Sharp two-sided bounds for the product of modified Bessel functions and for its logarithmic derivative

We prove that, for all $x>0$, the product of modified Bessel functions satisfies $$\frac{1}{2\sqrt{x^2+{\nu }^2+1\mathrm{/}5}}<I_{\nu }(x)K_{\nu }(x)<\frac{1}{2\sqrt{x^2+{\nu }^2-1\mathrm{/}4}},$$ where the lower bound holds for $\nu >-1$ and the upper bound for $\nu \ge 1\mathrm{/}2$, and that the sum of the logarithmic derivatives satisfies $$-\frac{x}{x^2+{\nu }^2-1}<\frac{I_{\nu }^{\mathrm{\prime }}(x)}{I_{\nu }(x)}+\frac{K_{\nu }^{\mathrm{\prime }}(x)}{K_{\nu }(x)}<-\frac{x}{x^2+{\nu }^2+7\mathrm{/}20},$$ 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 $I_{\nu -1}(x)\mathrm{/}{I}_{\nu }(x)$ and $K_{\nu +1}(x)\mathrm{/}{K}_{\nu }(x)$.