Simple bounds with best possible accuracy for ratios of modified Bessel functions (Q6155841)

The purpose of this paper is to find simple bounds for ratios of modified Bessel functions. First the sharpness of the bounds at \(x = 0\) and \(x = +\infty\) for the first and second function is classified in tables. Then, the best possible bounds of this type, not necessarily around \(x = 0\) and \(x = +\infty\) are given. Two auxiliary lemmas which will used later for proving the existence of the best algebraic bound are then proved by solving a quadratic equation. These bounds are sharp for \(x\to 0^+\). From the previous close to best bounds, the existence of best possible bounds which interpolate the function ratios at \(x = 0\) and \(x = +\infty\) and at an intermediate value \(x^{\star} > 0\) are then deduced.