= Second coefficient bound for normalized univalent functions
{title2=$|a_2|\le2$}
The <odd square-root transform of a normalized univalent function> has reciprocal exterior expansion $1/h(1/\zeta)=\zeta-(a_2/2)\zeta^{-1}+\cdots$. Its first negative Laurent coefficient has modulus at most one by the <area theorem for univalent functions>, proving the bound. If $w$ is omitted by $f$, the normalized map $wf/(w-f)$ has second coefficient $a_2+1/w$. Applying the bound twice gives $|1/w|\le4$, the <Koebe quarter theorem>.
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