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