Differential inequalities for the positive zeros of Bessel functions (Q914008)

The authors establish the following differential inequalities \[ \frac{dj_{\nu,k}}{d\nu}< \frac{j_{\nu,k}}{\nu+\ell+1}+ \frac{2}{j_{\nu,k}} \sum^{\ell-1}_{n=0} \{1- \frac{\nu+n+1}{\nu+\ell+1}\} h^ 2_{n,\nu+1} (\frac{1}{j_{\nu,k}}), \] \(\nu>-1\), \(k=1,2,...\), \(\ell =0,1,2,...\), where \(j_{\nu,k}\) and \(h_{n,\nu}(x)\), \(n\geq 0\), are the k-th zero of the Bessel function of the first kind and the Lommel polynomials, respectively. Some consequences of this result are studied. In particular, inequalities for \(j_{\nu,k}\) are obtained. Some of these are more stringent than the known bounds.