The Multivalent Class of Geometrically Close-to-Convex Functions

Abstract

The class of univalent close-to-convex functions, K, was introduced by Kaplan [4] and first studied by him. The first important extension to the class of multivalent close-to-convex functions, K(p) where p is a positive integer, was considered by Livingston [7]. Somewhat later, Styer [15] introduced the more general class, Kw(p), of weakly close-to-convex functions by simply taking the closure of Livingston's class K(p) in the topology of locally uniform convergence in B = {z: |z| ≤ 1}.In 1936 Biernacki [2] introduced his class of linearly accessible functions. A function f is linearly accessible if f is univalent in B, f(0) = 0, and C – f(B) where C is the complex plane, is a union of closed (Euclidean) rays with disjoint interiors. In an interesting result, Lewandowski [6] showed that the classes of univalent close-to-convex functions and linearly accessible functions are equal.