# Inverse of Strictly Increasing Concave Real Function is Convex

## Theorem

Let $f$ be a real function which is concave on the open interval $I$.

Let $J = f \sqbrk I$.

If $f$ is strictly increasing on $I$, then $f^{-1}$ is convex on $J$.

## Proof

Let $X = \map f x \in J, Y = \map f y \in J$.

From the definition of concave:

$\forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: \map f {\alpha x + \beta y} \ge \alpha \map f x + \beta \map f y$

Let $f$ be strictly increasing on $I$.

From Inverse of Strictly Monotone Function, it follows that $f^{-1}$ is strictly increasing on $J$.

Thus:

$\alpha \map {f^{-1} } X + \beta \map {f^{-1} } Y = \alpha x + \beta y \ge \map {f^{-1} } {\alpha X + \beta Y}$

Hence $f^{-1}$ is convex on $J$.

$\blacksquare$