On a generalization of Jentzsch´s theorem

Kurzfassung/Abstract

Let $E$ be a compact subset of $\Bbb C$ with connected, regular complement $\varOmega = \overline {\Bbb C} \setminus E$ and let $G(z)$ denote Green's function of $\varOmega $ with pole at $\infty $. For a sequence $(p_n)_{n\in \Lambda }$ of polynomials with deg $p_n = n$, we investigate the value-distribution of $p_n$ in a neighbourhood $U$ of a boundary point $z_{0}$ of $E$ if $G(z)$ is an exact harmonic majorant of the subharmonic functions $$\frac{1}{n}\log |p_n(z)|, n \in \Lambda$$ in $\overline {\Bbb C} \setminus E$. The result holds for partial sums of power series, best polynomial approximations, maximally convergent polynomials and can be extended to rational functions with a bounded number of poles.