2

1 , β i < 0, L i

⎧

⎨

⎩

λ β i /2

λ β i /2

m , β i < 0, i  1, , n. 3.9 Proof Let λ1 ≥ · · · ≥ λ mbe eigenvalues ofA and let Λ  diagλ1, , λ m There is a Hermitian matrixU that satisfies

Let

y  Ux y1, y2, , y mT

, p i y i2

m

i1 y i2, i  1, , m. 3.11 Then,

n

i1 x∗A β i x



x∗A β/2 x2 

n

i1 x∗U∗Λβ i Ux



x∗U∗Λβ/2 Ux2



n

i1 y∗Λβ i y



y∗Λβ/2 y2





y∗yn−2n i1m k1 λ β i

k p k

m

k1 λ β/2 k p k2





x∗xn−2n

i1m

k1 λ β i

k p k

m

k1 λ β/2 k p k2 .

3.12

What remains to show is that

n

i1m

k1 λ β i

k p k

m

k1 λ β/2 k p k2 ≤ 1

4n−1

n

k2



l1· · · l k  L1· · · L k2

l1· · · l k L1· · · L k , ∀p i ≥ 0, m

i1

p i  1. 3.13

We define the random variableζ, and assign Pζ  λ i   p i , i  1, , m Suppose ξ i 

ζ β i /2 , i  1, , n Notice that λ1andλ nare the upper and lower bounds of the random variable

ζ, so l iandL iare the lower and upper bounds ofξ i According toLemma 3.1, we know that

A2ξ1, , ξ k

G2ξ1, , ξ k ≤



1/2l1· · · l k  L1· · · L k2



l1· · · l k L1· · · L k2 . 3.14

Noticing that

Eξ1· · · ξ n

 Eζ β/2m

k1

we can use inequality2.8 to express inequality 3.13 as

Eξ2

1· · · Eξ2

n

E2

ξ1· · · ξ n ≤ n

k2



1/2l1· · · l k  L1· · · L k2



l1· · · l k L1· · · L k2 . 3.16

Remark 3.5 If n  2, β1 1, and β2 −1, this inequality takes the form

x∗Axx∗A−1x



x∗x2 ≤



λ1 λ m2

which is Kantorovich’s inequality7

References

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Journal of Multivariate Analysis, vol 24, no 2, pp 330–340, 1988.

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Mathematical Statistics, Academic Press, Boston, Mass, USA, 1994.

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1995.

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Journal of Inequalities and Applications, vol 1, no 1, pp 85–98, 1997.

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no 4, pp 755–763, 2002.

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2nd edition, 1952.