- Research Article
- Open Access

# Recurring Mean Inequality of Random Variables

- Mingjin Wang
^{1}Email author

**2008**:325845

https://doi.org/10.1155/2008/325845

© Mingjin Wang. 2008

**Received:**16 August 2007**Accepted:**9 May 2008**Published:**19 May 2008

## Abstract

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-Szegö and Kantorovich inequalities.

## Keywords

- Real Number
- Lower Bound
- Hermitian Matrix
- Minimum Eigenvalue
- Jensen Inequality

## 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 to random variables. This result can, in turn, be extended to establish other new inequalities, which include generalizations of the Polya-Szegö and Kantorovich inequalities [7].

We begin by introducing some preliminary concepts and known results which can also be found in [6].

Definition 1.1.

The *supremum and infimum of the random variable*
are defined as
and
, respectively, and denoted by
and
.

Definition 1.2.

Definition 1.3.

*independent arithmetic mean of the product of random variables*is given by

Definition 1.4.

*independent geometric mean of the product of random variables*to be

Remark 1.5.

The mean inequality of two random variables [6].

Theorem 1.6.

for .

## 2. Main Results

Our main results are given by the following theorem.

Theorem 2.1.

Proof.

from which (2.2) follows.

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

Corollary 2.2.

## 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.

Theorem 3.2 (the extensions of the inequality of *Polya-Szegö*).

Proof.

from which our result follows.

Remark 3.3.

where , and .

Theorem 3.4 (the extensions of *Kantorovich's* inequality).

Proof.

Remark 3.5.

which is *Kantorovich's* inequality [7].

## Authors’ Affiliations

## References

- Shaked M, Tong YL:
**Inequalities for probability contents of convex sets via geometric average.***Journal of Multivariate Analysis*1988,**24**(2):330–340. 10.1016/0047-259X(88)90043-7MATHMathSciNetView ArticleGoogle Scholar - Shaked M, Shanthikumar JG:
*Stochastic Orders and Their Applications, Probability and Mathematical Statistics*. Academic Press, Boston, Mass, USA; 1994:xvi+545.MATHGoogle Scholar - Shaked M, Shanthikumar JG, Tong YL:
**Parametric Schur convexity and arrangement monotonicity properties of partial sums.***Journal of Multivariate Analysis*1995,**53**(2):293–310. 10.1006/jmva.1995.1038MATHMathSciNetView ArticleGoogle Scholar - Tong YL:
**Some recent developments on majorization inequalities in probability and statistics.***Linear Algebra and Its Applications*1994,**199**(supplement 1):69–90.MATHMathSciNetView ArticleGoogle Scholar - Tong YL:
**Relationship between stochastic inequalities and some classical mathematical inequalities.***Journal of Inequalities and Applications*1997,**1**(1):85–98. 10.1155/S1025583497000064MATHMathSciNetGoogle Scholar - Wang M:
**The mean inequality of random variables.***Mathematical Inequalities & Applications*2002,**5**(4):755–763.MATHMathSciNetView ArticleGoogle Scholar - Hardy GH, Littlewood JE, Pólya G:
*Inequalities*. 2nd edition. Cambridge University Press, Cambridge, UK; 1952.MATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.