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+ν^2+1/5}}-1$ and the upper bound for $ν\ge 1/2$, and that the sum of the logarithmic derivatives satisfies \[