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Recurring Mean Inequality of Random Variables


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.

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.

If is bounded, the arithmetic mean of the random variable, , is given by


In addition, if , one defines the geometric mean of the random variable, , to be


Definition 1.3.

If are bounded random variables, the independent arithmetic mean of the product of random variables is given by


Definition 1.4.

If are bounded random variables with , one defines the independent geometric mean of the product of random variables to be


Remark 1.5.

If are independent, then


The mean inequality of two random variables [6].

Theorem 1.6.

Let and be bounded random variables. If and , then


Equality holds if and only if


for .

2. Main Results

Our main results are given by the following theorem.

Theorem 2.1.

Suppose that are bounded random variables, . Let be a sequence of real numbers. If





Let . We have




which implies that


Using the Jensen inequality [7] and assumption (2.1), we get




from which (2.2) follows.

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

Corollary 2.2.

Let be bounded random variables. If , , then


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 , then


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

Let , for and . Then,



This result is a consequence of inequality (2.8). Let have the distribution


We define functions as follows:


Let . Then,


Inequality (2.8) then becomes


from which our result follows.

Remark 3.3.

For , we can get the inequality of Polya-Szegö [7]:


where , and .

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

Let be an positive Hermitian matrix. Denote by and the maximum and minimum eigenvalues of , respectively. For real and , and any vector ,the following inequality is satisfied:





Let be eigenvalues of and let . There is a Hermitian matrix that satisfies






What remains to show is that


We define the random variable , and assign . Suppose Notice that and are the upper and lower bounds of the random variable , so and are the lower and upper bounds of . According to Lemma 3.1, we know that


Noticing that


we can use inequality (2.8) to express inequality (3.13) as


Remark 3.5.

If and , this inequality takes the form


which is Kantorovich's inequality [7].


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Correspondence to Mingjin Wang.

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Wang, M. Recurring Mean Inequality of Random Variables. J Inequal Appl 2008, 325845 (2008).

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