  We compute estimates for eigenvalues of a class of linear second-order
elliptic differential operators in divergence form (with Dirichlet boundary
condition) on a bounded domain in a complete Riemannian manifold. Our estimates
are based upon the Weyl's asymptotic formula. As an application, we find a
lower bound for the mean of the first k eigenvalues of the drifting Laplacian.
In particular, we have extended for this operator a partial solution given by
Cheng and Yang for the generalized conjecture of P\'olya. We also derive a
second-Yang type inequality due to Chen and Cheng, and other two inequalities
which generalize results by Cheng and Yang obtained for a domain in the unit
sphere and for a domain in the projective space.
