A mixed Hölder and Minkowski inequality.
In: Proceedings of the American Mathematical Society, Jg. 127 (1999-08-01), Heft 8, S. 2405-2415
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Zugriff:
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 $z=y|y|^{q-2}$: \[\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], ] \[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.] 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. [ABSTRACT FROM AUTHOR]
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A mixed Hölder and Minkowski inequality.
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Autor/in / Beteiligte Person: | Iusem, Alfredo N. ; Isnard, Carlos A. ; Butnariu, Dan |
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Zeitschrift: | Proceedings of the American Mathematical Society, Jg. 127 (1999-08-01), Heft 8, S. 2405-2415 |
Veröffentlichung: | 1999 |
Medientyp: | academicJournal |
ISSN: | 0002-9939 (print) |
DOI: | 10.1090/S0002-9939-99-04800-5 |
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