Geodesic
About a Pólya-Schiffer inequality
Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 65 (2011) no. 2.

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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 λ of the fixed membrane for any n the following inequality holds ∑_k=1^n1/λ_k≥∑_k=1^n1/λ_k^(σ), where λ_k^(σ) 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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Dittmar, Bodo; Hantke, Maren. About a Pólya-Schiffer inequality. Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 65 (2011) no. 2. http://geodesic.mathdoc.fr/item/AUM_2011_65_2_a1/

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