The universal eigenvalue bounds of Payne-Pólya-Weinberger, Hile-Protter, and H C Yang

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Abstract

In this paper we present a unified and simplified approach to the universal eigenvalue inequalities of Payne—Pólya—Weinberger, Hile—Protter, and Yang. We then generalize these results to inhomogeneous membranes and Schrödinger’s equation with a nonnegative potential. We also show that Yang’s inequality is always better than HileProtter’s (and hence also better than Payne—Pólya—Weinberger’s). In fact, Yang’s weaker inequality (which deserves to be better known), $$\lambda _{k + 1}< \left( {1 + \frac{4}{n}} \right)\frac{1}{k}\sum\limits_{i = 1}^k {\lambda _i } $$ , is also strictly better than Hile—Protter’s. Finally, we treat Yang’s (and related) inequalities for minimal submanifolds of a sphere and domains contained in a sphere by our methods.