Convex Real Function Composed with Increasing Convex Real Function is Convex

Theorem

Let $I$ be real interval.

Let $f : I \to \R$ be a convex function.

Let $J$ be a real interval containing the image of $f$.

Let $g : J \to \R$ be a increasing convex function.

Then $g \circ f : I \to \R$ is a convex function.

Proof

Let $x, y \in I$ and $t \in \closedint 0 1$.

Since $f$ is convex, we have:

$\map f {t x + \paren {1 - t} y} \le t \map f x + \paren {1 - t} \map f y$

Since $g$ is increasing, we then have:

$\map {\paren {g \circ f} } {t x + \paren {1 - t} y} \le \map g {t \map f x + \paren {1 - t} \map f y}$

Since $g$ is convex, we then have:

$\map g {t \map f x + \paren {1 - t} \map f y} \le t \map g {\map f x} + \paren {1 - t} \map g {\map f y}$

so that:

$\map {\paren {g \circ f} } {t x + \paren {1 - t} y} \le t \map {\paren {g \circ f} } x + \paren {1 - t} \map {\paren {g \circ f} } y$

That is:

$g \circ f : I \to \R$ is a convex function.

$\blacksquare$