Generalized Steffensen Inequalities and Their Optimal Constants

If $\Phi:[0,\infty )\rightarrow \mathbb{R}$ is convex and continuous with $\Phi (0)=0$ and if $q\in (1,\infty )$,\break $q^{\prime }:=\frac{q}{q-1}$, 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 $\Phi _{+}^{\prime} (0)=-\infty$.