Monotonicity and Convexity Properties of Zeros of Bessel Functions

Abstract

It is shown that ${{j_{\nu k} } / \nu }$ decreases as $\nu $ increases, $0 < \nu < \infty $, and that ${{j_{\nu k}^2 } / \nu }$ and ${{dj_{\nu k}^2 } / {d\nu }}$ increase with $\nu $ for sufficiently large $\nu $, where $j_{\nu k} $ is the kth positive zero of the Bessel function $J_\nu (x)$. In particular, ${{j_{\nu 1}^2 } / \nu }$ and ${{dj_{\nu 1}^2 } / {d\nu }}$ increase for $3 \leqq \nu < \infty $. Some related results are proved for zeros of $J'_\nu (x)$, of cross-product Bessel functions and of modified Bessel functions of purely imaginary order.