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

TL;DR

This paper characterizes the best possible bounds of a specific form for ratios of modified Bessel functions, ensuring sharpness at certain points and providing families of near-optimal bounds with proven properties.

Contribution

It introduces a method to determine optimal bounds for Bessel function ratios that are sharp at specific points and extends to families of bounds close to optimal.

Findings

Bounds are sharp at chosen points and are the best possible at those points.

Explicit expressions for bounds with maximal accuracy at 0+ and +∞ are provided.

Families of near-optimal bounds are constructed for finite positive x_*.

Abstract

The best bounds of the form $B(\alpha,\beta,\gamma,x)=(\alpha+\sqrt{\beta^2+\gamma^2 x^2})/x$ for ratios of modified Bessel functions are characterized: if $\alpha$, $\beta$ and $\gamma$ are chosen in such a way that $B(\alpha,\beta,\gamma,x)$ is a sharp approximation for $\Phi_{\nu}(x)=I_{\nu-1} (x)/I_{\nu}(x)$ as $x\rightarrow 0^+$ (respectively $x\rightarrow +\infty$) and the graphs of the functions $B(\alpha,\beta,\gamma,x)$ and $\Phi_{\nu}(x)$ are tangent at some $x=x_*>0$, then $B(\alpha,\beta,\gamma,x)$ is an upper (respectively lower) bound for $\Phi_{\nu}(x)$ for any positive $x$, and it is the best possible at $x_*$. The same is true for the ratio $\Phi_{\nu}(x)=K_{\nu+1} (x)/K_{\nu}(x)$ but interchanging lower and upper bounds (and with a slightly more restricted range for $\nu$). 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.