We present an abstract
approach to universal inequalities for the discrete spectrum of a self-adjoint
operator, based on commutator algebra, the Rayleigh–Ritz principle, and one
set of “auxiliary” operators. The new proof unifies classical inequalities of
Payne–Pólya–Weinberger, Hile–Protter, and H.C. Yang and provides a Yang type
strengthening of Hook’s bounds for various elliptic operators with Dirichlet boundary
conditions. The proof avoids the introduction of the “free parameters” of many
previous authors and relies on earlier works of Ashbaugh and Benguria, and,
especially, Harrell (alone and with Michel), in addition to those of the other
authors listed above. The Yang type inequality is proved to be stronger
under general conditions on the operator and the auxiliary operators. This
approach provides an alternative route to recent results obtained by Harrell and
Stubbe.
