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March 2010 Sharp distortion estimates for $p$-Bloch functions
Takao Terada, Hiroshi Yanagihara
Hiroshima Math. J. 40(1): 17-36 (March 2010). DOI: 10.32917/hmj/1270645080


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


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Takao Terada. Hiroshi Yanagihara. "Sharp distortion estimates for $p$-Bloch functions." Hiroshima Math. J. 40 (1) 17 - 36, March 2010.


Published: March 2010
First available in Project Euclid: 7 April 2010

zbMATH: 1238.30024
MathSciNet: MR2642967
Digital Object Identifier: 10.32917/hmj/1270645080

Primary: 30D45
Secondary: 30C80

Keywords: Bloch function , distortion estimate , variability region

Rights: Copyright © 2010 Hiroshima University, Mathematics Program

Vol.40 • No. 1 • March 2010
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