A Schwarz lemma for the modulus of a vector-valued analytic function

Abstract:

It is proved that \begin{equation*} |\nabla |f|(z)|\le \frac {1-|f(z)|^2}{1-|z|^2}, \quad z\in \mathbb D, \end{equation*} where $f:\mathbb D\mapsto \mathbb B_k$ is an analytic function from the unit disk $\mathbb D$ into the unit ball $\mathbb B_k\subset \mathbb C^k.$ Applications to the Lipschitz condition of the modulus of a $\mathbb C^k$-valued function are given.