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Foundation of Mathematics: Jensen inequality

2022-07-22 06:55:00 【NoBlackstone】

A catalogue of basic mathematical articles

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A catalogue of basic mathematical articles
Jensen inequality

Jensen inequality

There is a property in convex function , There are any two points in the definition field $x_1$ and $x_2$ , And there are $0\leqslant t\leqslant 1$ , obtain $t\times f(x_1) + (1-t)\times f(x_2)\geqslant f[tx_1 + (1 - t)x_2]$ , The geometric meaning is that any point on the secant of two points is greater than $x_1$ And $x_2$ Function value between , As shown in the figure ：

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The above formula , Namely Jensen Two point formula of inequality ;

Jensen Inequality is defined as , If for any point set , Yes $\lambda_i > 0$ And $\sum_i\lambda_i = 1$ , Then convex function $f (x)$ Satisfy $f(\sum_i\lambda_ix_i)\leqslant\sum_i\lambda_if(x_i)$ .