Exponential is Strictly Convex

Theorem

Let $x \in \R$ be a real number.

Let $\exp x$ be the exponential of $x$.

Then:

The function $f \left({x}\right) = \exp x$ is strictly convex.

Proof

By definition, the exponential function is the inverse of the natural logarithm function.

From Logarithm is Strictly Increasing, $\ln x$ is strictly increasing.

From Logarithm is Strictly Concave, $\ln x$ is strictly concave.

The result follows from Inverse of Strictly Increasing Strictly Concave Real Function is Strictly Convex.

$\blacksquare$