We consider the second order differential equation y'' + ω^(2)ρ(x)y = 0 where ω is a positive
parameter. The principal concern here is to find conditions on the function
ρ^(−1/2)(x), which ensure that the consecutive differences of sequences constructed
from the zeros of a nontrivial solution of the equation are regular in sign for ω
sufficiently large. In particular, if c_(νk)(α) denotes the k-th positive zero of the general
Bessel (cylinder) function C_(ν)(x; α) = J_(ν)(x) cos α−Y_(ν)(x) sin α of order ν, and if |ν| < 1/2,
we prove that
(−1)^(m)∆^(m+2)c_(νk)(α) > 0 (m = 0, 1, 2, ...; k = 1, 2, ...),
where ∆a_(k) = a_(k+1) − a_(k). This type of inequalities was conjectured by Lorch and Szego in 1963.
We also show that the differences of the zeros of various orthogonal polynomials with higher
degrees possess the sign-regularity.