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.
Let I be an interval of the real line and let f denote a real-valued, convex function defined on I.
[3] The finite form of Jensen's inequality is a special case of this result.
Consider the real numbers x1, …, xn ∈ I and let denote their arithmetic mean.
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).
To discuss the case of equality in (1), note that x1 > y1 by (3) and our assumption xi ≠ yi for all i ∈ {1, …, n − 1}.
Hence there is a strictly positive term in the sum on the right hand side of (7) and equality in (1) cannot hold.
The relaxed condition (5) means that An ≥ Bn, which is enough to conclude that cn(An−Bn) ≥ 0 in the last step of (7).
However, then there is a strictly positive term on the right hand side of (7) and equality in (1) cannot hold.
An explanation of Karamata's inequality and majorization theory can be found here.