@article{Fedynyak_Filevych_2020, title={Growth estimates for a Dirichlet series and its derivative}, volume={53}, url={http://matstud.org.ua/ojs/index.php/matstud/article/view/ms.53.1.3-12}, DOI={10.30970/ms.53.1.3-12}, abstractNote={<p>Let $A\in(-\infty,+\infty]$, $\Phi$ be a continuous function on $[a,A)$ such that for every $x\in\mathbb{R}$ we have<br>$x\sigma-\Phi(\sigma)\to-\infty$ 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 $\operatorname{Re}s&lt;A$, $M(\sigma,F)=\sup\{|F(s)|\colon \operatorname{Re}s=\sigma\}$ and $G(\sigma,F)=\sum |a_n|e^{\sigma\lambda_n}$ for each $\sigma&lt;A$. It is proved that if $\ln G(\sigma,F)\le(1+o(1))\Phi(\sigma)$, $\sigma\uparrow A$, then the inequality<br>$$<br>\varlimsup_{\sigma\uparrow A}\frac{M(\sigma,F’)}{M(\sigma,F)\overline{\Phi}\,^{-1}(\sigma)}\le1<br>$$<br>holds, and this inequality is sharp. % Abstract (in English)</p&gt;}, number={1}, journal={Matematychni Studii}, author={Fedynyak, S.I. and Filevych, P.V.}, year={2020}, month={Mar.}, pages={3-12} }