Let f(x) be a concave function of N variables defined over a convex set,S$\displaystyle \in R^N$Let g be an increasing and concave function of one var.
prove that g(f(x)) is concave over S
Let f(x) be a concave function of N variables defined over a convex set,S$\displaystyle \in R^N$Let g be an increasing and concave function of one var.
prove that g(f(x)) is concave over S
Probably obvious, but,...
It seems the second derivative should get us somehwere...
$\displaystyle g''(f(x)) \cdot f'(x) + g'(f(x)) \cdot f''(x)$
The information in the problem statement, relating to g'' and f'' is too compelling to pass up. Also, with f' positive, it seems we've only g' to argue. Will the convexity of S do that?
For what it's worth...
The proof by TKHunny is actually not true since we don't know that the functions are differentiable.
However, as he mentioned the proof is obvious.
Let me give it below.
Proof. Since $\displaystyle f:\mathbb{R}^{n}\to\mathbb{R}$ is concave, we have
$\displaystyle f\big(\lambda x+(1-\lambda)y\big)\geq\lambda f(x)+(1-\lambda)f(y)$ for all $\displaystyle x,y\in\mathbb{R}^{n}$ and all $\displaystyle \lambda\in[0,1]$. (*)
As $\displaystyle g:\mathbb{R}\to\mathbb{R}$ is increasing, we use (*) to get
$\displaystyle g\Big(f\big(\lambda x+(1-\lambda)y\big)\Big)\geq g\big(\lambda f(x)+(1-\lambda)f(y)\big)$.........($\displaystyle g$ is increasing)
...............................$\displaystyle \geq \lambda g\big(f(x)\big)+(1-\lambda)g\big(f(y)\big)$.....($\displaystyle g$ is concave)
for all $\displaystyle x,y\in\mathbb{R}^{n}$ and all $\displaystyle \lambda\in[0,1]$.
This completes the proof that $\displaystyle g\circ f$ is concave on $\displaystyle \mathbb{R}^{n}$.....$\displaystyle \rule{0.3cm}{0.3cm}$