Convexity of Negative Log Likelihood involving Gamma function

Question

Problem

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

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.

Answer 1

Hint:

$$\Gamma(x+1) = x\Gamma(x)$$ and $$u\mapsto- \log(a+u)$$ is convex.