Square-root improvement of a normalized conformal map ($|F^{\prime }(z_0)|/|f^{\prime }(z_0)|=(1+|a|)/(2\sqrt{|a|})$)

For a univalent function $f:\Omega \to \mathbb{D}$ with $f(z_0)=0$, suppose $a\ne 0$ 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.