
Journal of Convex Analysis 19 (2012), No. 2, 301321 Copyright Heldermann Verlag 2012 Generalized Steffensen Inequalities and Their Optimal Constants Patrick J. Rabier Dept. of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, U.S.A. rabier@imap.pitt.edu [Abstractpdf] If $\Phi :[0,\infty )\rightarrow \mathbb{R}$ is convex and continuous with $\Phi (0)=0$ and if $q\in (1,\infty )$,\break $q^{\prime }:=\frac{q}{q1}$, we first prove that the inequality $$ \Phi \left( \int_{0}^{\infty }f(r)dr\right) \leq C\int_{0}^{\infty }f(r)\Phi ^{\prime }(r^{1/q^{\prime }})dr $$ for every $f\in L^{q}(0,\infty)$, $f\geq 0$ with $f_{q}\leq 1$ holds when $C=1$. In general, both sides may be $\pm \infty$. Related inequalities for $f\in L^{1}(\mathbb{R}^{N})\cap L^{q}(\mathbb{R}^{N})$, $f\neq 0$ are derived. This inequality is independent of Jensen's inequality and, when $q=\infty$, it is an elaboration on an inequality of Steffensen which was discussed elsewhere by the author.\par The next goal of the paper is to identify the range of the admissible constants $C$ and, in particular, to characterize the optimal constant when $\Phi \geq 0$ or $\Phi \leq 0$. It turns out that $C=1$ is ``almost always'' optimal, at least in a restricted sense, but not always when $q<\infty$: Given $q$, the admissible constants lie on an interval containing $1$ whose left (right) endpoint is the supremum (infimum) of a function defined on some (left/right dependent) subset of $\mathbb{R}^{2}$.\par If $q=2$, these extrema can be calculated in a number of examples. Among other things, this reveals that $C=1$ need not be optimal when $\Phi \geq 0$ and $\Phi _{+}^{\prime }(0)=0$ or when $\Phi \leq 0$ and $\Phi _{+}^{\prime} (0)=\infty$. Keywords: Convexity, Steffensen's inequality, weighted integral. MSC: 26D15, 39B62 [ Fulltextpdf (219 KB)] for subscribers only. 