Higher monotonicity properties and inequalities for zeros of Bessel functions

Abstract:

L. Lorch and P. Szegö have considered the sign-regularity of the higher differences (with respect to the rank $k$) of the sequence $\{ {c_{\nu k}}\}$ of positive zeros of the Bessel function ${C_\nu }(x)$. Our main purpose here is to extend one of their main results to the higher derivatives with respect to $\kappa$ when ${c_{\nu k}}$ is appropriately defined as a function of a continuous variable $\kappa$ rather than the discrete variable $k$, and the difference operator is replaced by a derivative operator. We also present some inequalities arising from these and other results.