詹森不等式以丹麥數(shù)學家約翰·詹森（Johan Jensen）命名。它給出積分的凸函數(shù)值和凸函數(shù)的積分值間的關系。

In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was PRoven by Jensen in 1906.^[1] Given its generality, the inequality appears in many forms depending on the context, some of which are presented below. In its simplest form the inequality states that the convex transformation of a mean is less than or equal to the mean applied after convex transformation; it is a simple corollary that the opposite is true of concave transformations.

Jensen不等式是關于凸性(convexity)的不等式。凸性是非常好的性質(zhì)，在最優(yōu)化問題里面，線性和非線性不是本質(zhì)的區(qū)別，只有凸性才是。如果最優(yōu)化的函數(shù)是凸的，那么局部最優(yōu)就意味著全局最優(yōu)，否則無法推得全局最優(yōu)。有很多不等式都可以用Jensen不等式證得，從而可以把他們的本質(zhì)歸結(jié)為凸性。例如，均值不等式。

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