Extremal problems for close-to-convex functions

Abstract

Let C(β),β  0, denote the family of normalized close-to-convex functions of order β. For β=1 this is the usual set of close-to-convex functions, which had been defined by Kaplan.

We study the family Sub C(β) of functions which are subordinate to close-to-convex functions of order β. For β  1 it is shown that the extreme points of the closed convex hull of Sub C(β) are of the form

. Further for all β 0 the coefficient problem is solved. Also for the family C _m(β) of m-fold symmetric close-to-convex functions of order β an extreme point result is given, if β≥ 1. For all β≥0 and arbitrary , the pth integral means of the derivatives are shown to be maximized by the function f with

. This shows in particular that f has a rectifiable boundary curve if m>2/(1-β). On the other hand it is shown that if m>4/(1-β) then f has furthermore a quasiconformal extension.

AMS No.:

• 30C45
• 30C50
• 30C75
• 30C80