Distance between a maximum modulus point and the zero set of an analytic function

Author
S. I. Fedynyak1, P. V. Filevych2
1) Department of Applied Mathematics and Statistics Ukrainian Catholic University, Lviv, Ukraine; 2) Department of Computational Mathematics and Programming Lviv Polytechnic National University, Lviv, Ukraine
Abstract
Let $f$ be an analytic function in the disk $\mathbb{D}_R=\{z\in \mathbb{C}\colon |z|\le R\}$, $R\in (0,+\infty]$. A point $w\in\mathbb{D}_R$ is called a maximum modulus point of $f$ if $|f(w)|=M(|w|,f)$, where $M(r,f)=\max\{|f(z)|\colon |z|=r\}$. Denote by $d(w, f)$ the distance between a maximum modulus point $w$ and the zero set of $f$, i.e., $d(w,f)=\inf\{|w-z|\colon f(z)=0\}$. Let $\Phi$ be a continuous function on $[a,\ln R)$ such that $x\sigma-\Phi(\sigma)\to-\infty$, $\sigma\uparrow \ln R$, for every $x\in\mathbb{R}$. Let also $\widetilde{\Phi}$ be the Young-conjugate function of $\Phi$ and $\overline{\Phi}(x)=\widetilde{\Phi}(x)/x$ for all sufficiently large $x$. We prove that if $$ \ln M(r,f)\le (1+o(1))\Phi(\ln r),\quad r\uparrow R, $$ then $$ \varliminf_{|w|\uparrow R}d(w,f)\frac{\overline{\Phi}\,^{-1}(\ln |w|)}{|w|}\ge C_0, $$ where $C_0=0,5416\dots$. When the Taylor coefficients of $f$ are nonnegative, the constant $C_0$ can be replaced by $\pi$, and the inequality obtained in this case is sharp.
Keywords
analytic function; maximum modulus; maximum modulus point; zero set
DOI
doi:10.30970/ms.52.1.10-23
Reference
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Pages
10-23
Volume
52
Issue
1
Year
2019
Journal
Matematychni Studii
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