Karamata's inequality

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In mathematics, Karamata's inequality, also known as the Majorization Inequality, states that if f(x) is a convex function in x and the sequence

x1,x2,...,xn

majorizes

y1,y2,...,yn

then

f(x_1)+f(x_2)+...+f(x_n) \ge f(y_1)+f(y_2)+...+f(y_n).

The inequality is reversed if f(x) is concave.

Jensen's inequality is in fact a special case of this result of Jovan Karamata. Consider a sequence

x1,x2,...,xn

and let

A = \frac{x_1+x_2+...+x_n}{n}.

Then the sequence

x1,x2,...,xn

clearly majorizes the sequence

A,A,...,A (n times).

By Karamata's result,

f(x_1)+f(x_2)+...+f(x_n) \ge f(A)+f(A)+...+f(A) = nf(A),

and dividing by n produces the desired inequality. The sign is reversed if f(x) is concave, as in Jensen's inequality.

[edit] External links

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