Distortion theorems in the class Kn(D)

Abstract

This article presents the analysis of properties of functions belonging to the class Kn(D) earlier introduced by the author. This is a class of functions analytical in the domain D, for which the n-th divided di erence [F; zo; : : : ; zn] [...] 0 for any pairwise di erent points zo; : : : ; zn [...] D. For n = 1 the class K1(D) consisting of functions univalent in the domain D is obtained. The subclass ~Kn(E), formed by functions F(z) = zn + an+1zn+1 + [...] analytical in a unit circle E is separated from class Kn(D) and its properties are considered. The author touches classical and at the same time urgent questions, arising at the study of analytical functions, belonging to some class.