Amos-type bounds for modified Bessel function ratios

We systematically investigate lower and upper bounds for the modified Bessel function ratio R-nu = I nu+1/I-nu by functions of the form G(alpha,beta) (t) = t/(alpha+root t(2) + beta(2)) in case R-nu is positive for all t > 0, or equivalently, where nu >= -1 or nu is a negative integer. For nu >= -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 nu >= -1/2, the minimal elements are tangent to R-nu in exactly one point 0 <= t <= infinity, and have R, as their lower envelope. We also provide a new family of explicitly computable upper bounds. Finally, if nu is a negative integer, we explicitly describe the sets of lower and upper bounds, and give their greatest and least elements, respectively. (C) 2013 The Authors. Published by Elsevier Inc. All rights reserved.