The refinement of Fekete and Szegö problems for close-to-convex functions and close-to-quasi-convex mappings

Abstract

In this paper, we define the subclass Ck+1 of close-to-convex functions on the unit disk U in C, for which we use a condition weaker than the class of k-fold symmetric close-to-convex functions introduced by Koepf in [9]. Next, we establish the Fekete and Szegö inequality for the class Ck+1, and then we extend this result to the unit ball of a complex Banach space. The results presented here generalize some classical results.