Definition:Strictly Concave Real Function
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Definition
Let $f$ be a real function which is defined on a real interval $I$.
Definition 1
$f$ is strictly concave on $I$ if and only if:
- $\forall x, y \in I, x \ne y: \forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: f \left({\alpha x + \beta y}\right) > \alpha f \left({x}\right) + \beta f \left({y}\right)$
Definition 2
$f$ is strictly concave on $I$ if and only if:
- $\forall x_1, x_2, x_3 \in I: x_1 < x_2 < x_3: \dfrac {f \left({x_2}\right) - f \left({x_1}\right)} {x_2 - x_1} > \dfrac {f \left({x_3}\right) - f \left({x_2}\right)} {x_3 - x_2}$
Definition 3
$f$ is strictly concave on $I$ if and only if:
- $\forall x_1, x_2, x_3 \in I: x_1 < x_2 < x_3: \dfrac {f \left({x_2}\right) - f \left({x_1}\right)} {x_2 - x_1} > \dfrac {f \left({x_3}\right) - f \left({x_1}\right)} {x_3 - x_1}$
Also known as
A strictly concave function can also be referred to as:
- a strictly concave down function
- a strictly convex up function
Also see
- Results about concave real functions can be found here.