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New generalizations of Hardy’s integral inequality. (English) Zbl 0944.26021

Consider the Hardy-type inequality

a b a x f (t) d t p w(x)dxC a b f p (x)v(x)dx(1)

on the class of all non-negative and measurable functions f on (a,b), -a<b+. If s(1,), put s ' =s/(s-1). The classical Hardy inequality states that (1) holds with (a,b)=(0,), C=(p ' ) p , w(x)x -p and v(x)1, where p(1,). The authors of the paper under review prove three generalizations of the classical Hardy inequality. To illustrate their results, we mention one of them (Theorem  2.1): The inequality (1) holds if 0<a<b, C=q p/r ' (r ' ) -p/r ' [1-(a b) r ' /q ] p/r ' , w(x)x -p/r ' and v(x)1, where p,r(1,) and p -1 +q -1 +r -1 =1.

Reviewer: B.Opic (Praha)

MSC:
26D15Inequalities for sums, series and integrals of real functions