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Filevych, P. V.

doi:10.1023/a:1012378721807

Cited by 2 publications

(5 citation statements)

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“…In particular cases Problem 1 is solved in [2,4,5,8,6]. Denote by Ω * A the class of all function Φ ∈ Ω A , convex on [σ 0 , A).…”

Section: Problemmentioning

confidence: 99%

“…The solution of Problem 1, in the case of the sequence λ = (n) and an arbitrary function Φ ∈ Ω * A , was obtained practically in [2,4] for A = +∞ and in [5] for every A ∈ (−∞, +∞] (actually, the growth of power series was investigated in [2,4,5]). We state a result from [5] in the following equivalent formulation.…”

Section: Problemmentioning

confidence: 99%

“…and these inequalities are sharp (more precisely, for every A, B ∈ R such that A ≤ B ≤ A + τ(λ) there exists [3] a Dirichlet series F of the form (2) such that σ a (F) = A and β(F) = B).…”

Section: Introductionmentioning

confidence: 99%

“…If σ a (F) > −∞, then for each σ < σ a (F) let M(σ, F) = sup{|F(s)| : Re s = σ} be the maximum modulus of the series (2). If β(F) > −∞, then for each σ < β(F) let µ(σ, F) = max{|a n |e σλ n : n ∈ N 0 } be the maximal term of this series.…”

Section: Introductionmentioning

confidence: 99%

“…By D A (λ) we denote the class of all Dirichlet series of the form (2) A . The notion of Φ-type generalizes the classical notion of the type for entire Dirichlet series.…”

Section: Introductionmentioning

confidence: 99%

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Generalized types of the growth of Dirichlet series

Hlova

Filevych

2015

Carpathian Math. Publ.

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Let $A\in(-\infty,+\infty]$ and $\Phi$ be a continuously on $[\sigma_0,A)$ function such that $\Phi(\sigma)\to+\infty$ as $\sigma\to A-0$. We establish a necessary and sufficient condition on a nonnegative sequence $\lambda=(\lambda_n)$, increasing to $+\infty$, under which the equality$$\overline{\lim\limits_{\sigma\uparrow A}}\frac{\ln M(\sigma,F)}{\Phi(\sigma)}=\overline{\lim\limits_{\sigma\uparrow A}}\frac{\ln\mu(\sigma,F)}{\Phi(\sigma)},$$holds for every Dirichlet series of the form $F(s)=\sum_{n=0}^\infty a_ne^{s\lambda_n}$, $s=\sigma+it$, absolutely convergent in the half-plane ${Re}\, s<A$, where $M(\sigma,F)=\sup\{|F(s)|:{Re}\, s=\sigma\}$ and $\mu(\sigma,F)=\max\{|a_n|e^{\sigma\lambda_n}:n\ge 0\}$ are the maximum modulus and maximal term of this series respectively.

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“…In particular cases Problem 1 is solved in [2,4,5,8,6]. Denote by Ω * A the class of all function Φ ∈ Ω A , convex on [σ 0 , A).…”

Section: Problemmentioning

confidence: 99%

Section: Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…By D A (λ) we denote the class of all Dirichlet series of the form (2) A . The notion of Φ-type generalizes the classical notion of the type for entire Dirichlet series.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations