This article is cited in 1 paper

Growth of entire functions represented by Dirichlet series

V. A. Oskolkov^a, L. I. Kalinichenko<sup>b

a</sup> Moscow Institute of Municipal Economy and Construction
^b Rostov State University

Abstract: Let, $\displaystyle F(z)=\sum _{n=1}^\infty a_ne^{\lambda _nz}$ be an entire function represented in the whole of the plane by an absolutely convergent Dirichlet series such that
$$ 0\leqslant \lambda _1<\lambda _2<\dotsb ,\qquad \varlimsup _{n\to \infty }\frac {\ln n}{\lambda _n}=\mu \in [0,+\infty ). $$
The connection between the growth of the quantity
$$ M(F;x)=\sup \bigl \{|F(x+iy)|:|y|<+\infty \bigr \},\qquad x\to +\infty. $$
End the behaviour of $|a_n|$ and $\lambda_n$ as $n\to \infty$ is described in general form.

UDC: 517.5

MSC: 30D15, 30B50

Received: 29.06.1995

DOI: 10.4213/sm168

 English version: Sbornik: Mathematics, 1996, 187:10, 1545–1560

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