# Growth estimates for a Dirichlet series and its derivative

• S.I. Fedynyak Ukrainian Catholic University, Lviv, Ukraine
• P.V. Filevych Lviv Polytechnic National University, Lviv, Ukraine
Keywords: analytic function, maximum modulus, maximum modulus point, zero set

### Abstract

Let $A\in(-\infty,+\infty]$, $\Phi$ be a continuous function on $[a,A)$ such that for every $x\in\mathbb{R}$ we have
$x\sigma-\Phi(\sigma)\to-\infty$ as $\sigma\uparrow A$, $\widetilde{\Phi}(x)=\max\{x\sigma -\Phi(\sigma)\colon \sigma\in [a,A)\}$ be the Young-conjugate function of $\Phi$, $\overline{\Phi}(x)=\widetilde{\Phi}(x)/x$ for all sufficiently large $x$, $(\lambda_n)$ be a nonnegative sequence increasing to $+\infty$, $F(s)=\sum a_ne^{s\lambda_n}$ be a Dirichlet series absolutely convergent in the half-plane $\operatorname{Re}s<A$, $M(\sigma,F)=\sup\{|F(s)|\colon \operatorname{Re}s=\sigma\}$ and $G(\sigma,F)=\sum |a_n|e^{\sigma\lambda_n}$ for each $\sigma<A$. It is proved that if $\ln G(\sigma,F)\le(1+o(1))\Phi(\sigma)$, $\sigma\uparrow A$, then the inequality
$$\varlimsup_{\sigma\uparrow A}\frac{M(\sigma,F')}{M(\sigma,F)\overline{\Phi}\,^{-1}(\sigma)}\le1$$
holds, and this inequality is sharp. % Abstract (in English)

### References

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P.V. Filevich, On influence of the arguments of coefficients of a power series expansion of an entire function on the growth of the maximum of its modulus, Sib. Mat. Zh., 44 (2003), №3, 674–685. (in Russian) English translation in: Siberian Math. J., 44 (2003), №3, 529–538.

T.Ya. Hlova, P.V. Filevych, The growth of analytic functions in the terms of generalized types, J. Lviv Politech. Nat. Univ., Physical and mathematical sciences, (2014), №804, 75–83. (in Ukrainian)

Published
2020-03-17
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Fedynyak S, Filevych P. Growth estimates for a Dirichlet series and its derivative. Mat. Stud. [Internet]. 2020Mar.17 [cited 2021Nov.28];53(1):3-12. Available from: http://matstud.org.ua/ojs/index.php/matstud/article/view/ms.53.1.3-12
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