On a product of modified Bessel functions

Abstract:

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.