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

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 lim supr→ 1− T r, f / log 1/ 1 − r ∞. In this paper, we consider the class of meromorphic functions P in D for which lim supr→ 1− T r, f / log 1/ 1 − r < ∞, limr→ 1−T r, f ∞, 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 lim supr→ 1− T r, f / log 1/ 1 − r ∞ 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.