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

The best bounds of the form $B(\alpha ,\beta ,\gamma ,x)=(\alpha +\sqrt{{\beta }^2+{\gamma }^2{x}^2})\mathrm{/}x$ 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 ${\mathrm{\Phi }}_{\nu }(x)=I_{\nu -1}(x)\mathrm{/}{I}_{\nu }(x)$ as $x\to 0^{+}$ (respectively $x\to +\mathrm{\infty }$) and the graphs of the functions $B(\alpha ,\beta ,\gamma ,x)$ and ${\mathrm{\Phi }}_{\nu }(x)$ are tangent at some $x=x_{\ast }>0$, then $B(\alpha ,\beta ,\gamma ,x)$ is an upper (respectively lower) bound for ${\mathrm{\Phi }}_{\nu }(x)$ for any positive $x$, and it is the best possible at $x_{\ast }$. The same is true for the ratio ${\mathrm{\Phi }}_{\nu }(x)=K_{\nu +1}(x)\mathrm{/}{K}_{\nu }(x)$ but interchanging lower and upper bounds (and with a slightly more restricted range for $\nu$). Bounds with maximal accuracy at $0^{+}$ and $+\mathrm{\infty }$ are recovered in the limits $x_{\ast }\to 0^{+}$ and $x_{\ast }\to +\mathrm{\infty }$, and for these cases the coefficients have simple expressions. For the case of finite and positive $x_{\ast }$ we provide uniparametric families of bounds which are close to the optimal bounds and retain their confluence properties.