S. Yakubovich

A proof of Askey's convexity conjecture

For $-1<\alpha\leq1/2$, 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 $\beta''(\alpha)>0$ for $-1<\alpha\leq1/2$, including the left second deriv...