Maximum likelihood estimation of the concentration parameter of von Mises–Fisher distributions involves inverting the ratio $$R_\nu = I_{\nu +1} / I_\nu $$ R ν = I ν + 1 / I ν of modified Bessel functions and computational methods are required to invert these functions using approximative or iterative algorithms. In this paper we use Amos-type bounds for $$R_\nu $$ R ν to deduce sharper bounds for the inverse function, determine the approximation error of these bounds, and use these to propose a new approximation for which the error tends to zero when the inverse of $$R_\nu $$ R ν is evaluated at values tending to $$1$$ 1 (from the left). We show that previously introduced rational bounds for $$R_\nu $$ R ν which are invertible using quadratic equations cannot be used to improve these bounds.