Sharp two-sided bounds for the product of modified Bessel functions and for its logarithmic derivative
Year: 2026
Type: preprint
Abstract: We prove that, for all $x > 0$, the product of modified Bessel functions satisfies \[ \frac{1}{2\sqrt{x^2+ν^2+1/5}}-1$ and the upper bound for $ν\ge 1/2$, and that the sum of the logarithmic derivatives satisfies \[ -\frac{x}{x^2+ν^2-1}<\frac{I_ν'(x)}{I_ν(x)}+\frac{K_ν'(x)}{K_ν(x)}<-\frac{x}{x^2+ν^2+7/20}, \] where the upper bound holds for $ν>-1$ and the lower bound for $ν\ge 1$. 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 $K_{ν+1}(x)/K_ν(x)$.
Source: arXiv (Cornell University)
Authors: Segura, Javier, Suzuki, Soichiro
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221026682