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- Open Access
Recurring Mean Inequality of Random Variables
Journal of Inequalities and Applications volume 2008, Article number: 325845 (2008)
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.
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 , Shaked and Shanthikumar , Shaked et al. , 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 , 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 .
We begin by introducing some preliminary concepts and known results which can also be found in .
The supremum and infimum of the random variable are defined as and , respectively, and denoted by and .
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
If are bounded random variables, the independent arithmetic mean of the product of random variables is given by
If are bounded random variables with , one defines the independent geometric mean of the product of random variables to be
If are independent, then
The mean inequality of two random variables .
Let and be bounded random variables. If and , then
Equality holds if and only if
2. Main Results
Our main results are given by the following theorem.
Suppose that are bounded random variables, . Let be a sequence of real numbers. If
Let . We have
which implies that
Using the Jensen inequality  and assumption (2.1), we get
from which (2.2) follows.
Combining this result with Theorem 1.6, the following recurring inequalities are immediate.
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.
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.
For , we can get the inequality of Polya-Szegö :
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
we can use inequality (2.8) to express inequality (3.13) as
If and , this inequality takes the form
which is Kantorovich's inequality .
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Wang, M. Recurring Mean Inequality of Random Variables. J Inequal Appl 2008, 325845 (2008). https://doi.org/10.1155/2008/325845
- Real Number
- Lower Bound
- Hermitian Matrix
- Minimum Eigenvalue
- Jensen Inequality