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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!

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So you're asking if $I_\alpha(z)$ is log-convex in $\alpha$. – Stopple Feb 3 '11 at 16:15
to Stopple, what you said is exactly right. Do you have some advices? – user11936 Feb 3 '11 at 16:27
No, I've just been thinking about log-convexity recently. A quick plot in Mathematica (with $x=1$) makes you conjecture seem plausible. – Stopple Feb 3 '11 at 16:56
A google search on 'log convexity of bessel function' turns up a lot of hits. – Stopple Feb 4 '11 at 18:25
I don't think so. I have tried google search on keywords like that, but google can't give the results that I want. Fortunatly, I think I have proved the result, if you are interested in it, I can upload it when I finish to write it. – user11936 Feb 8 '11 at 10:07

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.

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