Recurring Mean Inequality of Random Variables

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

(1.1)

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

(1.2)

Definition 1.3.

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

(1.3)

Definition 1.4.

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

(1.4)

Remark 1.5.

If are independent, then

(1.5)

The mean inequality of two random variables [6].

Theorem 1.6.

Let and be bounded random variables. If and , then

(1.6)

Equality holds if and only if

(1.7)

for .

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

(2.1)

then

(2.2)

Proof.

Let . We have

(2.3)

So

(2.4)

which implies that

(2.5)

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

(2.6)

Hence,

(2.7)

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

(2.8)

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

(3.1)

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

Let , for and . Then,

(3.2)

Proof.

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

(3.3)

We define functions as follows:

(3.4)

Let . Then,

(3.5)

Inequality (2.8) then becomes

(3.6)

from which our result follows.

Remark 3.3.

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

(3.7)

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:

(3.8)

where

(3.9)

Proof.

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

(3.10)

Let

(3.11)

Then,

(3.12)

What remains to show is that

(3.13)

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

(3.14)

Noticing that

(3.15)

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

(3.16)

Remark 3.5.

If and , this inequality takes the form

(3.17)

which is Kantorovich's inequality [7].