Support Points of the Class of Close-to-Convex Functions

Abstract

Let H(U) be the linear space of holomorphic functions on U = {z:|z|<1} endowed with the topology of compact convergence, and denote by H′(U) its topological dual space. Let be a compact subset of H(U) and ƒ∈F. We say ƒ is a support point of if there exists an L∈H'(U), non-constant on , such that On the other hand, ƒ is an extreme point of if ƒ is not a proper convex combination of two other points of .

Support points of some classes of analytic and univalent functions

“…set of all support points of a compact family F is denoted by supp F Support points of the families of starlike, convex functions have been studied in [7] and support points of close-to-convex functions have been studied in [12] and [19]. Support points for starlike and convex function of order α have been studied in [9].…”

On extreme points and support points of the family of starlike functions of orderα

“…We begin as in [7]. Suppose that / is a support point of 2(l~a) is a support point of S(St(a)) then φ(z) = xz for some \x\ = 1.…”

New support points of $\mathcal{S}$ and extreme points of $\mathcal{HS}$

“…Later E. Grassman, W. Hengartner, and G. Schober [7] proved that each support point of G is a function of the form (1). In [8] D. R. Wilken and R. Hornblower showed that each extreme point of % Q, is a support point of C. A natural question arises as to whether the functions (1) are support points of S or extreme points of DCS.…”