Some Monotonicity Properties of Bessel Functions

It is proved that the sequence \[\left\{ {\int_{C_\nu k}^{C_\nu ,k + 1} {t^{\gamma - 1} \left| {\mathcal{C}_\nu (t)} \right|dt} } \right\}_{k = \kappa }^\infty \] is decreasing for all $\nu $, for $ - \infty < \gamma < \frac{3}{2}$, and for suitable $\varkappa $, where $C_\nu (t)$is an arbitrary Bessel function of order $\nu $ and $c_{\nu k} $ its kth positive zero. This subsumes and unifies results obtained by G. Szego and R. G. Cooke, extending and sharpening some. For one of his results Szego used a Sturm comparison theorem which is shown here to permit the requisite generalization and to incorporate and extend other results originally proved by quite different methods. Auxiliary results are derived. Various remarks are collected in the final section.