Computational Methods and Function Theory 8 (2008), No. 1, 243--259
Copyright Heldermann Verlag 2008
Vladimir V. Andrievskii
andriyev@math.kent.edu , Kent State University, Department of Mathematical Sciences, Kent, OH 44242, U.S.A.
Stephan Ruscheweyh
ruscheweyh@mathematik.uni-wuerzburg.de , Universität Würzburg, Institut für Mathematik, Am Hubland, 97074 Würzburg, Germany.
Let $G\in{\mathbb{C}}$ be a Jordan domain and $P$ a polynomial of degree at most $n$, satisfying $|P(z)|\leq1$ for $z\in G$ and $P(z_1)=0$, where $z_1\in\partial G$. If~$G$ is bounded by a quasiconformal curve we asymptotically estimate $|P(z_0)|$, where $z_0\in G$. In case $z_1$ is on a sufficiently smooth portion of $\partial G$, our results correspond to the previous ones by Hal\'asz for the special case $G=\mathbb{D}$, the unit disk. We also obtain complete results in case $z_1$ corresponds to a corner of~$\partial G$.
Keywords: Polynomial, zeros, quasiconformal curve.
MSC 2000: 30C10, 30C15, 41A10.
[FullText-pdf (309 K)] [FullText-ps (529 K)]