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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.
Highlights
Let D ⊂ R2 be a bounded connected domain
We prove a sharper version of this inequality for the fixed and free membrane problem and on the other hand, we are able to give formulas for the sum of all reciprocals containing only the coefficients of the series expansion of the conformal mapping
The aim of this section is to give a formula for the sum of all reciprocal eigenvalues of the fixed membrane problem for any bounded connected domain D
Summary
Let D ⊂ R2 be a bounded connected domain. We consider the following eigenvalue problems [1]: the eigenvalue problem of the fixed membrane. We prove a sharper version of this inequality for the fixed and free membrane problem and on the other hand, we are able to give formulas for the sum of all reciprocals containing only the coefficients of the series expansion of the conformal mapping. The eigenvalue problem of the fixed membrane (1) in a planar domain D is conformally equivalent to the following problem in the unit disk U. Let u(ko) be the eigenfunctions of the fixed membrane problem in the unit disk, λ(ko) the corresponding eigenvalues and let f (z) = z +a2z2 +. H2dA = 1 λn U λn for a function h satisfying the conditions given in the lemma. We will show that there exists a set of functions {hj}nj=1 satisfying the condition mentioned above and j hj =.
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