Asymptotic Location of the Zeros of the Faber Polynomials
DOI:
https://doi.org/10.5644/SJM.01.2.04Keywords:
Conformal mapping, exterior angle, capacityAbstract
Let $E$ be a compact set of the complex plane containing more than one point whose complement in the extended complex plane is simply connected. Let $\omega = \phi(z)$ map conformally Ext($E$) into $\vert\omega\vert>1$ and with $\phi(\infty)=\infty.$ The map $\phi(z)$ has the form $$\phi(z)=\frac{z}{c}+a_0+\frac{a_{-1}}{ z} +\frac{a_{-2}}{z^2}+ \cdots$$
The Faber polynomials for $E$, $\phi_n(z)$, consist of the polynomial part of $\phi(z)^n.$
For $\epsilon>0$ given let $$E_\epsilon:=\bigcup_{z\in
E}B(z,\epsilon),$$ where $B(z,\epsilon)$ denotes the disk of center $z$ and radius $\epsilon,$ and let $$B_\epsilon:=\mathrm{Br}(E_\epsilon).$$
Let
$$\alpha:=\inf_{z\in\mathrm{Br}(E)}~\left\{\theta;~\pi\theta\text{
is the exterior angle of $E$ at }z\right\}.$$ A typical result obtained in this work is the Theorem 3.2.
2000 Mathematics Subject Classification. 31E10, 41A10, 41A29, 41A40
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