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 ${z\colon |z| \leq r }$ 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 $\liminf_{r\to\infty} n(r)/A(r,f) \geq 1.07328$.