Asymptotic relation for zeros of cross-product of Bessel functions and applications

Let $a_{\nu,k}$ be the $k$-th positive zero of the cross-product of Bessel functions $J_\nu(R z) Y_\nu(z) - J_\nu(z) Y_\nu(R z)$, where $\nu\geq 0$ and $R>1$. We derive an initial value problem for a first order differential equation whose solution $\alpha(x)$ characterizes the limit behavior of $a_{\nu,k}$ in the following sense: $$ \lim_{k \to \infty} \frac{a_{kx,k}}{k} = \alpha(x), \quad x \geq 0. $$ Moreover, we show that $$ a_{\nu,k} < \frac{\pi k}{R-1} + \frac{\pi \nu}{2R}. $$ We use $\alpha(x)$ to obtain an explicit expression of the Pleijel constant for planar annuli and compute some of its values.