An order-interpolation inequality for Bessel functions

We show that $J_{\mu +\nu }(r{)}^2<J_{\nu -1\mathrm{/}2}(r{)}^2+J_{\nu +1\mathrm{/}2}(r{)}^2$ holds whenever $\mu \in (-1\mathrm{/}2,1\mathrm{/}2)$, $\nu \in [0,\mathrm{\infty })$, and $r\in (0,\mathrm{\infty })$. 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 Schrödinger equation. Briefly, the optimal constant on ${\mathbb{R}}^{d+1}$ is at most twice that on ${\mathbb{R}}^d$ for each $d\ge 2$.