  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.
