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

Abstract

The best bounds of the form $B(α,β,γ,x)=(α+\sqrt{β^2+γ^2 x^2})/x$ 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 $Φ_ν(x)=I_{ν-1} (x)/I_ν(x)$ as $x\rightarrow 0^+$ (respectively $x\rightarrow +\infty$) and the graphs of the functions $B(α,β,γ,x)$ and $Φ_ν(x)$ are tangent at some $x=x_*>0$, then $B(α,β,γ,x)$ is an upper (respectively lower) bound for $Φ_ν(x)$ for any positive $x$, and it is the best possible at $x_*$. The same is true for the ratio $Φ_ν(x)=K_{ν+1} (x)/K_ν(x)$ but interchanging lower and upper bounds (and with a slightly more restricted range for $ν$). Bounds with maximal accuracy at $0^+$ and $+\infty$ are recovered in the limits $x_*\rightarrow 0^+$ and $x_*\rightarrow +\infty$, 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.