Sharp distortion estimates for $p$-Bloch functions

Abstract

Let $p \in (0, \infty)$ and ${\mathfrak 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 $z_0 \not= z_1$ and $|w_1| \leq 1/(1-|z_1|^2)^p$, put $V^p(z_0 ; z_1,w_1) $ be the variability region of $f'(z_0)$ when $f$ ranges over the class ${\mathfrak B}_1^p$ with $f'(z_1) = w_1$, i.e., $V^p(z_0 ; z_1,w_1) = \{ f'(z_0) : f \in {\mathfrak B}_1^p \; \textrm{and} \; f'(z_1) = w_1 \}$. In 1988 M. Bonk 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 distortion theorems for ${\mathfrak B}_{1}^1$ as corollaries. In the present article we shall refine Bonk's method and explicitly determine $V^p(z_0 ; z_1,w_1)$.