We fix some parameter $0 < \alpha < 1$. We consider the function $$f_\alpha(x) = - \log (\alpha) (1 + e^x) - \log (1-\alpha) (1+e^{-x}) - \log \Gamma ( (1+e^x) + (1+e^{-x}))+ \log \Gamma (1+e^x) + \log \Gamma (1+e^{-x}),$$ where $\Gamma$ is the Gamma function. It seems empirically that this function is convex for any $\alpha$, but I have been unable to prove it from the properties of the Gamma or the log-Gamma function. Any help would be appreciated.
Context: this quantity appears naturally as a function to optimize in a statistical context with the $\beta$ random variable. The Gamma functions appear from normalization constants.