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Some conditions characterizing the “reverse” Hardy inequality. (English) Zbl 1170.26011

The authors obtain necessary and sufficient conditions for the “reverse” Hardy inequalities to hold, in the case \( -\infty < q \leq p < 0\). Let us precise that the “reverse” Hardy inequalities are given by \[ \begin{aligned} {\left( \int_a^b f^p(x)v(x) dx \right)}^{\frac{1}{p}} &\leq C {\left(\int_a^b {\left( \int_a^x f(t) dt \right)}^q u(x) dx \right)}^{\frac{1}{q}} \\ \noalign{\text{and}} \\ {\left( \int_a^b f^p(x)v(x) dx \right)}^{\frac{1}{p}} &\leq C {\left(\int_a^b {\left( \int_x^b f(t) dt \right)}^q u(x) dx \right)}^{\frac{1}{q}} \end{aligned} \] for \(f \geq 0\), and where the weight functions \(u, \, v\) has been completely characterized by P. R. Beesack and H. P. Heinig [Proc. Am. Math. Soc. 83, 532–536 (1981; Zbl 0484.26010)] for \(p, q < 1\), and by D. V. Prokhorov [Publ. Mat., Barc. 48, No. 2, 423–443 (2004; Zbl 1068.26022)] for \(p, q < 0 \) and \(p, q \in (0,1) \). A. Kufner and K. Kuliev [Adv. Alg. Anal. 1, No. 3, 219–228 (2006; Zbl 1170.26307)] also obtained conditions in the case \(- \infty < q \leq p < 0 \).

MSC:

26D10 Inequalities involving derivatives and differential and integral operators
26D15 Inequalities for sums, series and integrals
47B38 Linear operators on function spaces (general)
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