An upper bound for the zeros of the derivative of Bessel functions (Q1360191)

The aim of this paper is to establish a new upper bound for the positive zeros \(j_{\nu k}'\) contained in the following theorem: Let \(j_{\nu k}'\) denote the \(k\)-th positive zero of the derivative of Bessel functions \((d/dx)J_\nu(x)= J_\nu'(x)\) for \(\nu>0\), and \(k=1,2,\dots\;\). Then \(j_{\nu k}'<F_k(\nu)\), where \[ F_k(\nu)= \nu+a_k(\nu+A_k^3/a_k^3)^{1/3}+ {\textstyle\frac{3}{10}} a_k^2(\nu+A_k^3/a_k^3)^{-1/3}, \] and \[ A_k= {\textstyle\frac23} a_k\sqrt{2a_k}, \qquad a_k=2^{-1/3}x_k' \] and \(x_k'\) is the \(k\)-th positive zero of the derivative \(A_i'(x)\) of the Airy function. The bound is sharp for large values of \(\nu\) and improves known results. A similar inequality holds for the \(k\)-th positive zero \(y_{\nu k}'\) of \(Y_\nu'(x)\), too, where \(Y_\nu(x)\) denotes the Bessel function of the second kind.