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 \[ \frac{1}{2\sqrt{x^2+ν^2+1/5}}<I_ν(x)K_ν(x)<\frac{1}{2\sqrt{x^2+ν^2-1/4}} , \] where the lower bound holds for $\nu >-1$ and the upper bound for $\nu \ge 1/2$, 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 $\nu >-1$ and the lower bound for $\nu \ge 1$. 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 $I_{\nu -1}(x)/I_{\nu }(x)$ and $K_{\nu +1}(x)/K_{\nu }(x)$.

12 pages