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

The best bounds of the form B(alpha, beta, gamma, x) = (alpha + beta 2 + gamma 2x2)/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 (13,,(x) = I,,-1(x)/I,,(x) as x-+0+ (respectively x-+ +infinity) and the graphs of the functions B(alpha, beta, gamma, x) and (13,,(x) are tangent at some x = x* > 0, then B(alpha, beta, gamma, x) is an upper (respectively lower) bound for (13,,(x) for any positive x, and it is the best possible at x*. The same is true for the ratio (13,,(x) = K,,+1(x)/K,,(x) but interchanging lower and upper bounds (and with a slightly more restricted range for nu). Bounds with maximal accuracy at 0+ and +infinity are recovered in the limits x*-+ 0+ and x*-+ +infinity, and for these cases the coefficients have simple expressions. For the case of finite and positive x* we provide uniparametric families of bounds which are close to the optimal bounds and retain their confluence properties.(c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http:// creativecommons .org /licenses /by -nc -nd /4 .0/).

Automated summary

The study characterizes the best bounds for ratios of modified Bessel functions in the form of B(alpha, beta, gamma, x) = (alpha + beta 2 + gamma 2x2)/x. By choosing appropriate alpha, beta, and gamma, and ensuring that B(alpha, beta, gamma, x) is a sharp approximation for (13,,(x) = I,,-1(x)/I,,(x) as x-+0+ (respectively x-+ +infinity), and that the graphs of B(alpha, beta, gamma, x) and (13,,(x) are tangent at some x = x* > 0, B(alpha, beta, gamma, x) becomes an upper (respectively lower) bound for (13,,(x) for any positive x, and it is optimal at x*. The same holds true for the ratio (13,,(x) = K,,+1(x)/K,,(x), but with interchanged upper and lower bounds (and slightly more restricted range for nu). Bounds with maximum accuracy at 0+ and +infinity are obtained in the limits x*-+ 0+ and x*-+ +infinity, and for these cases, coefficients have simple expressions. For finite and positive x*, uniparametric families of bounds are provided, which closely approximate the optimal bounds and maintain their confluence properties.