Power Function is Convex Real Function

Theorem

Let $p \ge 1$ be a real number.

Define $f : \hointr 0 \infty \to \hointr 0 \infty$ by:

$\map f x = x^p$

for each $x \in \hointr 0 \infty$.

Then $f$ is a convex function.

Proof

Applying Derivative of Power twice, we have that:

$f$ is twice differentiable

with:

$\map {f} x = p \paren {p - 1} x^{p - 2}$

for each $x \in \hointr 0 \infty$.

Since $p \ge 1$, we have:

$p \paren {p - 1} \ge 0$

and so:

$\map {f} x \ge 0$

for each $x \in \hointr 0 \infty$.

From Real Function with Positive Derivative is Increasing:

$f'$ is increasing

and so from Real Function is Convex iff Derivative is Increasing:

$f$ is convex.

$\blacksquare$