Close-to-starlike holomorphic functions of several variables

Let X be a finite dimensional complex normed linear space with unit ball B = {x 6 X: \\ x \\ < 1}. In this paper the notion of a close-to-starlike holomorphic mapping from B into X is defined. The definition is a direct generalization of W. Kaplan's notion of one dimensional close-to-convex functions. It is shown that close-to-starlike mappings of B into X are univalent and these mappings are given an alternate characterization in terms of subordination chains.