Simple bounds with best possible accuracy for ratios of modified Bessel functions (2207.02713v3)

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.