On holomorphic functions on negatively curved manifolds

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