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PreprintCC0arXiv

A proof of Askey's convexity conjecture

  • K. Castillo
  • S. Yakubovich
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Abstract

For −1<α≤1/2, let Jα be the Bessel function of the first kind and let jα,2 be its second positive zero. Define β(α)<α+1 by ∫0jα,2u−β(α)Jα(u) du=0. We prove that β''(α)>0 for −1<α≤1/2, including the left second derivative at α=1/2. The continuous extension β(−1)=0 is strictly convex on [−1,1/2], strengthening Askey's 1993 convexity conjecture. The analytic argument uses a positive expansion of a primitive and a vanishing weighted sum. Increasing ratios of consecutive weights give a negative covariance term, while one comparison controls slope and curvature. The remaining algebraic step proves eight rational inequalities by finite exact polynomial calculations, reproduced in an appendix.

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Cite

BibTeXRIS
BibTeX
@misc{castillo2026proof,
  title = {{A proof of Askey's convexity conjecture}},
  author = {K. Castillo and S. Yakubovich},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.10209},
  eprint = {2609.10209},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.10209},
  note = {CC0},
}
RIS
TY  - UNPB
TI  - A proof of Askey's convexity conjecture
AU  - K. Castillo
AU  - S. Yakubovich
PY  - 2026
DA  - 2026/09/09
PB  - arXiv
DO  - 10.48550/arxiv.2609.10209
UR  - https://arxiv.org/abs/2609.10209
AB  - For −1<α≤1/2, let J_α be the Bessel function of the first kind and let j_α,2 be its second positive zero. Define β(α)<α+1 by ∫_0^j_α,2u^−β(α)J_α(u) du=0. We prove that β''(α)>0 for −1<α≤1/2, including the left second derivative at α=1/2. The continuous extension β(−1)=0 is strictly convex on [−1,1/2], strengthening Askey's 1993 convexity conjecture. The analytic argument uses a positive expansion of a primitive and a vanishing weighted sum. Increasing ratios of consecutive weights give a negative covariance term, while one comparison controls slope and curvature. The remaining algebraic step proves eight rational inequalities by finite exact polynomial calculations, reproduced in an appendix.
ER  -