In mathematics, a **concave function** is one for which the value at any convex combination of elements in the domain is greater than or equal to the convex combination of the values at the endpoints. Equivalently, a concave function is any function for which the hypograph is convex. The class of concave functions is in a sense the opposite of the class of a convex functions. A concave function is also synonymously called **concave downwards**, **concave down**, **convex upwards**, **convex cap**, or **upper convex**.

A real-valued function on an interval (or, more generally, a convex set in vector space) is said to be *concave* if, for any and in the interval and for any ,^{[1]}

A function is called *strictly concave* if

for any and .

For a function , this second definition merely states that for every strictly between and , the point on the graph of is above the straight line joining the points and .

A function is quasiconcave if the upper contour sets of the function are convex sets.^{[2]}

- A differentiable function f is (strictly) concave on an interval if and only if its derivative function f ′ is (strictly) monotonically decreasing on that interval, that is, a concave function has a non-increasing (decreasing) slope.
^{[3]}^{[4]} - Points where concavity changes (between concave and convex) are inflection points.
^{[5]} - If f is twice-differentiable, then f is concave if and only if f ′′ is non-positive (or, informally, if the "acceleration" is non-positive). If f ′′ is negative then f is strictly concave, but the converse is not true, as shown by
*f*(*x*) = −*x*^{4}. - If f is concave and differentiable, then it is bounded above by its first-order Taylor approximation:
^{[2]} - A Lebesgue measurable function on an interval
**C**is concave if and only if it is midpoint concave, that is, for any x and y in**C** - If a function f is concave, and
*f*(0) ≥ 0, then f is subadditive on . Proof:- Since f is concave and 1 ≥ t ≥ 0, letting
*y*= 0 we have - For :

- Since f is concave and 1 ≥ t ≥ 0, letting

- A function f is concave over a convex set if and only if the function −f is a convex function over the set.
- The sum of two concave functions is itself concave and so is the pointwise minimum of two concave functions, i.e. the set of concave functions on a given domain form a semifield.
- Near a strict local maximum in the interior of the domain of a function, the function must be concave; as a partial converse, if the derivative of a strictly concave function is zero at some point, then that point is a local maximum.
- Any local maximum of a concave function is also a global maximum. A
*strictly*concave function will have at most one global maximum.

- The functions and are concave on their domains, as their second derivatives and are always negative.
- The logarithm function is concave on its domain , as its derivative is a strictly decreasing function.
- Any affine function is both concave and convex, but neither strictly-concave nor strictly-convex.
- The sine function is concave on the interval .
- The function , where is the determinant of a nonnegative-definite matrix
*B*, is concave.^{[6]}

- Rays bending in the computation of radiowave attenuation in the atmosphere involve concave functions.
- In expected utility theory for choice under uncertainty, cardinal utility functions of risk averse decision makers are concave.
- In microeconomic theory, production functions are usually assumed to be concave over some or all of their domains, resulting in diminishing returns to input factors.
^{[7]}