On Harrell-Stubbe Type Inequalities for the Discrete Spectrum of a Self-Adjoint Operator

We produce a new proof and extend results by Harrell and Stubbe for the discrete spectrum of a self-adjoint operator. An abstract approach--based on commutator algebra, the Rayleigh-Ritz principle, and an ``optimal usage of the Cauchy-Schwarz inequality--is used to produce ``parameter-free, ``projection-free