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A mixed Hölder and Minkowski inequality

Authors:	Alfredo N Iusem Carlos A Isnard Dan Butnariu

Institution:	Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ, CEP 22460-320, Brazil ; Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ, CEP 22460-320, Brazil Dan Butnariu ; University of Haifa, Department of Mathematics and Computer Science, Mount Carmel, 31905 Haifa, Israel

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 . 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.

Keywords:	Banach spaces H\"{o}lder's inequality Minkowski's inequality

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