Bloch constant and variation of branch points (Q1089125)

Let F be the family of all homomorphic functions f in the unit disk D that are normalized by \(f(0)=0\) and \(f'(0)=1.\) The value B(f) is the least upper bound of the radii of all unramified disks contained in the Riemann image surface. The classical Bloch constant is defined by \(B=\inf \{B(f): f\in F\}\). The exact value of B is unknown, but Ahlfors and Grunsky obtained the upper bound \[ \Gamma (1/3)\Gamma (11/12)/(1+\sqrt{3})^{1/2}\Gamma (1/4)=0.47... \] by constructing the so-called Ahlfors-Grunsky function g. This upper bound is conjectured to be the actual value of B. By making use of a variation of branch points that is due to A. W. Goodman, the author proves that the Bloch radius B(f) attains a local minimum at the Ahlfors-Grunsky function g when its branch points are varied slightly.