Prove $f(x_1, x_2, x_3,x_4) = \frac{x_1}{x_1 x_3+x_2 x_4}$ is convex

Question

Prove that the function $f : \mathbb R_{>0}^4 \to \mathbb R$ defined by $$f(x_1,x_2,x_3,x_4) = \dfrac{x_1}{x_1 x_3+x_2 x_4}$$ is convex.

I am not sure if this assertion is correct but I do believe it is. Any help is appreciated!

Answer 1

If you sort $ x_1, x_2, x_3, x_4 $ in a vector, it will be of the form linear-over-quadratic. This function is not convex.

Answer 2

If your function were convex, so would sections through it, such as $x\mapsto f(x,1,1,1)=x/(x+1)$. This function has negative second derivative, and hence is not convex and neither is your original $f$.