An order-interpolation inequality for Bessel functions

We show that $J_{\mu + \nu}(r)^2 < J_{\nu-1/2}(r)^2 + J_{\nu+1/2}(r)^2$ holds whenever $\mu \in (-1/2, 1/2)$, $\nu \in [0, \infty)$, and $r \in (0, \infty)$. In fact, we prove a stronger version for any fixed non-trivial linear combination of the Bessel functions of the first and second kinds. This inequality can be regarded as a kind of interpolation with respect to order. As an application, we establish a dimension-comparison result for optimal constants of smoothing estimates for the free Schr\"{o}dinger equation. Briefly, the optimal constant on $\mathbb{R}^{d+1}$ is at most twice that on $\mathbb{R}^d$ for each $d \geq 2$.