ON THE UNIFORM CONVERGENCE OF THErn GENERALIZED BIEBERBACH POLYNOMIALS INrnREGIONS WITH K-QUASICONFORMAL BOUNDARY

摘要

Let G be a finite domain in the complex plane with K-quasicon formal boundary, Z0 be an arbitrary fixed rnpoint in G and p＞0. Let ψ(z) be the con formal mapping from G onto the disk with radius r0＞0 and centered rnat the origin 0, mormalized by ψ(Z0)＝0 and ψ(Z0)＝1. Let us set ψp(Z):＝ 0[ψ ( )]2/ψdξ, and let πn.p(Z) be rnthe generalized Bieberbach polynomial of degree n for the pair (G,Z0) that minimizes the integral － P ( Z ) ｜pdσz in the class П of all pol ynonials of degree ≤ n and satisfying the conditions Pn (Z0) ＝ 0 and rnP (Z0) ＝ 1. In this work we prove the uniform convergence of the generalized Bieberbach polynomials rnK2＋ 1rnXa,p(Z) to ψp(z) on Gin case of p ＞ 2-