Hölder's Inequality and Related Inequalities in Probability

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

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