A proof of Askey's convexity conjecture

Abstract

For $-1<\alpha \le 1/2$, let $J_{\alpha }$ be the Bessel function of the first kind and let $j_{\alpha ,2}$ be its second positive zero. Define $\beta (\alpha )<\alpha +1$ by $${\int }_0^{j_{\alpha ,2}}{u}^{-\beta (\alpha )}{J}_{\alpha }(u)du=0.$$ We prove that ${\beta }^{\prime \prime }(\alpha )>0$ for $-1<\alpha \le 1/2$, including the left second derivative at $\alpha =1/2$. The continuous extension $\beta (-1)=0$ 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.