monotonicity of functions related to modified Bessel function

Question

Dear colleagues, I recently met some problems related to the modified Bessel funtions of the first kind and the second kind. I want to know if there exist some results on the monotonicity of $\frac{I_\alpha^{'}(x)}{I_\alpha(x)}$ and $\frac{K_\alpha^{'}(x)}{K_\alpha(x)}$ with respect to $\alpha$. That is to say, I want the result that $\frac{I_{\alpha+\nu}^{'}(x)}{I_{\alpha+\nu}(x)}>\frac{I_\alpha^{'}(x)}{I_\alpha(x)}$ with any $\nu>0$ (I guess the result should be like this). For another part, $\frac{K_\alpha^{'}(x)}{K_\alpha(x)}$ should be decrease monotonicaly with respect to the order $\alpha$.

Thank you in advance!

Answer 1

The paper: Log-concavity for series in reciprocal gamma functions and applications by S. Kalmykov and D. Karp mentions (on page 9, Example 1) that $\nu \mapsto I_{\nu}(x)$ is log-concave on $(-1,\infty)$ for each fixed $x$. That should answer your question, as well as several other related questions on monotonicity of such ratios.