A nonlinear average is simply an average taken of a nonlinear function of some variable quantity. The value of such an average is generally different from the value of the nonlinear function at the average of the variable quantity. Jensen’s inequality is the result of nonlinear averaging a function that curves in a consistent direction, i.e. is concave up or concave down. For concave up functions, Jensen’s inequality says that the average of the function is greater than the function of the average, while for concave down functions, the opposite result is true.
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