Computational Methods and Function Theory 8 (2008), No. 1, 107--119
Copyright Heldermann Verlag 2008
Farit G. Avkhadiev
farit.avhadiev@ksu.ru , Kazan State University, Chebotarev Research Institute, 420008 Kazan, Russia.
Karl-Joachim Wirths
kjwirths@tu-bs.de , TU Braunschweig, Institut für Analysis und Algebra, 38106 Braunschweig, Germany.
Let $\Omega$ be a proper convex subdomain of the complex plane $\mathbb{C}$. Let further $\Pi_1\subset\mathbb{C}$ be a compact convex set containing more than one point and $\Pi=\overline{\mathbb{C}}\setminus \Pi_1$. We denote by $R_{\Omega}(z)$ and $R_{\Pi}(w)$ the conformal radius of $\Omega$ at $z$ and of $\Pi$ at finite points $w$, respectively. We are concerned with the set $A(\Omega,\Pi)$ of functions $f\colon \Omega\to\Pi$ meromorphic on $\Omega$. We prove that for $n\geq 2$, $f\in A(\Omega,\Pi)$, $z\in \Omega$ and $f(z)$ finite the inequalities \[ \frac{|f^{(n)}(z)|}{n!}\frac {(R_{\Omega}(z))^n}{R_{\Pi}(f(z))} \leq \frac{(1+p)^{n-2}}{p^{n-1}}\sum_{k=0}^np^k \] are valid, where $p$ is a measure for the distance between $f(z)$ and the point at infinity. We give examples showing that equality is possible in this estimate.
Keywords: Convex domain, concave domain, $n$th derivative, conformal radius, subordination.
MSC 2000: 30C80, 30C55, 30C20.
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