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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| Datum: | 1993 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Russisch Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
1993
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/5872 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Zusammenfassung: | 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. |
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