Some inequalities for gradients of harmonic functions
For a function u(x, y) harmonic in the upper half-plane y>0 and represented by the Poisson integral of a function v(t) ∈ L 2 (−∞,∞), we prove that the inequality \(grad u (x, y)|^2 {\text{ }} \leqslant {\text{ }}\frac{1}{{4\pi ^3 }}{\text{ }}\int\limits_{ - \infty }^\infty {v^2 } (t)dt\)...
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| Datum: | 1997 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Russisch Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
1997
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/5106 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Zusammenfassung: | For a function u(x, y) harmonic in the upper half-plane y>0 and represented by the Poisson integral of a function v(t) ∈ L 2 (−∞,∞), we prove that the inequality \(grad u (x, y)|^2 {\text{ }} \leqslant {\text{ }}\frac{1}{{4\pi ^3 }}{\text{ }}\int\limits_{ - \infty }^\infty {v^2 } (t)dt\) is true. A similar inequality is obtained for a function harmonic in a disk. |
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