Asymptotic estimates for analytic functions in strips and their derivatives

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