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About a Pólya-Schiffer inequality. (English) Zbl 1267.35141

Summary: For simply connected planar domains with the maximal conformal radius 1 it was proven in 1954 by G. Pólya and M. Schiffer [J. Anal. Math. 3, 245–245 (1954; Zbl 0056.32701)] that for the eigenvalues \(\lambda \) of the fixed membrane for any \(n\) the following inequality holds \[ \sum^n_{k=1}\frac{1}{\lambda_k}\geq \sum^n_{k=1}\frac{1}{\lambda^{(o)}_k}, \] where \(\lambda^{(o)}\) are the eigenvalues of the unit disk. The aim of the paper is to give a sharper version of this inequality and for the sum of all reciprocals to derive formulas which allow in some cases to calculate exactly this sum.

MSC:

35P15 Estimates of eigenvalues in context of PDEs
35P05 General topics in linear spectral theory for PDEs

Citations:

Zbl 0056.32701
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