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.