Ahlfors second fundamental theorem (${\sum }_j{n}_j\ge (q-2)S-hL$)

For a finite simply connected covering surface over the Riemann sphere and $q\ge 3$ fixed separated Jordan domains, let $n_j$ count its simply connected islands over the $j$th domain once each, regardless of mapping degree. Let $S$ be its average sheet number and $L$ its spherical boundary length. The displayed inequality holds with $h$ depending only on the target domains. For regular bordered surfaces $f:X\to Y$, a general formulation is $\max \{-\chi (X),0\}\ge -S\chi (Y)-hL$, where $L$ counts only the boundary mapped into the interior of $Y$ and $\chi$ is the modern Euler characteristic. The boundary error disappears for an actual branched covering. The covering-surface formulation is explained in www.math.purdue.edu/~eremenko/dvi/ahl.pdf ; the simply connected island estimate is also recorded in Lemma 2 of pdfs.semanticscholar.org/d247/562607e237af7c8fc6e683b77be6b94829fa.pdf .

Every nonconstant meromorphic function on the complex plane has a simple island over at least one of any five Jordan domains on the Riemann sphere with disjoint closures. The Ahlfors second fundamental theorem, area comparison over target domains, and a length-area exhaustion of the complex plane prove this: without a simple island every counted island has degree at least two, forcing $3S\le (5/2)S+O(L)$.