Results on meromorphic ϕ-normal functions

Abstract

Let ϕ : [0, 1) → (0, ∞) be an increasing function. A meromorphic function f in the unit disc is said to be ϕ-normal if its spherical derivative f #(z) ≔ |f′(z)|/(1 + |f(z)|2) satisfies f #(z) = 𝒪(ϕ(|z|)) as |z| → 1−. This article is devoted to the study of meromorphic ϕ-normal functions. In particular, an analogue of the Lohwater–Pommerenke theorem and several equivalent characterizations for ϕ-normal functions are established under certain regularity conditions on ϕ.