Generalized types of the growth of Dirichlet series

  • T.Ya. Hlova Institute for Applied Problems of Mechanics and Mathematics, 3b Naukova str., 79060, Lviv, Ukraine
  • P.V. Filevych Vasyl Stefanyk Precarpathian National University, 57 Shevchenka str., 76018, Ivano-Frankivsk, Ukraine
Keywords: Dirichlet series, maximum modulus, maximal term, generalized type
Published online: 2015-12-19

Abstract


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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How to Cite
(1)
Hlova T., Filevych P. Generalized Types of the Growth of Dirichlet Series. Carpathian Math. Publ. 2015, 7 (2), 172-187.