Inequalities involving Bessel and modified Bessel functions (Q921206)

The authors define a self-adjoint and compact operator whose eigenvalues are given by \(\pm 2/j_{\nu k}\), where \(j_{\nu k}\) is the kth positive zero of the Bessel function \(J_{\nu}(x)\) of the first kind. Using operator techniques the authors derive some lower and upper bounds for \(J_{\nu +1}(x)/J_{\nu}(x)\) and \(I_{\nu +1}(x)/I_{\nu}(x)\), where \(I_{\nu}(x)\) is a modified Bessel function of the first kind. The comparison between new and old inequalities shows the sharpness of the new bounds.