# Asymptotic estimates for analytic functions in strips and their derivatives

• G. I. Beregova Lviv Politechnic National University Lviv, Ukraine
• S. I. Fedynyak Ukrainian Catholic University Lviv, Ukraine
• P. V. Filevych Lviv Polytechnic National University
Keywords: analytic function, maximum modulus, Dirichlet series, convex function, Young-conjugate function

### Abstract

Let $-\infty\le A_0< A\le +\infty$, $\Phi$ be a continuous function on $[a,A)$ such that for every $x\in\mathbb{R}$ we have $x\sigma-\Phi(\sigma)\to-\infty$ as $\sigma\uparrow A$, $\widetilde{\Phi}(x)=\max\{x\sigma -\Phi(\sigma):\sigma\in [a,A)\}$ be the Young-conjugate function of $\Phi$, ${\Phi}_*(x)=\widetilde{\Phi}(x)/x$ for all sufficiently large $x$, and $F$ be an analytic function in the strip $\{s\in\mathbb{C}\colon A_0<\operatorname{Re}s<A\}$ such that the quantity $S(\sigma,F)=\sup\{|F(\sigma+it)|\colon t\in\mathbb{R}\}$ is finite for all $\sigma\in(A_0,A)$ and $F(s)\not\equiv0$. It is proved that if

\smallskip\centerline{$\ln S(\sigma,F)\le(1+o(1)\Phi(\sigma)$ as $\sigma\uparrow A$,}

\smallskip\noi then

\centerline{$\displaystyle \varlimsup_{\sigma\uparrow A}\frac{S(\sigma,F')}{S(\sigma,F){\Phi}_*^{-1}(\sigma)}\le c_0,$}

\smallskip\noi
where $c_0<1,1276$ is an absolute constant. From previously obtained results it follows that $c_0$ cannot be replaced by a constant less than $1$.

### Author Biographies

G. I. Beregova, Lviv Politechnic National University Lviv, Ukraine

Lviv Politechnic National University
Lviv, Ukraine

S. I. Fedynyak, Ukrainian Catholic University Lviv, Ukraine

Ukrainian Catholic University
Lviv, Ukraine

P. V. Filevych, Lviv Polytechnic National University

Lviv Polytechnic National University

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Published
2022-06-27
How to Cite
Beregova, G. I., Fedynyak, S. I., & Filevych, P. V. (2022). Asymptotic estimates for analytic functions in strips and their derivatives. Matematychni Studii, 57(2), 137-146. https://doi.org/10.30970/ms.57.2.137-146
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