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Proceedings of the American Mathematical Society

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


Author: Patrick J. Rabier
Journal: Proc. Amer. Math. Soc. 140 (2012), 665-675
MSC (2010): Primary 26D15, 39B62
DOI: https://doi.org/10.1090/S0002-9939-2011-10939-0
Published electronically: June 22, 2011
MathSciNet review: 2846336
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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.


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Additional Information

Patrick J. Rabier
Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
Email: rabier@imap.pitt.edu

Keywords: Convexity, Jensen’s inequality, Steffensen’s inequality, weighted integral.
Received by editor(s): June 16, 2010
Received by editor(s) in revised form: June 21, 2010, and December 5, 2010
Published electronically: June 22, 2011
Additional Notes: The useful comments of an anonymous referee are gratefully acknowledged.
Communicated by: Tatiana Toro
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.