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

2026/08/04 by Javier Segura
Mathematics · #math.CA #msc:33C10 #msc:26D07

paper · pdf

arxiv created 2026/08/04 · arxiv updated 2026/08/05

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) > (1)/(2)(x² + ν² + 1/5)^-1/2, which had been conjectured for x > 0 and ν> -1, and is here proved for large ν using Debye-type asymptotics. 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.

Citations

• Optimal constants of smoothing estimates for the 3D Dirac equation
• Monotonicity properties for ratios and products of modified Bessel functions and sharp trigonometric bounds
• A new type of sharp bounds for ratios of modified Bessel functions
• Eigenvalue bounds for radial magnetic bottles on the disk
• Casimir effect at nonzero temperature for wedges and cylinders
• Casimir Energy of a Ball and Cylinder in the Zeta Function Technique