Monotonicity of Ratio of Bessel Functions

Question

Let $$g(x)=\frac{xK_{n-1}(x)}{K_{n}(x)}$$ I want to prove that this function is increasing in $(0,+\infty)$ $\forall n=0,1,2\dots$. I tried to directly derive and using the recursion properties of $K_n$ but I had no luck. Thanks in advance for the help.

Answer 1

It is shown in Theorem $6$ of this paper that for any $\nu\geq 0$, the function $$ (0,+\infty) \ni x \mapsto \frac{{xK'_\nu (x)}}{{K_\nu (x)}} $$ is strictly decreasing. Then for all $\nu\geq 0$, the function $$ (0,+\infty) \ni x \mapsto \frac{{xK_{\nu - 1} (x)}}{{K_\nu (x)}} \equiv - \nu - \frac{{xK'_\nu (x)}}{{K_\nu (x)}} $$ is strictly increasing.