has an analytic extension around $\alpha$, then the proof of Theorem \[Main\] is absolutely trivial.

A norm estimate
===============

Several slightly different proofs of Julia-Wolff-Carathéodory Theorem can be found in the literature. An essential element in one of them is the Schwarz-Pick Lemma: an analytic self-map of the upper half-plane is a contraction with respect to a “good” metric on $\mathbb C^+$. In the next proposition, we obtain a similar result for noncommutative functions. We think that there is a rather striking resemblance between our result below and [@Mellon Corollary 3.3], but it is not clear to us yet whether the two results can be obtained from each other, or even to what extent they are related. We intend to pursue this question later.

\[prop:3.1\] Let $\mathcal A,\mathcal C$ be two von Neumann algebras and $f\colon H^+(\mathcal A)
\to H^+(\mathcal C)$ be a noncommutative map. For any $n\in\mathbb N$ and $a,c\in 
H^+(M_n(\mathcal A))$, the linear map $$M_n(\mathcal A)\ni b\mapsto\left(\Im f(a)\right)^{-\frac12}\Delta f
(a,c)\left((\Im a)^{\frac12}b(\Im c)^\frac12\right)
\left(\Im f(c)\right)^{-\frac12}\in M_n(\mathcal C)$$ is a complete contraction. In particular, $$\left\|\left(\Im f(a)\right)^{-\frac12}\Delta f
(a,c)(b)\left(\Im f(c)\right)^{-\frac12}\right\|_\mathcal C\leq
\left\|\left(\Im a\right)^{-\frac12}b\left(\Im c\right)^{-\frac12}\right\|_\mathcal A,$$ so that, by Equation , for $b=a-c$, $$\left\|\left(\Im f(a)\right)^{-\frac12}(f(a)-f(c))
\left(\Im f(c)\right)^{-\frac12}\right\|_\mathcal C\leq\left\|\left(
\Im a\right)^{-\frac12}(a-c)\left(\Im c\right)^{-\frac12}\right\|_\mathcal A.$$

