Hölder's Inequality and Related Inequalities in Probability

Hölder's Inequality and Related Inequalities in Probability

Cheh-Chih Yeh (Lunghwa University of Science and Technology, Taiwan)
Copyright: © 2011 |Pages: 8
DOI: 10.4018/jalr.2011010106
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Abstract

In this paper, the author examines Holder’s inequality and related inequalities in probability. The paper establishes new inequalities in probability that generalize previous research in this area. The author places Beckenbach’s (1950) inequality in probability, from which inequalities are deduced that are similar to Brown’s (2006) inequality along with Olkin and Shepp (2006).
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1. Introduction

Yeh, Yeh, and Chan (2008) link some equivalent probability inequalities in a common probability space, such as Hölder, Minkowski, Radon, Cauchy, and so on. In this paper, we will establish some new inequalities in probability which generalize some inequalities (Sun, 1997; Wan, Su, & Wang, 1967; Wang & Wang, 1987; Yeh, Yeh, & Chan, 2008). We also establish Beckenbach's (1950) inequality in probability, from which we deduce some inequalities which look like Brown's (2006) inequality along with Olkin and Shepp (2006) and related results (Beckenbach & Bellman, 1984; Casella & Berger, 2002; Danskin, 1952; Dresher, 1953; Gurland, 1968; Hardy, Littlewood, & Polya,1952; Kendall & Stuart; Loeve, 1998; Marshall & Olkin, 1979; Mullen, 1967; Persson, 1990; Sclove, Simons, & Ryzin, 1967; Yang & Zhen, 2004).

For convenience, throughout this paper, we let be a positive integer and define

where denote the expected value of a nonnegative random variable . And we consider only the random variables which have finite expected values.

To establish our results, we need the following two lemmas: Lemma 1 (Yeh, Yeh, & Chang, 2008) and Lemma 2 due to Radon (Hardy, Littlewood, & Polya,1952).

Lemma 1.

Let and be nonnegative random variables on a common probability space. Then the following inequalities are equivalent:

if with and ; if with and ; if with ; if with ; if or , if ; Minkowski's inequality: if , if ; Radon's inequality: if or , if .
  • 2000 Mathematics Subject Classification: Primary 26D15.

  • Lemma 2. Let , for . Then,

  • if or ;

  • if ;

  • Jensen’ inequality:if or

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