Sharp two-sided bounds for the product of modified Bessel functions and for its logarithmic derivative
Abstract
We prove that, for all x > 0, the product of modified Bessel functions satisfies 1/(2√(x2+ν2+1/5))<Iν(x)Kν(x)<1/(2√(x2+ν2−1/4)) , where the lower bound holds for ν>−1 and the upper bound for ν≥ 1/2, and that the sum of the logarithmic derivatives satisfies −x/(x2+ν2−1)<Iν'(x)/Iν(x)+Kν'(x)/Kν(x)<−x/(x2+ν2+7/20), where the upper bound holds for ν>−1 and the lower bound for ν≥ 1. All four constants are best possible. The four proofs share the same elementary structure: in each case the relevant auxiliary function satisfies a second-order differential inequality, and the inequality follows from the strong maximum principle. Among other applications, we obtain sharp monotonicity properties of the product and improved bounds for the ratios Iν−1(x)/Iν(x) and Kν+1(x)/Kν(x).
Related
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- Improved bounds for the two-point logarithmic Chowla conjecture
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BibTeX
@misc{segura2026sharp,
title = {{Sharp two-sided bounds for the product of modified Bessel functions and for its logarithmic derivative}},
author = {Javier Segura and Soichiro Suzuki},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07933},
eprint = {2610.07933},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07933},
}
RIS
TY - UNPB
TI - Sharp two-sided bounds for the product of modified Bessel functions and for its logarithmic derivative
AU - Javier Segura
AU - Soichiro Suzuki
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.07933
UR - https://arxiv.org/abs/2610.07933
AB - We prove that, for all x > 0, the product of modified Bessel functions satisfies 1/(2√(x^2+ν^2+1/5))<I_ν(x)K_ν(x)<1/(2√(x^2+ν^2−1/4)) , where the lower bound holds for ν>−1 and the upper bound for ν≥ 1/2, and that the sum of the logarithmic derivatives satisfies −x/(x^2+ν^2−1)<I_ν'(x)/I_ν(x)+K_ν'(x)/K_ν(x)<−x/(x^2+ν^2+7/20), where the upper bound holds for ν>−1 and the lower bound for ν≥ 1. All four constants are best possible. The four proofs share the same elementary structure: in each case the relevant auxiliary function satisfies a second-order differential inequality, and the inequality follows from the strong maximum principle. Among other applications, we obtain sharp monotonicity properties of the product and improved bounds for the ratios I_ν−1(x)/I_ν(x) and K_ν+1(x)/K_ν(x).
ER -