Amos-type bounds for modified Bessel function ratios

We systematically investigate lower and upper bounds for the modified Bessel function ratio R ν = I ν + 1 / I ν by functions of the form G α , β ( t ) = t / ( α + t 2 + β 2 ) in case R ν is positive for all t > 0 , or equivalently, where ν ≥ − 1 or ν is a negative integer. For ν ≥ − 1 , 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 ν ≥ − 1 / 2 , the minimal elements are tangent to R ν in exactly one point 0 ≤ t ≤ ∞ , and have R ν as their lower envelope. We also provide a new family of explicitly computable upper bounds. Finally, if ν is a negative integer, we explicitly describe the sets of lower and upper bounds, and give their greatest and least elements, respectively.