## Steffensenâs inequality and $L^{1}-L^{\infty }$ estimates of weighted integrals

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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 Jensenâs inequality, but it is closely related to Steffensenâs inequality.## References

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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