Published online by Cambridge University Press: 03 June 2019
Assume a point $z$ lies in the open unit disk
$\mathbb{D}$ of the complex plane
$\mathbb{C}$ and
$f$ is an analytic self-map of
$\mathbb{D}$ fixing 0. Then Schwarz’s lemma gives
$|f(z)|\leq |z|$, and Dieudonné’s lemma asserts that
$|f^{\prime }(z)|\leq \min \{1,(1+|z|^{2})/(4|z|(1-|z|^{2}))\}$. We prove a sharp upper bound for
$|f^{\prime \prime }(z)|$ depending only on
$|z|$.