## 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 $$V^p(z_0; z_1, w_1)$$.

### MSC:

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

### Keywords:

Bloch functions; distortion estimates

Zbl 0663.30030