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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: 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?)

  • [1] Butnariu, D., Iusem, A.N. Local moduli of convexity and their application to finding almost common points of measurable families of operators Recent Developments in Optimization Theory and Nonlinear Analysis, edited by Y. Censor and S. Reich, AMS Contemporary Mathematics Series, vol. 204, 1997, 61-91. CMP 97:11
  • [2] Butnariu, D., Iusem A.N., Burachik, R. Iterative methods of solving stochastic convex feasibility problems and applications (to be published).
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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

Dan Butnariu
Affiliation: University of Haifa, Department of Mathematics and Computer Science, Mount Carmel, 31905 Haifa, Israel

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

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