On a generalization of Jentzsch’s theorem

Abstract

Let E be a compact subset of C with connected, regular complement Ω = C ¯ ∖ E and let G ( z ) denote Green’s function of Ω with pole at ∞ . For a sequence ( p n ) n ∈ Λ 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 1 n log | p n ( z ) | , n ∈ Λ in C ¯ ∖ 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.