Still more generalizations of Hardy's inequality

Abstract

These generalizations of Hardy's Integral Inequality are generalizations of some inequalities of B. G. Pachpatte.1991 Mathematics subject classification {Amer. Math. Soc): 26D15.In [5] B. G. Pachpatte added one more to the multitude of papers generalizing Hardy's Inequality; and the present paper, inspired by [5], is yet another. The proof of our leading theorem is modelled on that of Pachpatte, and our theorem generalizes his. We have added iterated versions of our theorems. Our reference list mentions a few related papers. THEOREM 1. Let 0 < a < b < oo, c > 0, p > 0 and q > 0 be constants. Let r(x) be positive and locally absolutely continuous in [a, b), and f(x) be almost everywhere non-negative and measurable on (a,b). LetIf 0 < p < 1 and the reverse inequality (3) holds, then the reverse inequality (4) holds also.

Hardy inequality on time scales and its application to half-linear dynamic equations

“…Let us mention at least a few papers [5,11,15], among many others dealing with various types of HI's, and nice monographs [12,13,14]. All above facts seem to prove that there is no possibility of a last word on Hardy inequality.…”

Weighted hardy type integral inequalities involving many functions

“…Due to this, over the years much efforts and time has been devoted to the improvement and generalization of Hardy's inequalities (1) and (2). These includes, among others, the works in [1,6,7,8,10,11,12,13,14] and by Hussain, Pečarić in [2,3,4,5]. The main aim of this paper is to establish new weighted Hardy type integral inequalities involving two or more functions and to derive some weighted classical Hardy type inequalities by means of fairly elementary analysis.…”