Ratios of Bessel functions and roots of \(\alpha J_v(x)+xJ_v'(x)=0\)
The author shows that \(\zeta J_\nu(\nu \zeta)/J_{\nu+1} (\nu \zeta)\) is stricty decreasing in \(\nu\) on any interval not containing a singularity and obtaining an upper bound on its derivative: \[ {\partial \over \partial\nu} \left[{J_\nu (\nu\zeta)\over J_{\nu+1} (\nu\zeta)} \right]\leq-{2 \over \nu^2}. \] He also shows that \(x{\partial\over \partial v}[{J\nu(x) \over J_{\nu+1} (x)}]\geq 2\) where \(xJ_\nu(x)/J \nu+1(x)\) is strictly increasing in \(\nu\). The graph of \(J_\nu(x)/J_{\nu+1}(x)\) as a function of \(x\) is illustrated in figures for various values of \(\nu\). These results generalize and sharpen previously known results, and allow the deduction of a more complete description, including monotonicity and multiplicity, of the positive roots of the equation \(\alpha J\nu(x)+xJ_\nu'(x)=0\) for all real \(\nu\) and \(\alpha\).
scientific article; zbMATH DE number 1398604
scientific article