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

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