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Top1. 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
.