Ahlfors theory

Remarks

Example: for $f(z) = z$, $S(r) = \frac{1}{\pi}\int_0^r \frac{2\pi t}{(1+t^2)^2} dt = \frac{r^2}{1+r^2}$, which tends to $1$, the sheet number of a map that covers the sphere once. Here $L(r) = 2\pi r/(1 + r^2)$ tends to $0$, so the boundary becomes negligible. The Ahlfors–Shimizu characteristic $T_0(r) = \int_0^r S(t)/t \, dt$ differs from the Nevanlinna characteristic $T(r, f)$ by a bounded function. The main results compare, for disjoint regions $D_1, \dots, D_q$ of the sphere, the number of times the surface covers each $D_j$ with $S(r)$, with an error proportional to $L(r)$. For a transcendental meromorphic function on $\C$ the ratio $L(r)/S(r)$ tends to $0$ along a sequence of radii. This leads to the Ahlfors five islands theorem. Let $f$ be a transcendental meromorphic function on $\C$, and let $D_1, \dots, D_5$ be Jordan domains on the sphere with pairwise disjoint closures. Then $f$ maps some bounded simply connected domain in $\C$ conformally onto one of the $D_j$.