= Boundary-distance derivative bound for a conformal bijection
{title2=$\widetilde d/(4d)\leq|f'(z)|\leq4\widetilde d/d$}
For a <conformal bijection> $f:D\to\widetilde D$ between proper planar <domains>, put $d=\operatorname{dist}(z,\partial D)$ and $\widetilde d=\operatorname{dist}(f(z),\partial\widetilde D)$. Apply the <Koebe quarter theorem> to $(f(z+dw)-f(z))/(df'(z))$ on the <unit disc> to obtain $d|f'(z)|/4\leq\widetilde d$. Apply it again to $f^{-1}$ to get the opposite bound. Thus $\boxed{\widetilde d/(4d)\leq|f'(z)|\leq4\widetilde d/d}$. The <domains> need not be <simply connected domains>. For the whole plane, the <boundary> distances are infinite and these ratios are undefined.
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