  We develop a new method for obtaining bounds on the negative eigenvalues of
self-adjoint operators B in terms of a Schatten norm of the difference of the
semigroups generated by A and B, where A is an operator with non-negative
spectrum. Our method is based on the application of the Jensen identity of
complex function theory to a suitably constructed holomorphic function, whose
roots are in one-to-one correspondence with the negative eigenvalues of B.
Applying our abstract results, together with bounds on Schatten norms of
semigroup differences obtained by Demuth and Van Casteren, to Schroedinger
operators, we obtain inequalities on moments of the sequence of negative
eigenvalues, which are different from the Lieb-Thirring inequalities.
