Definition:Concave Function

Definition

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

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

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

wherever $x, y \in I$.

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

Alternative Definition

A real function $f$ defined on a real interval $I$ is concave on $I$ iff:

$\displaystyle \forall x_1, x_2, x_3 \in I: x_1 < x_2< x_3: \frac {f \left({x_2}\right) - f \left({x_1}\right)} {x_2 - x_1} \ge \frac {f \left({x_3}\right) - f \left({x_2}\right)} {x_3 - x_2}$

or:

$\displaystyle \forall x_1, x_2, x_3 \in I: x_1 < x_2< x_3: \frac {f \left({x_2}\right) - f \left({x_1}\right)} {x_2 - x_1} \ge \frac {f \left({x_3}\right) - f \left({x_1}\right)} {x_3 - x_1}$.

Hence a geometrical interpretation:

• In the left hand image above, the slope of $P_1 P_2$ is greater than that of $P_2 P_3$.
• In the right hand image above, the slope of $P_1 P_2$ is greater than that of $P_1 P_3$.

Equivalence of Definitions

These two definitions can be seen to be equivalent from Equivalence of Convex and Concave Definitions.

Also see

• Compare convex function. It is immediately obvious from the definition that $f$ is concave on $I$ iff $-f$ is convex on $I$.

• Concavity of Differentiable Functions

Sources

• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 12.13$