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),
, 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.
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Ashbaugh, M.S. The universal eigenvalue bounds of Payne-Pólya-Weinberger, Hile-Protter, and H C Yang. Proc. Indian Acad. Sci. (Math. Sci.) 112, 3–30 (2002). https://doi.org/10.1007/BF02829638
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DOI: https://doi.org/10.1007/BF02829638