Abstract. Chen [3] established a sharp inequality for the warping function of a warped product submanifold in a Riemannian space form in terms of the squared mean curvature. Later, in [5], he studied warped product submanifolds in complex hyperbolic spaces. In the present paper, we establish an inequality between the warping function $f$ (intrinsic structure) and the squared mean curvature $\|H\|^2$ and the holomorphic sectional curvature $c$ (extrinsic structures) for warped product submanifolds $M_1\times_fM_2$ with $J{\cal D}_1\perp{\cal D}_2$ (in particular, $CR$-warped product submanifolds and $CR$-Riemannian products) in any complex space form $\widetilde M(c)$. Examples of such submanifolds which satisfy the equality case are given.

AMS Subject Classification (1991): 53C40, 53C42, 53B25

Keyword(s): Warped products, CR, -warped products, CR, -products, warping function

Received May 30, 2003, and in revised form September 22, 2003. (Registered under 5822/2009.)