  By applying an average method in PDE, we obtain a dichotomy between
"constancy" and "infinity" of the warping functions on complete noncompact
Riemannian manifolds for an appropriate isometric immersion of a multiply
warped product manifold $N_1\times_{f_2} N_2 \times \cdots \times _{f_k} N_k\,
$ into a Riemannian manifold.
  Generalizing the earlier work of the authors in [{Glasg. Math. J. 51 (2009)
579-592], we establish sharp inequalities between the mean curvature of the
immersion and the sectional curvatures of the ambient manifold under the
influence of quantities of a purely analytic nature (the growth of the warping
functions). Several applications of our growth estimates are also presented.
