About a Pólya-Schiffer inequality

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