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We produce a new proof and extend results by Harrell and Stubbe for thediscrete spectrum of a self-adjoint operator. An abstract approach--based oncommutator algebra, the Rayleigh-Ritz principle, and an ``optimal'' usage ofthe Cauchy-Schwarz inequality--is used to produce ``parameter-free'',``projection-free'' versions of their theorems. We also analyze the strength ofthe various inequalities that ensue. The results contain classical bounds forthe eigenvalues. Extensions of a variety of inequalities \`a la Harrell-Stubbeare illustrated for both geometric and physical problems.

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