Univalence for convolutions (Q1907724)

Let \(S\), resp. \(C\) denote the class of univalent, resp. close-to-convex functions and \(*\) the Hadamard convolution of functions analytic in the unit disk. The author proves: 1) If \(f \in S\) and \(g \in C\) then \(f*g\) is univalent in \(\{z \mid |z |< 1 - \sqrt 3\}\). The result is sharp. 2) Denote by \(r_0\) the largest value for which \(f*g\) is univalent in \(\{z \mid |z |< r_0\}\) for all \(f,g \in S\). Then \((0.8) (2 - \sqrt 3) < r_0 \leq 2 - \sqrt 3\).