Askey's convexity conjecture

For $-1<\alpha\leq \tfrac12$, let $J_\alpha$ denote the Bessel function of the first kind, let $j_{\alpha,2}$ be its second positive zero, and define $\beta(\alpha)<\alpha+1$ by $\int_0^{j_{\alpha,2}}u^{-\beta(\alpha)}J_\alpha(u)\,du=0$. Askey's convexity conjecture asserts that $\beta$ is convex as a function of $\alpha$ on $(-1,\tfrac12]$, equivalently that $\beta''(\alpha)\geq 0$ wherever the second derivative exists. The endpoint formulation considers the continuous extension $\beta(-1)=0$ and asks for convexity on $[-1,\tfrac12]$.