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Recurring Mean Inequality of Random Variables
Journal of Inequalities and Applications volume 2008, Article number: 325845 (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.
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

then

Proof.
Let . We have

So

which implies that

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

Hence,

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,

Proof.
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:

where

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

Let

Then,

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].
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)90043-7
Shaked M, Shanthikumar JG: Stochastic Orders and Their Applications, Probability and Mathematical Statistics. Academic Press, Boston, Mass, USA; 1994:xvi+545.
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.1995.1038
Tong YL: Some recent developments on majorization inequalities in probability and statistics. Linear Algebra and Its Applications 1994,199(supplement 1):69–90.
Tong YL: Relationship between stochastic inequalities and some classical mathematical inequalities. Journal of Inequalities and Applications 1997,1(1):85–98. 10.1155/S1025583497000064
Wang M: The mean inequality of random variables. Mathematical Inequalities & Applications 2002,5(4):755–763.
Hardy GH, Littlewood JE, Pólya G: Inequalities. 2nd edition. Cambridge University Press, Cambridge, UK; 1952.
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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
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DOI: https://doi.org/10.1155/2008/325845
Keywords
- Real Number
- Lower Bound
- Hermitian Matrix
- Minimum Eigenvalue
- Jensen Inequality