Equivalence of Definitions of Concave Real Function

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Theorem

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

The following definitions of the concept of Concave Real Function are equivalent:

Definition 1

$f$ is concave on $I$ if and only if:

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

Definition 2

$f$ is concave on $I$ if and only if:

$\ds \forall x_1, x_2, x_3 \in I: x_1 < x_2< x_3: \frac {\map f {x_2} - \map f {x_1} } {x_2 - x_1} \ge \frac {\map f {x_3} - \map f {x_2} } {x_3 - x_2}$

Definition 3

$f$ is concave on $I$ if and only if:

$\forall x_1, x_2, x_3 \in I: x_1 < x_2< x_3: \dfrac {\map f {x_2} - \map f {x_1} } {x_2 - x_1} \ge \dfrac {\map f {x_3} - \map f {x_1} } {x_3 - x_1}$


Proof

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

That is:

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

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

As $\alpha + \beta = 1$, we have $x_2 = \alpha x_1 + \paren {1 - \alpha} x_3$.

Thus:

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

So:

\(\text {(1)}: \quad\) \(\ds \map f {x_2}\) \(\ge\) \(\ds \frac {x_3 - x_2} {x_3 - x_1} \map f {x_1} + \frac {x_2 - x_1} {x_3 - x_1} \map f {x_3}\) substituting for $\alpha$ and $\beta$
\(\ds \leadstoandfrom \ \ \) \(\ds \paren {x_3 - x_1} \map f {x_2}\) \(\ge\) \(\ds \paren {x_3 - x_2} \map f {x_1} + \paren {x_2 - x_1} \map f {x_3}\)
\(\ds \leadstoandfrom \ \ \) \(\ds \paren {x_3 - x_1} \map f {x_2}\) \(\ge\) \(\ds \paren {x_3 - x_1 - x_2 + x_1} \map f {x_1} + \paren {x_2 - x_1} \map f {x_3}\)
\(\ds \leadstoandfrom \ \ \) \(\ds \paren {x_3 - x_1} \map f {x_2} - \paren {x_3 - x_1} \map f {x_1}\) \(\ge\) \(\ds \paren {x_2 - x_1} \map f {x_3} - \paren {x_2 - x_1} \map f {x_1}\)
\(\ds \leadstoandfrom \ \ \) \(\ds \frac {\map f {x_2} - \map f {x_1} } {x_2 - x_1}\) \(\ge\) \(\ds \frac {\map f {x_3} - \map f {x_1} } {x_3 - x_1}\)


Again from $(1)$:

\(\text {(1)}: \quad\) \(\ds \map f {x_2}\) \(\ge\) \(\ds \frac {x_3 - x_2} {x_3 - x_1} \map f {x_1} + \frac {x_2 - x_1} {x_3 - x_1} \map f {x_3}\)
\(\ds \leadstoandfrom \ \ \) \(\ds \paren {x_3 - x_1} \map f {x_2}\) \(\ge\) \(\ds \paren {x_3 - x_2} \map f {x_1} + \paren {x_2 - x_1} \map f {x_3}\)
\(\ds \leadstoandfrom \ \ \) \(\ds \paren {x_3 - x_1} \map f {x_2}\) \(\ge\) \(\ds \paren {x_3 - x_1 - x_3 + x_2} \map f {x_3} + \paren {x_3 - x_2} \map f {x_1}\)
\(\ds \leadstoandfrom \ \ \) \(\ds \paren {x_3 - x_2} \map f {x_3} - \paren {x_3 - x_2} \map f {x_1}\) \(\ge\) \(\ds \paren {x_3 - x_1} \map f {x_3} - \paren {x_3 - x_1} \map f {x_2}\)
\(\ds \leadstoandfrom \ \ \) \(\ds \frac {\map f {x_3} - \map f {x_1} } {x_3 - x_1}\) \(\ge\) \(\ds \frac {\map f {x_3} - \map f {x_2} } {x_3 - x_2}\)


Thus:

$\dfrac {\map f {x_2} - \map f {x_1} } {x_2 - x_1} \ge \dfrac {\map f {x_3} - \map f {x_1} } {x_3 - x_1} \ge \dfrac {\map f {x_3} - \map f {x_2} } {x_3 - x_2}$


So:

$\dfrac {\map f {x_2} - \map f {x_1} } {x_2 - x_1} \ge \dfrac {\map f {x_3} - \map f {x_1} } {x_3 - x_1}$

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

$\dfrac {\map f {x_2} - \map f {x_1} } {x_2 - x_1} \ge \dfrac {\map f {x_3} - \map f {x_2} } {x_3 - x_2}$

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


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

$\blacksquare$

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