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

Let S denote the set of normalized univalent functions in the unit disk. We consider the problem of finding the radius of convexity r 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 α for α ∈ [0, 1] and prove the (sharp) estimate r α ≥ r 1 for α ∈ ℂ with |2 α – 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.