How should I prove whether this function is convex or concave?
$f(\textbf{x}) = \frac{1}{x_1 + \frac{1}{x_2 + \frac{1}{x_3 + \frac{1}{x_4}}}}\,\,\,\,\,\,,\,\,\,\,\,\,\textbf{x}\in\mathbb{R}^4_{>0}$
I tried to prove it by checking the definition of convexity:
$f(\lambda\textbf{x} + (1-\lambda\textbf{y})) \leq\lambda f(\textbf{x}) + (1-\lambda)f(\textbf{y})$.
But the definition of the function is so complicated that makes it hard to check the above inequality.
I even tried to check if $\nabla^2 f(\textbf{x})$ is positive semidefinite or not, but I didn't nail it.
Can someone bring any new and smart idea please?!
