I have a function of multiple variables, defined as a vector $x = (x_1,x_2,...,x_n) $ as follows: $f(x) = \frac{1-\sum_{i=1}^{n}x_i}{\alpha-(1-\sum_{i=1}^{n}x_i)}$ with $(\alpha-(1-\sum_{i=1}^n x_i)) \geq 0$.
How to show the convexity of this function by showing that the n-dimensional Hessian matrix is positive semi-definite? The elements of the Hessian matrix are given as follows: $\frac{\partial^2 f}{\partial x_i^2} = \frac{2\alpha}{(\alpha-(1-\sum_{i=1}^{n}x_i))^3}$, $\frac{\partial^2 f}{\partial x_j \partial x_i} = \frac{2\alpha}{(\alpha-(1-\sum_{i=1}^{n}x_i))^3}$.