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Hardy-type inequalities via convexity. (English) Zbl 1083.26013

It was noticed by S. Kaijser, L.-E. Persson and A. Öberg [see their paper, “On Carleman and Knopp’s inequalities”, J. Approximation Theory 117, No. 1, 140–151 (2002; Zbl 1049.26014)] that the Hardy-type inequalities can be extended to \[ \int_{0}^{\infty}\Phi\left( \frac{1}{x}\int_{0}^{x}f(t)\,dt\right) \frac {dx}{x}\leq\int_{0}^{\infty}\Phi\left( f(x)\right) \frac{dx}{x} \] where \(\Phi\) is a convex function on \(\left( 0,\infty\right) .\) The paper under review contains further developments in the weighted and multidimensional framework. Also, some reversed inequalities are pointed out.

MSC:

26D10 Inequalities involving derivatives and differential and integral operators
26D15 Inequalities for sums, series and integrals
26A51 Convexity of real functions in one variable, generalizations

Citations:

Zbl 1049.26014
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