Convexity of Negative Log Likelihood involving Gamma function


The negative log-likelihood of Dirichlet multinominal distribution is given by $ $ f(\alpha)=-L(\alpha)=\sum_{i=1}^n\sum_{j=1}^d \log\frac{\Gamma(\alpha_j)}{\Gamma(\alpha_j+\mathbf{x}_{ij})}+\text{constant} $ $ where $ \mathbf{x}_{ij}\in \{0, 1\}$ and $ \alpha\in \mathbb{R}^d, \alpha_j >0$

Then determined whether $ f(\alpha)$ is convex or not.

What I Have Done

It is natural to compute the Hessian $ \nabla_\alpha^2 f(\alpha)$ . However, when I checked Wikipedia, the derivative of Gamma function looks much too complicated.

Then I tried to find some counter examples for $ d=1$ case. It seems that this function is indeed convex according to several simulations I carried out.

I am wondering if we could analytically determine the convexity of this function.