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

The best bounds of the form B(α,β,γ,x)=(α+β2+γ2x2)/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 accuracy 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.