On the radius of convexity for a class of conformal maps

Abstract

Abstract. Let A denote the class of all analytic functions f in the open unit disc D in the complex plane satisfying f (0) = 0, fThe behaviour of functions in this class has been extensively studied in the literature. In this paper, we shall prove that no member of U 0 (λ) = {f ∈ U (λ) : f ′′ (0) = 0} is convex in D for any λ and obtain a lower bound for the radius of convexity for the family U 0 (λ). These results settle a conjecture proposed in the literature negatively. We also improve the existing lower bound for the radius of convexity of the family U 0 (λ).