I am trying to determine the convexity of a composition of functions, $c$, defined as,
$c(x) = f(g(x),h(x))$
where $f:\mathbb{R}^2\to\mathbb{R}$, $g,h:\mathbb{R}^n\to\mathbb{R}$, and $x\in\mathbb{R}^n$. I know that both $g$ and $h$ are convex in $x$, and that $f$ is increasing and convex in each of its (two) arguments. To establish convexity of $c$ do I need to construct the Hessian? How would I go about doing this for a multi-dimensional composition?
Thanks
$g(t x + (1 - t)y) \leq tg(x) + (1 - t)g(y)$ and similarly for $h,$ the hypothesis on $f$ being increasing in each coordinate yields $c(tx + (1-t)y) \leq tc(x) + 1 - t)c(y)$ (use increasing in the first coordinate with $g$ and the second with $h.$