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PreprintCC0arXiv

Quantitative Pólya inequalities for annular sectors

  • Guowei Dai
  • Yingxin Sun
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Abstract

We prove explicit positive corrections to the Dirichlet and Neumann Pólya inequalities at every positive eigenvalue index of every annular sector. The coefficients depend only on the radii and are uniform in the aperture. Real-order angular integration preserves the exact area term, while directional rounding estimates and radial trials control all frequencies. Optimized bounds attain the sharp circular-arc coefficient for the continuous count. We identify the full sector perimeter asymptotic and a low-index obstruction to using its coefficient throughout the spectrum. The strict Neumann inequality also passes to products with bounded Lipschitz domains.

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Cite

BibTeXRIS
BibTeX
@misc{dai2026quantitative,
  title = {{Quantitative Pólya inequalities for annular sectors}},
  author = {Guowei Dai and Yingxin Sun},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.19461},
  eprint = {2610.19461},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.19461},
  note = {CC0},
}
RIS
TY  - UNPB
TI  - Quantitative Pólya inequalities for annular sectors
AU  - Guowei Dai
AU  - Yingxin Sun
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.19461
UR  - https://arxiv.org/abs/2610.19461
AB  - We prove explicit positive corrections to the Dirichlet and Neumann Pólya inequalities at every positive eigenvalue index of every annular sector. The coefficients depend only on the radii and are uniform in the aperture. Real-order angular integration preserves the exact area term, while directional rounding estimates and radial trials control all frequencies. Optimized bounds attain the sharp circular-arc coefficient for the continuous count. We identify the full sector perimeter asymptotic and a low-index obstruction to using its coefficient throughout the spectrum. The strict Neumann inequality also passes to products with bounded Lipschitz domains.
ER  -