Angular derivatives for holomorphic self-maps of the disk
type: Item
instance of: scholarly article
title: Angular derivatives for holomorphic self-maps of the disk
zbMATH Open document ID: 1393.30040
author: Jochen Becker
author: Christian Pommerenke
published in: Computational Methods and Function Theory
publication date: 2017-12-22T00:00:00Z
review text: Let \(\varphi\) be a holomorphic function in the unit disk \(\mathbb{D}\) with \(\varphi(\mathbb{D})\subset \mathbb{D}\). Local properties of such functions near points where the angular derivative exists are studied. In particular, the following results are obtained. Theorem 2.4. Let \(\varphi\) have finite angular derivatives at \(\zeta_k\in \partial \mathbb{D}\), \(\omega_k:=\varphi(\zeta_k)\in \partial \mathbb{D}\), \(\alpha_k= |\varphi'(\alpha_k)|\), and let \[ q_k:=\frac1{\sqrt{\alpha_k\alpha_{k+1}}} \Bigl| \frac{\omega_k-\omega_{k+1}}{\zeta_k-\zeta_{k+1}} \Bigr|\quad\text{for}\quad k=1,2,3 \] with \(k=1\) instead of \(k=4\). Then, we have \[ q_1^2+q_2^2+q_3^2\leq 1+2q_1q_2q_3. \] Theorem 5.1. Let \(\zeta \in \partial \mathbb{D}\) and let the angular derivative \(\varphi'(\zeta)\neq \infty\) exist. If \(H\) is any horocycle at \(\zeta\), then \(\varphi(z)\to \varphi(\zeta)\) as \(z\to \zeta\), \(z\in H\). Theorem 5.1 generalized results by \textit{G. T. Cargo} [Can. J. Math. 14, 334--348 (1962; Zbl 0103.29802)] and \textit{P. R. Ahern} and \textit{D. N. Clark} [Mich. Math. J. 21, 115--127 (1974; Zbl 0277.30027)] proved for Blaschke products.
Mathematics Subject Classification ID: 30H05
Mathematics Subject Classification ID: 30E25
zbMATH DE Number: 6821108
zbMATH Keywords: bounded holomorphic functions
zbMATH Keywords: angular limits
zbMATH Keywords: angular derivatives
reviewed by: Igor Chyzhykov
MaRDI profile type: Publication
full work available at URL: s40315-017-0199-x
OpenAlex ID: W2608560347
cites work: On inner functions with \(H^p\)-derivative
cites work: Q4043323
cites work: On the images of horodisks under holomorphic self-maps of the unit disk
cites work: Inequalities for angular derivatives and boundary interpolation
cites work: Angular and Tangential Limits of Blaschke Products and their Successive Derivatives
cites work: Analytic flows on the unit disk: angular derivatives and boundary fixed points
cites work: Digons and angular derivatives of analytic self-maps of the unit disk
cites work: Q4888816
cites work: Inequalities for the Angular Derivative of an Analytic Function in the Unit Disk
cites work: Dynamic extension of the Julia-Wolff-Carathéodory theorem
cites work: Q4680828
cites work: On normal and automorphic functions
cites work: Q4000499
cites work: Angular derivatives at boundary fixed points for self-maps of the disk
cites work: Composition operators and classical function theory
cites work: On Distortion Under Bounded Univalent Functions with the Angular Derivative Fixed
DOI: 10.1007/S40315-017-0199-X
description: scientific article; zbMATH DE number 6821108