A sharp form of the Ahlfors’ distortion theorem, with applications

Abstract

The constant appearing in the asymptotic version of the Ahlfors’ distortion theorem is 1 1 . Also it is shown that for mean 1 1 -valent functions f = z + a 2 z 2 ⋯ ‖ a n + 1 | − | a n ‖ ⩽ 1 f = z + {a_2}{z^2} \cdots \left \| {{a_{n + 1}}| - |{a_n}} \right \| \leqslant 1 for n ⩾ N ( f ) n \geqslant N(f) .