Amos-type bounds for modified Bessel function ratios

作者：Hornik Kurt; Gruen Bettina

来源：Journal of Mathematical Analysis and Applications, 2013, 408(1): 91-101.

DOI：10.1016/j.jmaa.2013.05.070

• Bounds
• Modified Bessel function ratio
• Modified Bessel functions of the first kind
• Inequalities

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.

• 出版日期2013-12-1