In the present paper, we investigate the eigenvalues of an elliptic differential operator on compact Riemannian manifolds with boundary and derive a general inequality for these eigenvalues. Applying this inequality, we give universal estimates for eigenvalues on compact domains of complete submanifolds in an Euclidean space, and of complete manifolds admitting special functions. Finally, we fi nd universal bounds on the (k + 1)-th eigenvalue on such objects in terms of the fi rst k eigenvalues independent of the domains.
