engleski

Generalizations and refinements of Hardy's inequality

In this talk an integral operator with general non-negative kernel on measure spaces with positive $\sigma$-finite measure is considered and some new weighted Hardy type inequalities for convex functions and refinements of weighted Hardy type inequalities for superquadratic functions are obtained. Moreover, some refinements of weighted Hardy type inequalities for convex functions and some new refinements of discrete Hardy type inequalities are given. As special cases of our results, we obtain refinements of the classical one-dimensional Hardy's, Polya--Knopp's, Hardy-Hilbert's and related dual inequalities, as well as a generalization and refinement of the classical Godunova's inequality. We show that our results may be seen as generalizations of some recent results related to Riemann-Liuuville's and Weyl's operator. Furthermore, improvements and reverses of new weighted Hardy type inequalities with integral operators are stated and proved. New Cauchy type mean is introduced and monotonicity property of this mean is proved.

The Hardy inequality; The Hilbert inequality; Inequalities; Hardy type inequalities; convex function; kernel; The Hardy operator with general kernel.

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