Value Distribution for a Class of Small Functions in the Unit Disk

Abstract

If f is a meromorphic function in the complex plane, R. Nevanlinna noted that its characteristic function T(r, f) could be used to categorize f according to its rate of growth as |z | = r → ∞. Later H. Milloux showed for a transcendental meromorphic function in the plane that for each positive integer k, m(r, f(k)/f) = o(T(r, f)) as r → ∞, possibly outside a set of finite measure where m denotes the proximity function of Nevanlinna theory. If f is a meromorphic function in the unit disk D = {z:|z | < 1}, analogous results to the previous equation exist when . In this paper, we consider the class of meromorphic functions 𝒫 in D for which , , and m(r, f′/f) = o(T(r, f)) as r → 1. We explore characteristics of the class and some places where functions in the class behave in a significantly different manner than those for which holds. We also explore connections between the class 𝒫 and linear differential equations and values of differential polynomials and give an analogue to Nevanlinna′s five‐value theorem.