Hölder's Inequality and Related Inequalities in Probability

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). For convenience, throughout this paper, we let n be a positive integer and define E X EX p E X p p p p = ≠ =      ( ) , exp( ln ), , / 1 0 0 where EX denote the expected value of a nonnegative random variable X . 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 X and Y be nonnegative random variables on a common probability space. Then the following inequalities are equivalent: DOI: 10.4018/jalr.2011010106 International Journal of Artificial Life Research, 2(1), 54-61, January-March 2011 55 Copyright © 2011, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. ( ) a 1 EX Y EX EY h k h k £ ( ) ( ) if h k + = 1 with h > 0 and k > 0 ; ( ) a 2 EX Y EX EY h k h k £ ( ) ( ) if h k + ≤ 1 with h > 0 and k > 0 ; ( ) b 1 EX Y EX EY h k h k 3 ( ) ( ) if h k + = 1 with hk < 0 ; ( ) b 2 EX Y EX EY h k h k 3 ( ) ( ) if h k + ≥ 1 with hk < 0 ; ( ) c EX EX p p 3 ( ) if p 3 1 or p £ 0 , EX EX p p £ ( ) if 0 1 < < p ; ( ) d Minkowski’s inequality: ( ) M 1 E X Y E X E Y p p p | | | | | | + ≤ + if p 3 1 , ( ) M 2 E X Y E X E Y p p p | | | | | | + ≥ + if p £ 1 ; ( ) e Radon’s inequality: E X Y EX EY p