Steffensen’s inequality and $L^{1}-L^{\infty }$ estimates of weighted integrals

Abstract:

Let $\Phi :[0,\infty )\rightarrow \mathbb {R}$ be a continuous convex function with $\Phi (0)=0.$ We prove that $\Phi \left ( \frac {||f||_{1}}{\omega _{N}||f||_{\infty }}\right ) \leq \frac {1}{\omega _{N}||f||_{\infty }}\int _{ \mathbb {R}^{N}}|f(x)|\Phi ^{\prime }(|x|^{N})dx$ for every $f\in L^{1}(\mathbb {R}^{N})\cap L^{\infty }(\mathbb {R}^{N}),f\neq 0,$ where $\omega _{N}$ is the measure of the unit ball of $\mathbb {R}^{N}.$ This can be used to obtain lower or upper bounds for weighted integrals $\int _{\mathbb {R}^{N}}|f(x)|\eta (|x|)dx$ in terms of the $L^{1}$ and $L^{\infty }$ norms of $f,$ which are often much sharper than crude estimates that may be obtained, if at all, by a visual inspection of the integrand. The basic inequality is essentially independent of Jensen’s inequality, but it is closely related to Steffensen’s inequality.