Inverse of Strictly Increasing Strictly Convex Real Function is Strictly Concave

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Theorem

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

Let $J = f \left[{I}\right]$.


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


Proof

Let:

$X = f \left({x}\right) \in J$
$Y = f \left({y}\right) \in J$.

From the definition of strictly convex:

$\forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: f \left({\alpha x + \beta y}\right) < \alpha f \left({x}\right) + \beta f \left({y}\right)$


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

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

Thus:

$\alpha f^{-1} \left({X}\right) + \beta f^{-1} \left({Y}\right) = \alpha x + \beta y < f^{-1} \left({\alpha X + \beta Y}\right)$

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

$\blacksquare$


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