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In this book some general aspects of generalizations, refinements, and variants of famous Hardy's inequality are presented. An integral operator with general non- negative kernel on measure spaces with positive σ-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. Furthermore, improvements and reverses of new weighted Hardy type inequalities with integral operators are stated and proved. By using the concept of the subdifferential of a convex function, the general Boas-type inequality is given and some new inequalities for superquadratic and subquadratic functions as well as for functions which can be bounded by non-negative convex or superquadratic function are obtained. The Boas functional and related inequality allow us to adjust Lagrange and Cauchy mean value theorems to the context and in that way define a new class of two-parametric means of the Cauchy-type. We also give some interesting, one-dimensional and multidimensional, examples related to balls and cones in R^n.
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