Inverse of Strictly Increasing Convex Real Function is Concave

Theorem

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

Let $J = f \sqbrk I$.

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

Proof

Let:

$X = \map f x \in J$

$Y = \map f y \in J$.

From the definition of convex:

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

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

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

Thus:

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

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

$\blacksquare$

Also see

• Inverse of Strictly Increasing Strictly Convex Real Function is Strictly Concave

• Inverse of Strictly Increasing Concave Real Function is Convex
• Inverse of Strictly Increasing Strictly Concave Real Function is Strictly Convex

• Inverse of Strictly Decreasing Convex Real Function is Convex
• Inverse of Strictly Decreasing Strictly Convex Real Function is Strictly Convex

• Inverse of Strictly Decreasing Concave Real Function is Concave
• Inverse of Strictly Decreasing Strictly Concave Real Function is Strictly Concave

Sources

• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 12.21 \ (3)$