Some refinements of the Fekete and Szegö inequalities in one and higher dimensions

Abstract

In this paper, we introduce a class of holomorphic functions on unit disk . Let be a normalized holomorphic function on such that , and has a zero of order k + 1 at . By using the formula of Faà di Bruno for the kth derivative of a composite function, we establish the Fekete and Szegö inequality for , and then we extend this result to higher dimensions. The results presented here generalize some known results. Finally, a certain problem is also considered.