On a property of the entire dirichlet series with decreasing coefficients
The class $S_{Ψ}^{ *} (A)$ of the entire Dirichlet series $F(s) = \sum\nolimits_{n = 0}^\infty {a_n exp(s\lambda _n )}$ is studied, which is defined for a fixed sequence $A = (a_n ),\; 0 < a_n \downarrow 0,\sum\nolimits_{n = 0}^\infty {a_n< + \infty } ,$ by the conditions $0 ≤...
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| Дата: | 1993 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
1993
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/5872 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512094120574976 |
|---|---|
| author | Sheremeta, M. M. Шеремета, М. М. Шеремета, М. М. |
| author_facet | Sheremeta, M. M. Шеремета, М. М. Шеремета, М. М. |
| author_sort | Sheremeta, M. M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-19T09:19:41Z |
| description | The class $S_{Ψ}^{ *} (A)$ of the entire Dirichlet series $F(s) = \sum\nolimits_{n = 0}^\infty {a_n exp(s\lambda _n )}$ is studied, which is defined for a fixed sequence $A = (a_n ),\; 0 < a_n \downarrow 0,\sum\nolimits_{n = 0}^\infty {a_n< + \infty } ,$ by the conditions $0 ≤ λ_n ↗ +∞$ and $λ_n ≤ (1n^+(1/a_n ))$ imposed on the parameters $λ_n$, where $ψ $ is a positive continuous function on $(0, +∞)$ such that $ψ(x) ↑ +∞$ and $x/ψ(x) ↑ +∞$ as $x →+ ∞$.
In this class, the necessary and sufficient conditions are given for the relation $ϕ(\ln M(σ, F)) ∼ ϕ(\ln μ(σ, F))$ to hold as $σ → +∞$, where $M(\sigma ,F) = sup\{ |F(\sigma + it)|:t \in \mathbb{R}\} ,\mu (\sigma ,F) = max\{ a_n exp(\sigma \lambda _n ):n \in \mathbb{Z}_ + \}$, and $ϕ$ is a positive continuous function increasing to $+∞$ on $(0, +∞)$, forwhich $\ln ϕ(x)$ is a concave function and $ϕ(\ln x)$ is a slowly increasing function. |
| first_indexed | 2026-03-24T03:23:19Z |
| format | Article |
| fulltext |
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| id | umjimathkievua-article-5872 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T03:23:19Z |
| publishDate | 1993 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/14/d960425a2512f32c4e3edaac8f4b8014.pdf |
| spelling | umjimathkievua-article-58722020-03-19T09:19:41Z On a property of the entire dirichlet series with decreasing coefficients Об одном свойстве целых рядов Дирихле с убывающими коэффициентами Sheremeta, M. M. Шеремета, М. М. Шеремета, М. М. The class $S_{Ψ}^{ *} (A)$ of the entire Dirichlet series $F(s) = \sum\nolimits_{n = 0}^\infty {a_n exp(s\lambda _n )}$ is studied, which is defined for a fixed sequence $A = (a_n ),\; 0 < a_n \downarrow 0,\sum\nolimits_{n = 0}^\infty {a_n< + \infty } ,$ by the conditions $0 ≤ λ_n ↗ +∞$ and $λ_n ≤ (1n^+(1/a_n ))$ imposed on the parameters $λ_n$, where $ψ $ is a positive continuous function on $(0, +∞)$ such that $ψ(x) ↑ +∞$ and $x/ψ(x) ↑ +∞$ as $x →+ ∞$. In this class, the necessary and sufficient conditions are given for the relation $ϕ(\ln M(σ, F)) ∼ ϕ(\ln μ(σ, F))$ to hold as $σ → +∞$, where $M(\sigma ,F) = sup\{ |F(\sigma + it)|:t \in \mathbb{R}\} ,\mu (\sigma ,F) = max\{ a_n exp(\sigma \lambda _n ):n \in \mathbb{Z}_ + \}$, and $ϕ$ is a positive continuous function increasing to $+∞$ on $(0, +∞)$, forwhich $\ln ϕ(x)$ is a concave function and $ϕ(\ln x)$ is a slowly increasing function. У класі $S_{Ψ}^{ *} (A)$ цілих рядів Діріхле $F(s) = \sum\nolimits_{n = 0}^\infty {a_n exp(s\lambda _n )}$, який при фіксованій послідовності $A = (a_n ),\; 0 < a_n \downarrow 0,\sum\nolimits_{n = 0}^\infty {a_n< + \infty } ,$ визначається умовами $0 ≤ λ_n ↗ +∞$ і $λ_n ≤ (1n^+(1/a_n ))$ на показники $λ_n$, де $ψ $ - додатна неперервна на $(0, +∞)$ функція, $ψ(x) ↑ +∞$ і $x/ψ(x) ↑ +∞$ при $x →+ ∞$ вказані необхідна і достатня умови виконання співвідношення $ϕ(\ln M(σ, F)) ∼ ϕ(\ln μ(σ, F))$ при $σ → +∞$, де $M(\sigma ,F) = sup\{ |F(\sigma + it)|:t \in \mathbb{R}\} ,\mu (\sigma ,F) = max\{ a_n exp(\sigma \lambda _n ):n \in \mathbb{Z}_ + \}$, a $ϕ$ — додатна неперервна зростаюча до $+∞$ на $(0, +∞)$ функція така, що $\ln ϕ(x)$ вгнута, а $ϕ(\ln x)$ — повільно зростаюча функція. Institute of Mathematics, NAS of Ukraine 1993-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/5872 Ukrains’kyi Matematychnyi Zhurnal; Vol. 45 No. 6 (1993); 843–853 Український математичний журнал; Том 45 № 6 (1993); 843–853 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/5872/8427 https://umj.imath.kiev.ua/index.php/umj/article/view/5872/8428 Copyright (c) 1993 Sheremeta M. M. |
| spellingShingle | Sheremeta, M. M. Шеремета, М. М. Шеремета, М. М. On a property of the entire dirichlet series with decreasing coefficients |
| title | On a property of the entire dirichlet series with decreasing coefficients |
| title_alt | Об одном свойстве целых рядов Дирихле с убывающими коэффициентами |
| title_full | On a property of the entire dirichlet series with decreasing coefficients |
| title_fullStr | On a property of the entire dirichlet series with decreasing coefficients |
| title_full_unstemmed | On a property of the entire dirichlet series with decreasing coefficients |
| title_short | On a property of the entire dirichlet series with decreasing coefficients |
| title_sort | on a property of the entire dirichlet series with decreasing coefficients |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/5872 |
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