Journal of Inequalities and Applications
Volume 5 (2000), Issue 2, Pages 191-199
doi:10.1155/S1025583400000102
Spherical derivative of meromorphic function with image of finite spherical area
Department of Mathematics, Tokyo Metropofitan University, Minami-Osawa, Hachioji, Tokyo 192-0397, Japan
Received 21 April 1999; Revised 18 May 1999
Copyright © 2000 Shinji Yamashita. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Let Ω be a domain in the complex plane C with the Poincare metric PΩ(z)|dz| which is |dz|/(1−|z|2) if Ω is the open unit disk. Suppose that the Riemann sphere C∪{∞} of radius 1/2, so that it has the area π and let 0<β<π. Let αΩ,β(z), z∈Ω, be the supremum of the spherical derivative |f′(z)|/(1+|f(z)|2) of f meromorphic in Ω such that the spherical area of the image f(Ω)⊂C∪{∞} is not greater than β. Then
αΩ,β(z)≤βπ−βPΩ(z),
z∈Ω.
The equality holds if and only if Ω is simply connected.