  Let M be a simply-connected complete Kahler manifold whose sectional
curvature is bounded between two negative numbers. In this paper we prove the
existence of non-constant bounded holomorphic functions on M if the complex
dimension of M is greater or equal to three. Our proof uses bounded
plurisubharmonic exhaustion functions, the Cauchy-Riemann equations and uniform
Holder estimates for CR functions on geodesic spheres.
