About a Pólya-Schiffer inequality

Bodo Dittmar, Maren Hantke

Abstract


For simply connected planar domains with the maximal conformal radius 1 it was proven in 1954 by G. Pólya and M. Schiffer that for the eigenvalues \(\lambda\) of the fixed membrane for any \(n\) the following inequality holds \[\sum_{k=1}^n\frac{1}{\lambda_k}\geq \sum_{k=1}^n\frac{1}{\lambda_k^{(\sigma)}},\] where \(\lambda_k^{(\sigma)}\) 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.

Keywords


Membrane eigenvalues; sums of reciprocal eigenvalues

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References


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DOI: http://dx.doi.org/10.17951/a.2011.65.2.29-44
Data publikacji: 2016-07-27 21:54:08
Data złożenia artykułu: 2016-07-26 07:54:18

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