Some conditions characterizing the ``reverse'' Hardy inequality
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 \textit{P. R. Beesack} and \textit{H. P. Heinig} [Proc. Am. Math. Soc. 83, 532--536 (1981; Zbl 0484.26010)] for \(p, q < 1\), and by \textit{D. V. Prokhorov} [Publ. Mat., Barc. 48, No. 2, 423--443 (2004; Zbl 1068.26022)] for \(p, q < 0 \) and \(p, q \in (0,1) \). \textit{A. Kufner} and \textit{K. Kuliev} [Adv. Alg. Anal. 1, No. 3, 219--228 (2006; Zbl 1170.26307)] also obtained conditions in the case \(- \infty < q \leq p < 0 \).
scientific article; zbMATH DE number 5291268
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