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I was searching for some results for the log-concavity of the modified Bessel function of a second type, but I failed. Has there been any known work on this? I am not even sure if it is the modified Bessel function of a second kind is indeed log-concave or not.

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Theorem 2(b) in [1] is equivalent to log-convexity of $K_\nu$ for every $\nu$. This is said to be "well-known", and three references are given.

[1] Árpád Baricz, Saminathan Ponnusamy, Matti Vuorinen, Functional inequalities for modified Bessel functions, DOI:10.1016/j.exmath.2011.07.001, arXiv:1009.4814

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