Higher monotonicity properties of certain Sturm-Liouville functions

A Bessel function inequality connected with stability of least square smoothing, II

“…The question arises as to whether the inequality (1) can be extended to additional values of v or improved in other ways. That it is valid not only for v ^ f, as shown here, but also for v ^ \ can be established, but to do so requires a substantial extension of certain results of [1] used here. This has been done and will be published elsewhere.…”

“…The proof here, valid for all v jg $, is based on higher monotonicity properties of Bessel functions as discussed in [1] and [2]. As there, let…”

Higher monotonicity properties and inequalities for zeros of Bessel functions

“…Introduction L. Lorch and P. Szegö [6] initiated the study of the sign-regularity of higher differences with respect to the rank k, of the sequence {cpk} = {cvk(a)) of positive zeros of the cylinder function %(x) = %(x> a) = cosaJv(x) -sinaYv(x), 0 < a < n.…”

On the zeros of some special functions: Differential equations and nicholson-type formulas

“…Other methods are related to integral representations for the functions and seem to be provable, though not easily discoverable, by differential equations methods. Among these are methods based on Nicholson's formula [13, Formula (1) was used by L. Lorch and P. Szego [9] to show some remarkable sign-regularity properties of the higher k-differences of the sequence (c(v,k,a)} in the case \v\ > -y . Beyond its obvious use to show that c increases with v , (3) has been used to get further information about these zeros; see [10,11] 3.…”