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In mathematics, **Karamata's inequality**,^{[1]} named after Jovan Karamata,^{[2]} also known as the **majorization inequality**, is a theorem in elementary algebra for convex and concave real-valued functions, defined on an interval of the real line. It generalizes the discrete form of Jensen's inequality, and generalizes in turn to the concept of Schur-convex functions.

Let *I* be an interval of the real line and let *f* denote a real-valued, convex function defined on *I*. If *x*_{1}, …, *x _{n}* and

(1) |

Here majorization means that *x*_{1}, …, *x _{n}* and

and | (2) |

and we have the inequalities

for all i ∈ {1, …, n − 1}. | (3) |

and the equality

(4) |

If *f* is a strictly convex function, then the inequality (**1**) holds with equality if and only if we have *x _{i}* =

- If the convex function
*f*is non-decreasing, then the proof of (**1**) below and the discussion of equality in case of strict convexity shows that the equality (**4**) can be relaxed to**(5)** - The inequality (
**1**) is reversed if*f*is concave, since in this case the function −*f*is convex.

The finite form of Jensen's inequality is a special case of this result. Consider the real numbers *x*_{1}, …, *x _{n}* ∈

denote their arithmetic mean. Then (*x*_{1}, …, *x _{n}*) majorizes the

Dividing by *n* gives Jensen's inequality. The sign is reversed if *f* is concave.

We may assume that the numbers are in decreasing order as specified in (**2**).

If *x _{i}* =

If *x _{i}* =

It is a property of convex functions that for two numbers *x* ≠ *y* in the interval *I* the slope

of the secant line through the points (*x*, *f* (*x*)) and (*y*, *f* (*y*)) of the graph of *f* is a monotonically non-decreasing function in *x* for *y* fixed (and vice versa). This implies that

(6) |

for all *i* ∈ {1, …, *n* − 1}. Define *A*_{0} = *B*_{0} = 0 and

for all *i* ∈ {1, …, *n*}. By the majorization property (**3**), *A _{i}* ≥

(7) |

which proves Karamata's inequality (**1**).

To discuss the case of equality in (**1**), note that *x*_{1} > *y*_{1} by (**3**) and our assumption *x _{i}* ≠

If the convex function *f* is non-decreasing, then *c _{n}* ≥ 0. The relaxed condition (

If the function *f* is strictly convex and non-decreasing, then *c _{n}* > 0. It only remains to discuss the case

**^**Kadelburg, Zoran; Đukić, Dušan; Lukić, Milivoje; Matić, Ivan (2005), "Inequalities of Karamata, Schur and Muirhead, and some applications" (PDF),*The Teaching of Mathematics*,**8**(1): 31–45, ISSN 1451-4966**^**Karamata, Jovan (1932), "Sur une inégalité relative aux fonctions convexes" (PDF),*Publ. Math. Univ. Belgrade*(in French),**1**: 145–148, Zbl 0005.20101

An explanation of Karamata's inequality and majorization theory can be found here.