In what way knowing that $f$ is increasing helps me here?

Question

I have to show that: given $f: A\to \mathbb{R}$ and $g: B\to A$ two real functions, then if $f$ is concave and increasing in $A$ and $g$ is concave in $B$, the composition $f\circ g$ is concave in $B$.

Here is what I did: since $g$ is concave, by definition:

$$g(\lambda x + (1-\lambda) y) \geq \lambda g(x) + (1-\lambda) g(y)$$

Whence

$$f(g(\lambda x + (1-\lambda) y)) \geq f( \lambda (g(x)) + (1-\lambda)g(y))$$

Now, knowing that $f$ is concave too I have

$$f( \lambda (g(x)) + (1-\lambda)g(y)) \geq \lambda f(g(x)) + (1-\lambda)f(g(y))$$

• Question: in what way knowing that $f$ is increasing helps me here? I cannot even "see" that $f$ is increasing. For what I know, I could have done this even if $f$ were decreasing...

But for sure I'm missing something...

Thank you!

Answer 1

Think about how you get your "Whence" line from the previous line again. Hint: it is not automatically true without certain conditions on $f$.