Steffensen’s inequality and 𝐿¹-𝐿^{∞} estimates of weighted integrals

Abstract

Let Φ : [ 0 , ∞ ) → R \Phi :[0,\infty )\rightarrow \mathbb {R} be a continuous convex function with Φ ( 0 ) = 0. \Phi (0)=0. We prove that Φ ( | | f | | 1 ω N | | f | | ∞ ) ≤ 1 ω N | | f | | ∞ ∫ R N | f ( x ) | Φ ′ ( | x | N ) d x \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 ∈ L 1 ( R N ) ∩ L ∞ ( R N ) , f ≠ 0 , f\in L^{1}(\mathbb {R}^{N})\cap L^{\infty }(\mathbb {R}^{N}),f\neq 0, where ω N \omega _{N} is the measure of the unit ball of R N . \mathbb {R}^{N}. This can be used to obtain lower or upper bounds for weighted integrals ∫ R N | f ( x ) | η ( | x | ) d x \int _{\mathbb {R}^{N}}|f(x)|\eta (|x|)dx in terms of the L 1 L^{1} and L ∞ L^{\infty } norms of f , 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.