Estimates of spherical derivative of meromorphic functions

Abstract

The spherical derivative f # = ‖f'‖/(1 + ‖f‖ 2 ) of f meromorphic in D = {‖z‖ < 1} is estimated from above and below in terms of various geometrical quantities, for example, δ # (z, f), p(z, f), and ρ a u (z, f), in several theorems. A necessary and sufficient condition for (1 - ‖z‖ 2 )f # (z) to be bounded in D is that there exists r, 0 < r < 1, such that f(w) ≠ -1/f(z) for all z, w ∈ D satisfying ‖w-z‖/‖1-zw‖ < r. Also, (1-‖z‖ 2 )f # (z) is bounded in D if and only if δ # (z, f)/ρ a u (z, f) is bounded in D minus the points z where f # (z) = 0. Applications to evaluating the Poincare density in a plane domain will be considered.