= Ahlfors second fundamental theorem
{c}
{title2=$\sum_j n_j\ge(q-2)S-hL$}
For a finite simply connected covering surface over the <Riemann sphere> and $q\ge3$ 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 https://www.math.purdue.edu/~eremenko/dvi/ahl.pdf ; the simply connected island estimate is also recorded in Lemma 2 of https://pdfs.semanticscholar.org/d247/562607e237af7c8fc6e683b77be6b94829fa.pdf .
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