Value distribution for a class of small functions in the unit disk (Q539376)

Summary: If \(f\) is a meromorphic function in the complex plane, R. Nevanlinna noted that its characteristic function \(T(r,f)\) can be used to categorize \(f\) according to its rate of growth as \(|z|=r\to \infty\). Later, H. Milloux showed for transcendental meromorphic functions in the plane that, for each positive integer \(k\), \[ m\bigg(r,\frac{f^{(k)}}{f}\bigg)= o\big(T(r,f)\big)\quad\text{as}\quad r\to \infty, \] 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 \[ \limsup_{r\to 1^-}\frac{T(r,f)}{\log\frac{1}{1-r}}=+\infty. \] In this paper, we consider the class of meromorphic functions \(\mathcal P\) in \(D\) for which \[ \limsup_{r\to 1^-}\frac{T(r,f)}{\log\frac{1}{1-r}}<+\infty,\quad \limsup_{r\to 1^-}T(r,f)=+\infty \] and \[ m\bigg(r,\frac{f}{f'}\bigg)=o\big(T(r,f)\big)\quad \text{as}\quad r\to 1. \] We explore the characteristics of this class and some places where functions in the class behave in a significantly different manner than those for which \[ \limsup_{r\to 1^-}\frac{T(r,f)}{\log\frac{1}{1-r}}=+\infty \] holds. We also explore connections between the class \(\mathcal P\) and linear differential equations and values of differential polynomials, and give an analogue to Nevanlinna's five-value theorem.