Generalized Steffensen inequalities and their optimal constants
Let \(F: [0,\infty)\to\mathbb R\) be convex and continuous, with \(F(0)= 0\). For \(p> 1\) put \(q= p/(p-1)\). It is proved that, NEWLINE\[NEWLINEF\Biggl(\int^\infty_0 f\,dr\Biggr)\leq C.NEWLINE\]NEWLINE \(\int^\infty_0 f(f) F'(r^{1/q})\,dr\) holds for every \(f\geq 0\) in the unit of \(L^p(0,\infty)\) and with the constant \(C= 1\). Here, in general, both sides may be \(\pm\infty\). Related inequalities for \(f\) in \(L^1(\mathbb R^N)\cap L^p(\mathbb R^N)\), \(f\geq 0\) are also derived. The next aim of the paper is to identify the range of admissible constants \(C\) and, in particular, to characterize the optimal constant when \(F\geq 0\) or \(F\leq 0\). The results are a little complicated to be stated here.
scientific article; zbMATH DE number 6064738