Equivalence of Definitions of Concave Real Function

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

The following definitions of a concave real function are equivalent:

Proof
Let $f$ be concave real function on $I$ according to definition 1.

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

Make the substitutions $x_1 = x, x_2 = \alpha x + \beta y, x_3 = y$.

As $\alpha + \beta = 1$, we have $x_2 = \alpha x_1 + \left({1 - \alpha}\right) x_3$.

Thus:
 * $\alpha = \dfrac {x_3 - x_2} {x_3 - x_1}, \beta = \dfrac {x_2 - x_1} {x_3 - x_1}$

So:

Again from $(1)$:

Thus:
 * $\dfrac {f \left({x_2}\right) - f \left({x_1}\right)} {x_2 - x_1} \ge \dfrac {f \left({x_3}\right) - f \left({x_1}\right)} {x_3 - x_1} \ge \dfrac {f \left({x_3}\right) - f \left({x_2}\right)} {x_3 - x_2}$

So:
 * $\dfrac {f \left({x_2}\right) - f \left({x_1}\right)} {x_2 - x_1} \ge \dfrac {f \left({x_3}\right) - f \left({x_1}\right)} {x_3 - x_1}$

demonstrating that $f$ is concave on $I$ according to definition 3, and:


 * $\dfrac {f \left({x_2}\right) - f \left({x_1}\right)} {x_2 - x_1} \ge \dfrac {f \left({x_3}\right) - f \left({x_2}\right)} {x_3 - x_2}$

demonstrating that $f$ is concave on $I$ according to definition 2.

As each step is an equivalence, the argument reverses throughout.