On a product of modified Bessel functions

Let $I_{\nu}$ and $K_{\nu}$ denote the modified Bessel functions of the first and second kinds of order $\nu.$ In this note we prove that the monotonicity of $u\mapsto I_{\nu}(u)K_{\nu}(u)$ on $(0,\infty)$ for all $\nu\geq -1/2$ is an almost immediate consequence of the corresponding Turán type inequalities for the modified Bessel functions of the first and second kinds of order $\nu.$ Moreover, we show that the function $u\mapsto I_{\nu}(u)K_{\nu}(u)$ is strictly completely monotonic on $(0,\infty)$ for all $\nu\in[-1/2,1/2].$ At the end of this note, a conjecture is stated.