Recurring Mean Inequality of Random Variables
© Mingjin Wang. 2008
Received: 16 August 2007
Accepted: 9 May 2008
Published: 19 May 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 .
The mean inequality of two random variables .
2. Main Results
Our main results are given by the following theorem.
from which (2.2) follows.
Combining this result with Theorem 1.6, the following recurring inequalities are immediate.
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.
Theorem 3.2 (the extensions of the inequality of Polya-Szegö).
from which our result follows.
where , and .
Theorem 3.4 (the extensions of Kantorovich's inequality).
which is Kantorovich's inequality .
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