= Square-root improvement of a normalized conformal map
{title2=$|F'(z_0)|/|f'(z_0)|=(1+|a|)/(2\sqrt{|a|})$}
For a <univalent function> $f:\Omega\to\mathbb D$ with $f(z_0)=0$, suppose $a\ne0$ is omitted. Apply the disk automorphism $T_a$ so the resulting image omits zero, take a <holomorphic square root>, then apply the disk automorphism taking the value at $z_0$ to zero. If $\Omega$ is a <simply connected domain>, the root exists. The resulting normalized univalent map has derivative magnitude multiplied by $(1+|a|)/(2\sqrt{|a|})>1$, since $(1-\sqrt{|a|})^2>0$. Thus a map maximizing the derivative cannot omit an interior disk point.
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