Higher Monotonicity Properties for Zeros of Certain Sturm-Liouville Functions

In this paper, we consider the differential equation $y^{″}+{\omega }^2\rho \left(x\right)y=0$, where $\omega$ is a positive parameter. The principal concern here is to find conditions on the function ${\rho }^{-1/2}\left(x\right)$ which ensure that the consecutive differences of sequences constructed from the zeros of a nontrivial solution of the equation are regular in sign for sufficiently large $\omega$. In particular, if $c_{\nu k}\left(\alpha \right)$ denotes the kth positive zero of the general Bessel (cylinder) function $C_{\nu }(x;\alpha )=J_{\nu }\left(x\right)cos\alpha -Y_{\nu }\left(x\right)sin\alpha$ of order $\nu$ and if $\left|\nu \right|<1/2$, we prove that ${(-1)}^m{\Delta }^{m+2}{c}_{\nu k}\left(\alpha \right)>0(m=0,1,2,\dots ;k=1,2,\dots ),$ where $\Delta a_k=a_{k+1}-a_k.$ This type of inequalities was conjectured by Lorch and Szego in 1963. In addition, we show that the differences of the zeros of various orthogonal polynomials with higher degrees possess sign regularity.