On the Meromorphic Functions in the Punctured Plane Without Multiple Values

Abstract

The study of meromorphic functions without multiple values in the plane started by F. Nevanlinna is extended to meromorphic functions in the punctured plane ${\mathbb {C}}^{{*}}.$ It is a classical result that a meromorphic function $f\left( z\right)$ can be obtained as quotient of solutions of the second order differential equation $u^{{\prime \prime }}+\left\{ f\left( z\right) ,z\right\} u=0,$ where $\left\{ f\left( z\right) ,z\right\}$ is the Schwarzian derivative of $f\left( z\right)$. In our hypothesis of meromorphic functions of finite order without multiple values in the puntured plane, the Schwarzian derivative $\left\{ f\left( z\right) , z\right\}$ turns out to be a rational function with only possible poles at 0 and $\infty$. In these conditions the asymptotic behaviour of $f\left( z\right)$ can be described by a result of Hille (Lectures on Ordinary Differential Equations in the complex plane, Addison Wesley, Boston, 1969) on ordinary differential equations in the complex plane. The results obtained are framed in the value distribution theory of meromorphic functions, in particular in the punctured plane we shall consider the work of Khrystiyanin and Kondratyuk (Mat Stud 23(1):19–30, 2005; Mat Stud 23(1):57–68, 2005) and Korhonen (Nevanlinna theory on an annulus. Value distribution theory and related topics. Adv. Complex analysis and applications, Kluwer Academic Publishers, Dordrecht, 2004).