title: Amos-type bounds for modified Bessel function ratios.
abstract: We systematically investigate lower and upper bounds for the modified Bessel function ratio [Formula: see text] by functions of the form [Formula: see text] in case [Formula: see text] is positive for all [Formula: see text], or equivalently, where [Formula: see text] or [Formula: see text] is a negative integer. For [Formula: see text], we give an explicit description of the set of lower bounds and show that it has a greatest element. We also characterize the set of upper bounds and its minimal elements. If [Formula: see text], the minimal elements are tangent to [Formula: see text] in exactly one point [Formula: see text], and have [Formula: see text] as their lower envelope. We also provide a new family of explicitly computable upper bounds. Finally, if [Formula: see text] is a negative integer, we explicitly describe the sets of lower and upper bounds, and give their greatest and least elements, respectively.
authors.affiliation: Institute for Statistics and Mathematics, WU Wirtschaftsuniversität Wien, Augasse 2-6, 1090 Vienna, Austria.
authors.forename: Kurt
authors.initials: K
authors.lastname: Hornik
authors.affiliation: Department of Applied Statistics, Johannes Kepler University Linz, Altenbergerstraße 69, 4040 Linz, Austria.
authors.forename: Bettina
authors.initials: B
authors.lastname: Grün
keywords: Bounds; Inequalities; Modified Bessel function ratio; Modified Bessel functions of the first kind