  Let $M=X\times Y$ be the product of two complex manifolds of positive
dimensions.
  In this paper, we prove that there is no complete K\"ahler metric $g$ on $M$
such that: either (i) the holomorphic bisectional curvature of $g$ is bounded
by a negative constant and the Ricci curvature is bounded below by $-C(1+r^2)$
where $r$ is the distance from a fixed point; or (ii) $g$ has nonpositive
sectional curvature and the holomorphic bisectional curvature is bounded above
by $-B(1+r^2)^{-\delta}$ and the Ricci curvature is bounded below by
$-A(1+r^2)^\gamma$ where $A, B, \gamma, \delta$ are positive constants with
$\gamma+2\delta<1$. These are generalizations of some previous results, in
particular the result of Seshadri and Zheng.
