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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 [...] where λ(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.

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 $$\sum _{k=1}^n\frac{1}{{\lambda }_k}\ge \sum _{k=1}^n\frac{1}{{\lambda }_k^{\left(\sigma \right)}},$$ where ${\lambda }_k^{\left(\sigma \right)}$ 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.