HĂ¶lder s inequality and related inequalities in probability

Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.. Keywords: Beckenbach’s Inequality, Holder’s Inequality, Inequalities, Probab

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54 International Journal of Artificial Life Research, 2(1), 54-61, January-March 2011

is prohibited.

Keywords: Beckenbach’s Inequality, Holder’s Inequality, Inequalities, Probablity, Proof

1 INTRODUCTION

Yeh, Yeh, and Chan (2008) link some equivalent

probability inequalities in a common probability

space, such as Hölder, Minkowski, Radon,

Cau-chy, and so on In this paper, we will establish

some new inequalities in probability which

gen-eralize 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)

Hölder’s Inequality and Related Inequalities in Probability

Cheh-Chih Yeh, Lunghwa University of Science and Technology, Taiwan

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).

For convenience, throughout this paper,

we let n be a positive integer and define

E X p p

p p

=











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 fol-lowing 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 fol-lowing inequalities are equivalent:

DOI: 10.4018/jalr.2011010106

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International Journal of Artificial Life Research, 2(1), 54-61, January-March 2011 55

is prohibited.

( )a1 EX Y h k £ (EX) (h EY)k if h+ = 1k with

h > 0 and k > 0;

( )a2 EX Y h k £ (EX) (h EY)k if h+ ≤ 1k

with h > 0 and k > 0;

( )b1 EX Y h k ³ (EX) (h EY)k if h+ = 1k

with hk < 0;

( )b2 EX Y h k ³ (EX) (h EY)k if h+ ≥ 1k

with hk < 0;

( )c EX p ³ (EX)p if p ³ 1 or p £ 0,

EX p £ (EX)p if 0< <p 1;

( )d Minkowski’s inequality:

(M1)E p |X+Y |≤E p |X |+E Y p | | if

p ³ 1,

(M2)E p |X+Y |≥E p |X |+E Y p | | if

p £ 1;

( )e Radon’s inequality: E X

Y

EX EY

p p

−1 ≥ −1

if p ³ 1 or p £ 0, E X

Y

EX EY

p p

−1 ≤ −1 if

0< <p 1

2000 Mathematics Subject Classification:

Primary 26D15.

Lemma 2 Let a i, bi∈( , )0∞ for i= 1 2, , , n

Then,

b

a b k

p

k

p

k p k n

n

−

=

−

=

∑

1 1 1 1

( ) if 1 < p or

p < 0;

( )b a

b

a b k

p

k

p

k p k n

k p k n n

−

=

−

=

∑

1 1 1 1

( ) if 0< <p 1;

( )c Jensen’ inequality:

(a s a s a ) (a a a )

n r r

1 2

1

1 2

1

+ +  + < + +  +

if 0 < <r s or r < < 0.s

2 HÖLDER’S INEQUALITY

Theorem 2.1 Let X X1, 2, , X n be nonnega-tive random variables on a common probability space Then the following two results hold:

( )R1

E X X q q X EX EX EX

n

n q

( 11 22  ) £ ( 1) ( 1 2) 2  ( )

if q q1, ,2  Îq n ( , )0 1 with q i

i

n

=

1

;

( )R2

E X X r r X EX EX EX

n

r

r r n r r r n

n

( 11 22 ) ( 1) ( 2 ) ( )

if r r1, ,2  Îr n ( , )0 1 with r r i

i

n

=

∑

1

In particular, if r £ 1, then

(R2*)

E X X r r X EX EX EX

n

n r

( 11 22  ) £ ( 1) ( 1 2) 2  ( )

Proof Case ( )R1 : Inequality ( )R1 can be proved

by mathematical induction using ( )a1 of Lemma 1 But for the sake of complete-ness, we give its proof here It follows from ( )a1 of Lemma 1 that ( )R1 holds for n = 2 Suppose that ( )R1 holds for

n =k Thus, for n= + 1k , let q1+q2++q k =q

q

i i k

=

∑ 1 1 and q+q k+1=1 Hence:

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