Amos-type bounds for modified Bessel function ratios

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.