On the mean value theorem for analytic functions (Q1086382)

The author looks for a certain analogue of the mean value theorem for analytic functions defined in the unit disk \({\mathbb{D}}\) of the complex plane \({\mathbb{C}}\). Therefore, he defines for some region \(R\subset {\mathbb{C}}\) the family B(R) of analytic functions on \({\mathbb{D}}\) with f'(\({\mathbb{D}})\subset R\), and m(R) to be the supremum of all \(r<1\) for which \(f\in B(R)\) implies that \[ (f(b)-f(a))/(b-a)\in R,\quad | a|,| b| <r. \] One gets easily that \(m(R)=1\) if and only if R is convex, i.e. in this case the situation is quite similar to the real one. The author gives a necessary and sufficient condition on R for \(m(R)>0\). He shows that for R simply connected m(R)\(\geq 2-\sqrt{3}\) (which is the radius of convexity for univalent functions). In the further paper a variational method is developed which gives some information about the nature of a function \(f\in B(R)\) for which m(R) is actually the ''mean value radius''.