On the radius of univalence of certain regular functions (Q1209010)

If \(f\) is regular, normalized and univalent in the unit disc then \(f\in S^*_ c\) if \({\mathcal R}(zf'(z))/(f(z)+\overline{f(\bar z)})>0\). Further \(f\in S^*_{sc}\) if \({\mathcal R}(zf'(z))/(f(z)-\overline{f(\bar z)})>0\). Similar definitions hold for discs of other radii. It is shown that \(F\in S^*_ c\) (or \(S^*_{sc})\) implies that \(f(z)={1\over 2}(zF'(z))\) belongs to \(S^*_ c\) (or \(S^*_{sc})\) for \(| z|<{1\over 2}\), both results being sharp.