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 |
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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 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860511300662067200 |
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| author | Grigor'ev, Yu. A. Григорьев, Ю. А. Григорьев, Ю. А. |
| author_facet | Grigor'ev, Yu. A. Григорьев, Ю. А. Григорьев, Ю. А. |
| author_sort | Grigor'ev, Yu. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:24:43Z |
| description | 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. |
| first_indexed | 2026-03-24T03:10:42Z |
| format | Article |
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| id | umjimathkievua-article-5106 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T03:10:42Z |
| publishDate | 1997 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/d1/c1d949b7614ef1fa17f01bcf9a8adcd1.pdf |
| spelling | umjimathkievua-article-51062020-03-18T21:24:43Z Some inequalities for gradients of harmonic functions Некоторые неравенства для градиентов гармонических функций Grigor'ev, Yu. A. Григорьев, Ю. А. Григорьев, Ю. А. 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. Доведено, що для функції u(x, y) гармонічної у верхній півплощині y>0 і зображуваної інтегралом Пуассона від функції v(t) ∈ L 2 (−∞,∞). справедлива нерівність \(grad u (x, y)|^2 {\text{ }} \leqslant {\text{ }}\frac{1}{{4\pi ^3 }}{\text{ }}\int\limits_{ - \infty }^\infty {v^2 } (t)dt\) . Подібна нерівність одержана також для функції, яка гармонічна в крузі. Institute of Mathematics, NAS of Ukraine 1997-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/5106 Ukrains’kyi Matematychnyi Zhurnal; Vol. 49 No. 8 (1997); 1135–1136 Український математичний журнал; Том 49 № 8 (1997); 1135–1136 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/5106/6908 https://umj.imath.kiev.ua/index.php/umj/article/view/5106/6909 Copyright (c) 1997 Grigor'ev Yu. A. |
| spellingShingle | Grigor'ev, Yu. A. Григорьев, Ю. А. Григорьев, Ю. А. Some inequalities for gradients of harmonic functions |
| title | Some inequalities for gradients of harmonic functions |
| title_alt | Некоторые неравенства для градиентов гармонических функций |
| title_full | Some inequalities for gradients of harmonic functions |
| title_fullStr | Some inequalities for gradients of harmonic functions |
| title_full_unstemmed | Some inequalities for gradients of harmonic functions |
| title_short | Some inequalities for gradients of harmonic functions |
| title_sort | some inequalities for gradients of harmonic functions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/5106 |
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