Lemniscate zone and distortion theorems for multivalent functions. II
type: Item
instance of: scholarly article
title: Lemniscate zone and distortion theorems for multivalent functions. II
zbMATH Open document ID: 1417.30017
DOI: 10.1134/S0001434618110081
author: V. N. Dubinin
published in: Mathematical Notes
publication date: 2019-03-19T00:00:00Z
review text: The paper continues earlier author's works on the relationship between the connectivity of lemniscates of a holomorphic function and its distortion. Let $f$ be a nonconstant meromorphic function on the unit disk $U=\{\zeta:|\zeta|<1\}$, and let $n(w,f)$ denote the number of roots of the equation $f(\zeta)=w$ in $U$. The function $f$ is circumferentially mean $p$-valent if \[ \frac{1}{2\pi}\int_0^{2\pi}n(\rho e^{i\varphi},f)d\varphi\leq p \] for any $\rho\in(0,1)$. The set $L_f(t)=\{\zeta\in U:|f(\zeta)|=t\}$, $0\leq t\leq\infty$, is a lemniscate of $f$. The union $L_f(t_1,t_2)=\bigcup_{t\in(t_1,t_2)}L_f(t)$, $0\leq t_1<t_2$, is a lemniscate zone of the first kind if each $L_f(t)$, $t\in(t_1,t_2)$, is connected and its image under $f$ is a $p$-fold cover of $\gamma(t):=\{w:|w|=t\}$. The set $L_f(t_1,t_2)$ is a lemniscate zone of the second kind if each $L_f(t)$, $t\in(t_1,t_2)$, contains no connected components compactly embedded in $U$. Finally, $L_f(t_1,t_2)$ is a lemniscate zone of the third kind if each $L_f(t)$ either contains no connected components compactly embedded in $U$ or is connected and its $f$-image forms a $p$-fold cover of $\gamma(t)$. For a positive integer $p$ and $\varkappa\in(0,1)$, let $Z_p(z)\equiv Z_p(z;\varkappa)$ be defined as \[ Z_p(\text{sn}(u;k);\varkappa):=\text{sn}(u\mathbf{K}(\varkappa)/\mathbf{K}(k);\varkappa),\;\;u\in\mathbb C,\;\;0<k<1, \] where the module $k$ is determined by $\mathbf{K}'(k)\mathbf{K}(\varkappa)=p\mathbf{K}'(\varkappa)\mathbf{K}(k)$, and $\mathbf{K}(\cdot)$ and $\mathbf{K}'(\cdot)$ are complete elliptic integrals of the first kind. Denote $F(z)\equiv F(z;p,\tau)=\Phi(Z_p(z;\varkappa))$, $p>1$, $\tau>1$, \[ \Phi(v)=\sqrt{\tau}\frac{v\sqrt{\varkappa}+1}{1-v\sqrt{\varkappa}},\;\;\;\sqrt{\varkappa}=\frac{\sqrt{\tau}-1}{\sqrt{\tau}+1}. \] All zeros of $F$ are simple and negative. Let $\alpha$ and $\beta$ denote the least and the greatest zero of $F$, respectively. The zeros of $F$ are numbered in the order of their occu…
reviewed by: Dmitri V. Prokhorov
Mathematics Subject Classification ID: 30D30
Mathematics Subject Classification ID: 30F25
zbMATH DE Number: 7037783
zbMATH Keywords: meromorphic function
zbMATH Keywords: \(p\)-valent function
zbMATH Keywords: lemniscate
zbMATH Keywords: Zolotarev fraction
zbMATH Keywords: symmetrization
MaRDI profile type: Publication
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full work available at URL: s0001434618110081
OpenAlex ID: W2907545856
Wikidata QID: Q128688952
description: scientific article; zbMATH DE number 7037783