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

If is a meromorphic function in the complex plane, R. Nevanlinna noted that its characteristic function (,) could be used to categorize according to its rate of growth as ||=→∞. Later H. Milloux showed for a transcendental meromorphic function in the plane that for each positive integer , (,()/)=((,)) as →∞, possibly outside a set of finite measure where denotes the proximity function of Nevanlinna theory. If is a meromorphic function in the unit disk ={∶||<1}, analogous results to the previous equation exist when limsup→1−((,)/log(1/(1−)))=+∞. In this paper, we consider the class of meromorphic functions in for which limsup→1−((,)/log(1/(1−)))<∞, lim→1−(,)=+∞, and (,′/)=((,)) as →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 limsup→1−((,)/log(1/(1−)))=+∞ holds. We also explore connections between the class and linear differential equations and values of differential polynomials and give an analogue to Nevanlinna's five-value theorem