On a subclass of close to convex functions

Abstract

Let C′(α,β) be the class of functions f ( z ) = z + ∑ n = 2 ∞ a n z n analytic in D = { z : | z | < 1 } , satisfying for some convex function g( z) with g(0)= g′(0) − 1 = 0 and for all z in D the condition | z f ' ( z ) g ( z ) - 1 z f ' ( z ) g ( z ) + ( 1 - 2 α ) | < β for some α, β (0≤α<1, 0<β≤ 1). A sharp coefficient estimate, distortion theorems and radius of convexity are determined for the class C′(α,β). The results extend the work of C. Selvaraj.