On a Subclass of Close-to-Convex Mappings

In this paper, the class of close-to-convex mappings of order \(\alpha \) is introduced in the unit ball of a complex Banach space, and then, we give the sharp distortion theorems for this class of mappings in the unit ball of a complex Hilbert space \(X\) or the unit polydisc in \(\mathbb {C}^n\) . As an application, a sharp growth theorem for close-to-convex mappings of order \(\alpha \) is obtained.