Definition:Concave Real Function/Definition 1/Strictly

Definition
Let $f$ be a real function which is defined on a real interval $I$.

$f$ is strictly concave on $I$ iff:


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


 * ConcaveFunction1.png

The geometric interpretation is that any point on the chord drawn on the graph of any strictly concave function always lies below the graph.

Also defined as
By setting $\alpha = t$ and $\beta = 1 - t$, this can also be written as:


 * $\forall x, y \in I, x \ne y: \forall t \in \left({0 \,.\,.\, 1}\right) : f \left({t x + \left({1 - t}\right) y}\right) > t f \left({x}\right) + \left({1 - t}\right) f \left({y}\right)$

Also see

 * Equivalence of Definitions of Strictly Concave Real Function