Extremal functions for Boch constants (Q1101576)

The author establishes an intriguing inequality for a function f that is holomorphic in the open unit disk \({\mathbb{D}}\) and extremal for any one of the following: the Bloch, Landau, locally schlicht Bloch or univalent Bloch constant. If f is such an extremal function, then \[ | \lim_{\epsilon \to 0}\frac{1}{\pi}\int_{\epsilon <| z| <1}\frac{\overline{f'(z)}}{z\quad 2f'(z)}dx dy| \leq 1. \] This inequality is established by making use of a quasiconformal variation of the domain of f and an affine transformation of the range.