Some inequalities for eigenvalues of an elliptic differential operator

‎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 find universal bounds on  the (k+1)-th eigenvalue on such objects in terms of the first k eigenvalues independent of  the domains.