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

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.