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 Iν(x)Kν(x) > 1/2(x2 + ν2 + 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.
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BibTeX
@misc{segura2026optimized,
title = {{Optimized bounds for the product and the ratios of modified Bessel functions}},
author = {Javier Segura and Soichiro Suzuki},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.03633},
eprint = {2608.03633},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.03633},
}
RIS
TY - UNPB
TI - Optimized bounds for the product and the ratios of modified Bessel functions
AU - Javier Segura
AU - Soichiro Suzuki
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2608.03633
UR - https://arxiv.org/abs/2608.03633
AB - 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(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.
ER -