= Extremal derivative of a disc map into a symmetric annulus
{title2=$\max|f\prime(0)|=4L/\pi$}
Let $L>0$ and consider <holomorphic functions> $f$ from the <unit disc> into $e^{-L}<|w|<e^L$, normalized by $f(0)=1$. Their largest possible <derivative> modulus is $4L/\pi$. Lift $f$ to its <holomorphic logarithm> $h$ with $h(0)=0$, so $|\operatorname{Re}h|<L$. The function $\tan(\pi h/(4L))$ maps the disc into itself and fixes zero. The <Schwarz lemma> gives $|h'(0)|\le4L/\pi$. Equality holds precisely for
$$
f_\theta(z)=\exp\left(\frac{4L}{\pi}\arctan(e^{i\theta}z)\right),\qquad\theta\in\mathbb R.
$$
Here $\arctan$ is its <holomorphic> branch vanishing at zero; $(4L/\pi)\arctan z$ is a <conformal map> from the disc onto the vertical strip. Thus the maximal <derivative> modulus is attained, with a circle of distinct extremizers rather than a unique map.
Back to article page
