= Carathéodory derivative bound for the right half-plane
{c}
{title2=$|F'(0)|\le2\operatorname{Re}F(0)$}
If a <holomorphic function> $F$ on the <unit disc> has positive real part, the <Cayley transform between the half-plane and disk> $\phi=(F-F(0))/(F+\overline{F(0)})$ maps the disc into itself and vanishes at zero. The <Schwarz lemma> gives the displayed derivative bound. Equality holds exactly when $\phi(z)=\omega z$ with $|\omega|=1$, equivalently $F(z)=i\operatorname{Im}F(0)+\operatorname{Re}F(0)(1+\omega z)/(1-\omega z)$.
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