Convexity of a surface of conformal radius and estimates for the coefficients of a mapping function (Q2387104)

Let \(F(\omega)\), \(F(\infty)=\infty\), \(F'(\infty)>0\), be holomorphic in \(E^-\setminus\{\infty\}\), \(E^-=\{\omega: | \omega| >1\}\), and map \(E^-\) on a domain \(D\). The main result of the article is proved in Theorem 1: If a surface \(\Omega\) of a conformal radius \(R(D,F(\omega))=| F'(\omega)| (| \omega| ^2-1)\) is convex downwards over \(E^-\), then the boundary \(\partial D\) of \(D\) is a convex curve. A counterexample is given to show that the converse is not true.