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

摘要

Let S denote the set of normalized univalent functions in the unit disk. We consider the problem of finding the radius of convexity r{sub}α of the set {(1 - α)f(z) +αzf'(z): f ∈ S} for fixed α∈ C. Using a linearization method we find the exact value of r{sub}αfor α∈[0,1] and prove the (sharp) estimate r{sub}α≥r{sub}1 for a α∈C with |2a - 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{the square root of }6. The case α = 1/2 is related to an old conjecture of Robinson dating back to 1947.