From Hardy to Carleman and general mean-type inequalities (Q2712738)

The authors establish conditions on weight functions \(v,w\) in order that the inequality NEWLINE\[NEWLINE \left(\int_{0}^{\infty}\left(A(f,x)\right)^qw(x) dx\right)^{1/q} \leq C \left(\int_{0}^{\infty}f(x)^pv(x) dx\right)^{1/p} NEWLINE\]NEWLINE hold for all positive \(f\) on \((0,\infty)\), where \(0<p\leq q<\infty\), and \(A(f,x)\) is either the generalized averaging operator \(((\int_{0}^{x}u(t) dt)^{-1}\int_{0}^{x}u(t)f^\alpha(t) dt)^{1/\alpha}\), where \(\alpha\neq 0\), or the corresponding geometric mean operator \(\exp((\int_{0}^{x}u(t) dt)^{-1}\int_{0}^{x}u(t)\log f(t) dt)\).NEWLINENEWLINEFor the entire collection see [Zbl 0958.00028].