Summary: We systematically investigate lower and upper bounds for the modified Bessel function ratio \(R_\nu=I_{\nu+1}/I_\nu\) by functions of the form \(G_{\alpha,\beta}(t)=t/(\alpha+\sqrt{t^2+\beta^2})\) in case \(R_\nu\) is positive for all \(t>0\), or equivalently, where \(\nu\geq -1\) or \(\nu\) is a negative integer. For \(\nu\geq -1\), we give an explicit description of the set of lower bounds and show that it has a greatest element. We also characterize the set of upper bounds and its minimal elements. If \(\nu\geq -1/2\), the minimal elements are tangent to \(R_\nu\) in exactly one point \(0\leq t\leq\infty\), and have \(R_\nu\) as their lower envelope. We also provide a new family of explicitly computable upper bounds. Finally, if \(\nu\) is a negative integer, we explicitly describe the sets of lower and upper bounds, and give their greatest and least elements, respectively.