  Based on a well known Sh.-T. Yau theorem we obtain that the real part of a
holomorphic function on a K\"{a}hler manifold with the Ricci curvature bounded
from below by $-1$ is contractive with respect to the distance on the manifold
and the hyperbolic distance on $(-1,1)$ inhered from the domain
$(-1,1)\times\mathbb{R}$. Moreover, in the case of bounded holomorphic
functions we prove that the modulus is contractive with respect to the distance
on the manifold and the hyperbolic distance on the unit disk.

Complex Variables