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This research demonstrates the log-convexity and log-concavity of the modified Bessel function of the first kind and the related functions. The method of coefficient is used to verify such properties. One of our results contradicts the conjecture proposed by Neumann in 2007 which states that modified Bessel function of the first kind Iν$I_{\nu }$ is log-concave in (0,∞)$(0,\infty )$ given ν>0$\nu >0$. The log-concavity holds true in some bounded domain. The application of the other results in Kibble’s bivariate gamma distribution is also demonstrated.

The logarithmic concavity of modified Bessel functions of the first kind and its related functions