Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles 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.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Steffensen’s inequality and $L^{1}-L^{\infty }$ estimates of weighted integrals
HTML articles powered by AMS MathViewer

by Patrick J. Rabier PDF
Proc. Amer. Math. Soc. 140 (2012), 665-675 Request permission


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.
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 26D15, 39B62
  • Retrieve articles in all journals with MSC (2010): 26D15, 39B62
Additional Information
  • Patrick J. Rabier
  • Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
  • Email:
  • 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:
  • MathSciNet review: 2846336