On the radius of convexity of linear combinations of univalent functions and their derivatives

Abstract

Key words Robinson's conjecture, Hadamard product, univalent functions, radius of convexity MSC (2000) Primary: 30C75 Let S denote the set of normalized univalent functions in the unit disk. We consider the problem of finding the radius of convexity rα of the setfor fixed α ∈ C. Using a linearization method we find the exact value of rα for α ∈ [0, 1] and prove the (sharp) estimate rα ≥ r1 for α ∈ C with |2α − 1| ≤ 1. As an application of these results the sharp lower bound for the radius of convexity of the convolution f * g where f, g ∈ S and g is close-to-convex is found to be 5 − 2 √ 6. The case α = 1/2 is related to an old conjecture of Robinson dating back to 1947.