Sharp distortion estimates for \(p\)-Bloch functions. (English) Zbl 1238.30024

Let \(p\in (0,\infty)\) and \(\mathcal B_1^p\) be the class of analytic functions \(f\) in the unit disk \(\mathbb D\) with \( f(0)=0\) satisfying \(|f'(z)|\leq 1/(1-|z|^2)^p\). For \(z_0, z_1\in \mathbb D\), \(w_1\in \mathbb C\) with \(z_0\neq z_1\) and \(|w_1|\leq 1/(1-|z|^2)^p\), let \(V^p(z_0; z_1, w_1)\) be the variability region of \(f'(z_0)\) when \(f\) ranges over the class \(\mathcal B_1^p\) with \(f'(z_1)=w_1\), i.e., \(V^p(z_0; z_1, w_1)=\{f'(z_0): f\in \mathcal B_1^p\text{ and } f'(z_1)=w_1\}\). In 1988, M. Bonk [Extremalprobleme bei Bloch-Funktionen. Braunschweig (FRG): Technische Univ. Braunschweig, Naturwissenschaftliche Fakultät (1988; Zbl 0663.30030)] showed that \(V^1(z_0; z_1, w_1)\) is a convex closed Jordan domain and determined it by giving a parametrization of the simple closed curve \(\partial V^1(z_0; z_1, w_1)\). He also derived a distortion theorem for \(\mathcal B_1^1\) as a corollary. In the paper under review, the authors refine Bonk’s method and determine \(V^p(z_0; z_1, w_1)\).


30D45 Normal functions of one complex variable, normal families
30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination


Zbl 0663.30030