# About a Pólya-Schiffer inequality

Annales UMCS, Mathematica (2011)

- Volume: 65, Issue: 2, page 29-44
- ISSN: 2083-7402

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topBodo Dittmar, and Maren Hantke. "About a Pólya-Schiffer inequality." Annales UMCS, Mathematica 65.2 (2011): 29-44. <http://eudml.org/doc/267623>.

@article{BodoDittmar2011,

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 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.},

author = {Bodo Dittmar, Maren Hantke},

journal = {Annales UMCS, Mathematica},

keywords = {Membrane eigenvalues; sums of reciprocal eigenvalues; membrane eigenvalues},

language = {eng},

number = {2},

pages = {29-44},

title = {About a Pólya-Schiffer inequality},

url = {http://eudml.org/doc/267623},

volume = {65},

year = {2011},

}

TY - JOUR

AU - Bodo Dittmar

AU - Maren Hantke

TI - About a Pólya-Schiffer inequality

JO - Annales UMCS, Mathematica

PY - 2011

VL - 65

IS - 2

SP - 29

EP - 44

AB - 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.

LA - eng

KW - Membrane eigenvalues; sums of reciprocal eigenvalues; membrane eigenvalues

UR - http://eudml.org/doc/267623

ER -

## References

top- Bandle, C., Isoperimetric Inequalities and Applications, Pitman Publ., London, 1980. Zbl0436.35063
- Dittmar, B., Sums of reciprocal eigenvalues of the Laplacian, Math. Nachr. 237 (2002), 45-61. Zbl1200.35199
- Dittmar, B., Sums of free membrane eigenvalues, J. Anal. Math. 95 (2005), 323-332. Zbl1077.30008
- Dittmar, B., Eigenvalue problems and conformal mapping, R. Kühnau (ed.), Handbook of Complex Analysis: Geometric Function Theory. Vol. 2, Elsevier, Amsterdam, 2005, pp. 669-686. Zbl1091.35045
- Dittmar, B., Free membrane eigenvalues, Z. Angew. Math. Phys. 60 (2009), 565-568.[Crossref] Zbl1169.74027
- Hantke, M., Summen reziproker Eigenwerte, Dissertation Martin-Luther-Universität, Halle-Wittenberg, 2006.
- Henrot, A., Extremum problems for eigenvalues of elliptic operators, Birkäuser, Basel-Boston-Berlin, 2006. Zbl1109.35081
- Luttinger, J. M., Generalized isoperimetric inequalities, J. Mathematical Phys. 14 (1973), 586-593, ibid. 14 (1973), 1444-1447, ibid. 14 (1973), 1448-1450. Zbl0261.52006
- Pólya, G., Schiffer, M., Convexity of functionals by transplantation, J. Analyse Math. 3 (1954), 245-345. Zbl0056.32701
- Pólya, G., Szegö, G., Isoperimetric Inequalities in Mathematical Physics, Princeton University Press, Princeton, N. J., 1951. Zbl0044.38301

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