# Convexity of Negative Log Likelihood involving Gamma function

## Problem

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