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 .
Functions of a single variable
1. 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. 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 its second derivative is negative then it is strictly concave, but the opposite is not true, as shown by f(x) = −x4.
6. 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 :
Functions of n variables
2. 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.
3. Near a 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.
- 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.
- 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.
- Concave polygon
- Jensen's inequality
- Logarithmically concave function
- Quasiconcave function
- Lenhart, S.; Workman, J. T. (2007). Optimal Control Applied to Biological Models. Mathematical and Computational Biology Series. Chapman & Hall/ CRC. ISBN 978-1-58488-640-2.
- Varian, Hal (1992). Microeconomic Analysis (Third ed.). New York: Norton. ISBN 0-393-95735-7.
- Cover, Thomas M.; Thomas, J. A. (1988). "Determinant inequalities via information theory". SIAM Journal on Matrix Analysis and Applications. 9 (3): 384&ndash, 392. doi:10.1137/0609033.
- Pemberton, Malcolm; Rau, Nicholas (2015). Mathematics for Economists: An Introductory Textbook. Oxford University Press. pp. 363–364. ISBN 978-1-78499-148-7.
- Crouzeix, J.-P. (2008). "Quasi-concavity". In Durlauf, Steven N.; Blume, Lawrence E. The New Palgrave Dictionary of Economics (Second ed.). Palgrave Macmillan. doi:10.1057/9780230226203.1375.
- Rao, Singiresu S. (2009). Engineering Optimization: Theory and Practice. John Wiley and Sons. p. 779. ISBN 0-470-18352-7.