Complex product manifolds and bounds of curvature

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Let M=X× Y be the product of two complex manifolds of positive dimensions. In this paper, we prove that there is no complete Kähler 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+r²) 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+r²)⁻δ and the Ricci curvature is bounded below by -A(1+r²)γ where A, B, γ, δ are positive constants with γ+2δ<1. These are generalizations of some previous results, in particular the result of Seshadri and Zheng.