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

Abstract EN

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 limsupr→1-(T(r,f)/log(1/(1-r)))=+∞.

In this paper, we consider the class of meromorphic functions P in D for which limsupr→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 limsupr→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.