Computational Methods and Function Theory 3 (2003), No. 1, 127--141
Copyright Heldermann Verlag 2003
Farit G. Avkhadiev
nina@dionis.kfti.kcn.ru, Chebotarev Research Institute, Kazan State University, 420008 Kazan, Russia.
Karl-Joachim Wirths
kjwirths@tu-bs.de, Institut für Analysis, Technische Universität Braunschweig, 38106 Braunschweig, Germany.
Let $\Omega$ and $\Pi$ be two simply connected domains in the complex plane $\mathbb{C}$ which are not equal to the whole plane $\mathbb{C}$. We consider functions $f\colon \Omega \to \Pi$ analytic in $\Omega$ and we get estimates for $|f^{(n)}(z)|$, $z \in \Omega$, which are sharp in the following sense. Let $\lambda_{\Omega}(z)$ and $\lambda_{\Pi}(w)$ denote the reciprocal of the conformal radius of $\Omega$ in $z$ and of $\Pi$ in $w$, respectively. Inequalities of the type $$ \frac{|f^{(n)}(z)|}{n!}\leq M_n(z,\Omega,\Pi)\frac{(\lambda_{\Omega}(z))^n}{\lambda_{\Pi}(f(z))}, \qquad z\in \Omega, $$ are considered where $M_n(z,\Omega, \Pi)$ does not depend on $f$ and represents the smallest value possible at this place. We especially consider cases where $ \Omega$ or~$\Pi$ is an angular domain $H_{\alpha}$ with opening angle $\alpha \pi$, $1\leq \alpha \leq 2$. We determine $M_n(z,\Delta,H_{\alpha})$ where $\Delta$ denotes the unit disk. Furthermore, we prove identities and inequalities for $$ C_n(\Omega, \Pi) := \sup\{M_n(z,\Omega, \Pi) \mid\, z\in \Omega\} $$ for several cases where $H_{\alpha}$ plays the role of $\Omega$ or $\Pi$.
Keywords: Angular domains, derivatives of arbitrary order, conformal radius.
MSC 2000: Primary 30C80; Secondary 30C55.
[FullText-pdf (356 K)] [FullText-ps (284 K)]