 Extremal derivative of a disc map into a symmetric annulus

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 $|\mathrm{Re}h|<L$. The function $\tan (\pi h/(4L))$ maps the disc into itself and fixes zero. The Schwarz lemma gives $|h^{\prime }(0)|\le 4L/\pi$. Equality holds precisely forHere $\arctan$ 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.