On a generalization of Jentzsch’s theorem

Let EE be a compact subset of CC with connected, regular complement Ω=C¯∖E and let G(z)G(z) denote Green’s function of ΩΩ with pole at ∞∞. For a sequence (pn)n∈Λ(pn)n∈Λ of polynomials with degpn=ndegpn=n, we investigate the value-distribution of pnpn in a neighbourhood UU of a boundary point z0z0 of EE if G(z)G(z) is an exact harmonic majorant of the subharmonic functions 1nlog|pn(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.