Simple bounds with best possible accuracy for ratios of modified Bessel functions

Abstract

The best bounds of the form B(α, β, γ, x) = (α + ✓β2 + γ2 x2)/x for ratios of modified Bessel functions are characterized: if α, β and γ are chosen in such a way that B(α, β, γ, x) is a sharp approximation for Φν (x) = Iν−1(x)/Iν(x) as x → 0+ (respectively x → +∞) and the graphs of the functions B(α, β, γ, x) and Φν (x) are tangent at some x = x∗ > 0, then B(α, β, γ, x) is an upper (respectively lower) bound for Φν (x) for any positive x, and it is the best possible at x∗. The same is true for the ratio Φν (x) = Kν+1(x)/Kν (x) but interchanging lower and upper bounds (and with a slightly more restricted range for ν). Bounds with maximal accu- racy at 0+ and +∞ are recovered in the limits x∗ → 0+ and x∗ → +∞, and for these cases the coefficients have simple expressions. For the case of finite and positive x∗ we provide uniparametric families of bounds which are close to the optimal bounds and retain their confluence properties.