On the Growth of the Maximum of the Modulus of an Entire Function on a Sequence
Let M f(r) and μf(r) be, respectively, the maximum of the modulus and the maximum term of an entire function f and let Φ be a continuously differentiable function convex on (−∞, +∞) and such that x = o(Φ(x)) as x → +∞. We establish that, in order that the equality \(\lim \inf \limits_{r \to + \inf...
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| Дата: | 2002 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2002
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/4154 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510287869771776 |
|---|---|
| author | Filevych, P. V. Філевич, П. В. |
| author_facet | Filevych, P. V. Філевич, П. В. |
| author_sort | Filevych, P. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:23:18Z |
| description | Let M f(r) and μf(r) be, respectively, the maximum of the modulus and the maximum term of an entire function f and let Φ be a continuously differentiable function convex on (−∞, +∞) and such that x = o(Φ(x)) as x → +∞. We establish that, in order that the equality \(\lim \inf \limits_{r \to + \infty} \frac{\ln M_f (r)}{\Phi (\ln r)} = \lim \inf \limits_{r \to + \infty} \frac{\ln \mu_f (r)}{\Phi (\ln r)}\) be true for any entire function f, it is necessary and sufficient that ln Φ′(x) = o(Φ(x)) as x → +∞. |
| first_indexed | 2026-03-24T02:54:36Z |
| format | Article |
| fulltext |
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| id | umjimathkievua-article-4154 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:54:36Z |
| publishDate | 2002 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/87/db5cc0f648fca7a53491cda221787187.pdf |
| spelling | umjimathkievua-article-41542020-03-18T20:23:18Z On the Growth of the Maximum of the Modulus of an Entire Function on a Sequence Про зростання максимума модуля цілої функції на послідовності Filevych, P. V. Філевич, П. В. Let M f(r) and μf(r) be, respectively, the maximum of the modulus and the maximum term of an entire function f and let Φ be a continuously differentiable function convex on (−∞, +∞) and such that x = o(Φ(x)) as x → +∞. We establish that, in order that the equality \(\lim \inf \limits_{r \to + \infty} \frac{\ln M_f (r)}{\Phi (\ln r)} = \lim \inf \limits_{r \to + \infty} \frac{\ln \mu_f (r)}{\Phi (\ln r)}\) be true for any entire function f, it is necessary and sufficient that ln Φ′(x) = o(Φ(x)) as x → +∞. Нехай $Mf(r)$ і $μf(r)$ — відповідно максимум модуля та максимальний член цілої функції $f$, а $Φ$ — неперервно диференційовна опукла на $(−∞, +∞)$ функція така, що $x = o(Φ(x)) Institute of Mathematics, NAS of Ukraine 2002-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4154 Ukrains’kyi Matematychnyi Zhurnal; Vol. 54 No. 8 (2002); 1149-1153 Український математичний журнал; Том 54 № 8 (2002); 1149-1153 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/4154/5017 https://umj.imath.kiev.ua/index.php/umj/article/view/4154/5018 Copyright (c) 2002 Filevych P. V. |
| spellingShingle | Filevych, P. V. Філевич, П. В. On the Growth of the Maximum of the Modulus of an Entire Function on a Sequence |
| title | On the Growth of the Maximum of the Modulus of an Entire Function on a Sequence |
| title_alt | Про зростання максимума модуля цілої функції на послідовності |
| title_full | On the Growth of the Maximum of the Modulus of an Entire Function on a Sequence |
| title_fullStr | On the Growth of the Maximum of the Modulus of an Entire Function on a Sequence |
| title_full_unstemmed | On the Growth of the Maximum of the Modulus of an Entire Function on a Sequence |
| title_short | On the Growth of the Maximum of the Modulus of an Entire Function on a Sequence |
| title_sort | on the growth of the maximum of the modulus of an entire function on a sequence |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4154 |
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