Optimized bounds for the product and the ratios of modified Bessel functions

Abstract

New sharp bounds for the product and the ratios of modified Bessel functions are presented. Most bounds for the product are derived as direct consequences of previously established bounds for the ratios of consecutive orders, except for the lower bound $I_ν(x)K_ν(x) > \frac{1}{2}(x^2 + ν^2 + 1/5)^{-1/2}$, which had been conjectured for $x > 0$ and $ν> -1$ and we prove in the present paper, showing that the constant $1/5$ can not be lowered. Moreover, very sharp bounds are obtained for the ratios (and consequently for the product) by asymptotically optimizing certain uniparametric inequalities. These optimized bounds are remarkably accurate: they remain extremely sharp for both small and large $x$ with fixed $ν$, and for large $ν$ with fixed $x$ or fixed $z = x/ν$. As a consequence, they provide precise upper and lower estimates across a wide range of parameters.