Reversed Hölder Type Inequalities for Monotone Functions of Several Variables

✍ Scribed by Sorina Barza; Josip Pečarlć; Lars-Erik Person

• Publisher: John Wiley and Sons
• Year: 2009
• Tongue: English
• Weight: 512 KB
• Volume: 186
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.3211860104

✦ Synopsis

Some reversed Hiilder type inequalities yielding for monotone or quasimonotone functions of one variable have recently been obtained and applied (see e.g. [l], (21, (31, [5], [S], [12], [14], [17]).

In this paper some inequalities of this type are proved for the more general case with n functions of m variables ( m , n E Z+). Some of these results seem to be new also for the case n = 1 or m = 1.

1. Introduction

Various variants of Holder type inequalities are very useful in Mathematical Analysis and its applications. Moreover, in several cases these inequalities are applied to functions satisfying additional conditions e. g. that the involved functions are monotone or quasimonotone. A function f on 10, m[ is said to be quasimonotone if, for some real number a, f ( t ) ~-~ is a decreasing or an increasing function o f t . More precisely, we say that f E Qp if and only if f(z)z-P is decreasing, and f E QP if and only if f(z)z-P is increasing. Moreover, we consider the class QC =: Qa QP of functions endowed with two quasimonotonicity conditions. (Examples of functions belonging to such classes are the decreasing rearrangement f k ( t ) , the averaged decreasing rearrangement p(t), the I< -Jand E-functionals, the modulus of continuity, etc., see e.g. [4]). Here we only recall the following recently proved reversed Holder type inequalities (see [l], [2], [3], [5]): Theorem A. (a) Led QO < Q and f E Q"". If 0 < p 5 q < 00, then 1991 Mafhemaiics Subjecf Classificafion. 26D 15, 26DO7, 260 20. Keywords and phrases. Inequalities, reversed Hiilder inequalities, quasimonotone functions of heveral variables, best constants, Cebysev inequality, applications.