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

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 jalr.2011010106.m01 be a positive integer and define

jalr.2011010106.m02
where jalr.2011010106.m03 denote the expected value of a nonnegative random variable jalr.2011010106.m04. 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 jalr.2011010106.m05 and jalr.2011010106.m06 be nonnegative random variables on a common probability space. Then the following inequalities are equivalent:

jalr.2011010106.m07jalr.2011010106.m08 if jalr.2011010106.m09 with jalr.2011010106.m10 and jalr.2011010106.m11; jalr.2011010106.m12jalr.2011010106.m13 if jalr.2011010106.m14 with jalr.2011010106.m15 and jalr.2011010106.m16; jalr.2011010106.m17jalr.2011010106.m18 if jalr.2011010106.m19 with jalr.2011010106.m20; jalr.2011010106.m21jalr.2011010106.m22 if jalr.2011010106.m23 with jalr.2011010106.m24; jalr.2011010106.m25jalr.2011010106.m26 if jalr.2011010106.m27 or jalr.2011010106.m28, jalr.2011010106.m29 if jalr.2011010106.m30; jalr.2011010106.m31Minkowski's inequality: jalr.2011010106.m32jalr.2011010106.m33 if jalr.2011010106.m34, jalr.2011010106.m35jalr.2011010106.m36 if jalr.2011010106.m37; jalr.2011010106.m38Radon's inequality: jalr.2011010106.m39 if jalr.2011010106.m40 or jalr.2011010106.m41, jalr.2011010106.m42 if jalr.2011010106.m43.
  • 2000 Mathematics Subject Classification: Primary 26D15.

  • Lemma 2. Let jalr.2011010106.m44, jalr.2011010106.m45 for jalr.2011010106.m46. Then,

  • jalr.2011010106.m47jalr.2011010106.m48 if jalr.2011010106.m49 or jalr.2011010106.m50;

  • jalr.2011010106.m51jalr.2011010106.m52 if jalr.2011010106.m53;

  • jalr.2011010106.m54Jensen’ inequality:jalr.2011010106.m55if jalr.2011010106.m56 or jalr.2011010106.m57

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