- Karamata's inequality
In
mathematics , Karamata's inequality, also known as the Majorization Inequality, states that if f(x) is aconvex function in x and the sequence:x_1, x_2, ..., x_n
:y_1, y_2, ..., y_n
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 ofJovan Karamata . Consider a sequence:x_1, x_2, ..., x_n
and let
:A = frac{x_1+x_2+...+x_n}{n}.
Then the sequence
:x_1, x_2, ..., x_n
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.
External links
An explanation of Karamata's Inequality and majorization theory can be found [http://www.artofproblemsolving.com/Forum/viewtopic.php?highlight=majorization+karamata&t=14975 here] .
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