Maximum and average valence of meromorphic functions

Abstract

If $f$ is a meromorphic function from the complex plane $\mathbb{C}$ to the extended complex plane $\overline{\mathbb{C}}$, for $r>0$ let $n(r)$ be the maximum number of solutions in $\{z:|z|\le r\}$ of $f(z)=a$ for $a\in \overline{\mathbb{C}}$, and let $A(r,f)$ be the average number of such solutions. Using a technique introduced by Toppila, we exhibit a meromorphic function for which ${\mathrm{liminf}}_{r\to \infty }n(r)/A(r,f)\ge 1.07328$.