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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Steffensen’s inequality and $L^{1}-L^{\infty }$ estimates of weighted integrals
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by Patrick J. Rabier PDF
Proc. Amer. Math. Soc. 140 (2012), 665-675 Request permission

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
  • 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
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2846336