Bloch and normal functions on general planar regions

Abstract

Let Ω be a hyperbolic region in the complex plane λ_Ω the density of the hyperbolic metric on Ω Set δ_Ω (z) = dist(z, ∂Ω); l/δ_Ω is called the quasihyperbolic density on Ω. Roughly speaking, we show that holomorphic functions cannot distinguish between λ_Ω and l/δ_Ω while meromorphic functions sometimes can. More precisely, for a holomorphic function f on Ω the quantities |f′ (z)|/(z)/λ_Ω (z) and |f′ (z)|δ _Ω(z) are both either uniformly bounded on Ω (that is, fis a Bloch function) or unbounded. With the Euclidean derivative |f′| replaced by the spherical derivative f ^# = | f′|/(1+|f|²), Lehto and Virtanen have observed that the analogous result is generally false. However, we characterize those regions for which there exists a finite constant n = n(Ω) such that f ^#(z)δ_Ω(z) ≤ nf ^#(z)λ_Ω(z)≤ nf ^# (z)δ_Ω(z), z ∈ Ω, for any meromorphic function on Ω. In addition, we present another characterization of Bloch functions. A holomorphic function f on Ω is not a Bloch function if and only if there is a sequence {z _n} ^221E_n=1 zin Ω and a sequence {ρn} ^221E_n=1 of positive numbers such that ρn/δ_Ω (zn) → 0 and f (zn + ρnς) — f(zn) →aς where |a| = 1, locally uniformly son C.