Computational Methods and Function Theory 3 (2003), No. 2, 413--424
Copyright Heldermann Verlag 2003
Pavel Gumenuk
gumenuk@sgu.ru , Department of Mechanics and Mathematics, Saratov State University, Astrakhanskaya 83, 410012 Saratov, Russia.
Let $f$ be a meromorphic function, and $z_0\in\mathbb{C}$ its attracting fixed point with multiplier $\lambda\neq0$. In this paper we consider the problem of finding a lower estimate for the largest number $R(z_0,f)$, such that if the function $f$ is univalent and has no poles in the disk of radius $r_u$ centered at $z_0$, then $\lim_{n\to+\infty}f^n(z)=z_0$ whenever $|z-z_0|< r_u R(z_0,f)$. For every $\varsigma\in e^{i\mathbb{R}}$, $\varsigma\neq1$, and sufficiently small $\lambda\in\{\varsigma\mu:\,0<\mu<1\}$, the sharp estimate is obtained.
Keywords: Basin of attraction, attracting fixed point, univalence radius.
MSC 2000: Primary 37F10; Secondary 30D05, 30C55.
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