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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A mixed Hölder and Minkowski inequality


Authors: Alfredo N. Iusem, Carlos A. Isnard and Dan Butnariu
Journal: Proc. Amer. Math. Soc. 127 (1999), 2405-2415
MSC (1991): Primary 46B10, 46B25, 46E30
Published electronically: April 9, 1999
MathSciNet review: 1487373
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Abstract | References | Similar Articles | Additional Information

Abstract: Hölder's inequality states that $\left \Vert x\right \Vert _{p}\left \Vert y\right \Vert _{q}-\left \langle x,y\right \rangle \ge 0$ for any $(x,y)\in \mathcal{L}^{p}(\Omega )\times \mathcal{L}^{q}(\Omega )$ with $1/p+1/q=1$. In the same situation we prove the following stronger chains of inequalities, where $z=y|y|^{q-2}$:

\begin{displaymath}\left \Vert x\right \Vert _{p}\left \Vert y \right \Vert _{q}-\left \langle x,y\right \rangle \ge (1/p)\big [\big (\left \Vert x \right \Vert _{p}+\left \Vert z\right \Vert _{p}\big )^{p} -\left \Vert x+z\right \Vert _{p}^{p}\big ]\ge 0 \quad\text{if }p\in (1,2], \end{displaymath}

\begin{displaymath}0\le \left \Vert x\right \Vert _{p}\left \Vert y\right \Vert _{q}-\left \langle x,y\right \rangle \le (1/p)\big [\big (\left \Vert x \right \Vert _{p}+\left \Vert z\right \Vert _{p}\big )^{p} -\left \Vert x+z\right \Vert _{p}^{p}\big ] \quad \text{if }p\ge 2.\end{displaymath}

A similar result holds for complex valued functions with Re$(\left \langle x,y\right \rangle )$ substituting for $\left \langle x,y\right \rangle $. We obtain these inequalities from some stronger (though slightly more involved) ones.


References [Enhancements On Off] (What's this?)

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Additional Information

Alfredo N. Iusem
Affiliation: Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ, CEP 22460-320, Brazil
Email: iusp@impa.br

Dan Butnariu
Affiliation: University of Haifa, Department of Mathematics and Computer Science, Mount Carmel, 31905 Haifa, Israel
Email: dbutnaru@mathcs2.haifa.ac.il

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04800-5
PII: S 0002-9939(99)04800-5
Keywords: Banach spaces, H\"{o}lder's inequality, Minkowski's inequality
Received by editor(s): December 5, 1996
Received by editor(s) in revised form: November 13, 1997
Published electronically: April 9, 1999
Additional Notes: The first author’s research for this paper was partially supported by CNPq grant no. 301280/86.
Work by the third author was done during his visit to the Department of Mathematics of the University of Texas at Arlington.
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1999 American Mathematical Society