On an inequality suggested by Littlewood

Gao

Description

We study an inequality suggested by Littlewood, our result refines a result of Bennett. 2000 Mathematics Subject Classification . Primary 26D15. We study an inequality suggested by Littlewood, our result refines a result of Bennett. 2000 Mathematics Subject Classification . Primary 26D15.

Sujets

• Inequality (mathematics)
• Series (mathematics)

Informations

• Publié par: biomed
• Publié le: 01 janvier 2011
• Nombre de lectures: 29
• Langue: English

Extrait

GaoJournal of Inequalities and Applications2011,2011:5 http://www.journalofinequalitiesandapplications.com/content/2011/1/5

R E S E A R C H

On

an

Peng Gao

inequality

Correspondence: penggao@ntu. edu.sg Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 637371 Singapore

suggested

by

Open Access

Littlewood

Abstract We study an inequality suggested by Littlewood, our result refines a result of Bennett. 2000Mathematics Subject Classification. Primary 26D15. Keywords:Inequalities, infinite series, nonnegative sequences

Introduction In connection with work on the general theory of orthogonal series, Littlewood [1] raised some problems concerning elementary inequalities for infinite series. One of them asks to decide whether an absolute constantKexists such that for any nonnega  a n ) with A tive sequence ( n = = a , 1 2 ∞ ∞ ∞    3/2 2 2 4 a n A a≤AK a (1:1) n k n n n=1 k= n n=1

The above problem was solved by Bennett [2], who proved the following more gen eral result: Theorem 1.1([2, Theorem 4]). Letp≥1,q> 0,r> 0satisfying(p(q+r)  q)/p≥1 be fixed.Let K(p,q,r)be the best possible constant such that for any nonnegative  n sequence(a n )with A n =a , =1   r ∞ ∞ ∞    1+r/q p q 1+p/q p q a n A n a≤K(p,q,r) (a n A n ) (1:2) k n=1 =n=1

Then

K(p,q,r)≤

r p(q+r)−q

The special casep= 1,q=r= 2 in (1.2) leads to inequality (1.1) withK= 4 and Theorem 1.1 implies thatK(p,q,r) is finite for anyp≥1,q> 0,r> 0 satisfying (p(q +r) q)/p≥1, a fact we shall use implicitly throughout this article. We note that Bennett only proved Theorem 1.1 forp,q,r≥1 but as was pointed out in [3], Ben nett’s proof actually works for thep,q,r’ssatisfying the condition in Theorem 1.1. Another proof of inequality (1.2) for the special caser=qwas provided by Bennett [4] and a close look at the proof there shows that it in fact can be used to establish Theorem 1.1.

© 2011 Gao; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.