I am trying to prove that the following function is an increasing function $$ f(x) = \int_0^{2\pi} \log \left(\exp(-\frac{4+4x \cos \theta}{\sigma^2}) \frac{I_0(\frac{2\sqrt{x^2 + 4x \cos \theta + 4}}{\sigma^2})}{I_0(\frac{2x}{\sigma^2})} + 1\right) d \theta $$ where $\sigma$ is a positive constant, $I_0(\cdot)$ is the modified Bessel function of the first kind of order $0$.
I used Matlab to compute the function numerically. The results are shown below.
I have been working on this problem for one week but have no idea. Can anyone provide any thoughts on this problem?
