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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 proved by Jensen in 1906, [ 1 ] building on an earlier proof of the same inequality for doubly-differentiable functions by Otto Hölder in 1889. [ 2 ]
Using the finite form of Jensen's inequality for the natural logarithm, we can prove the inequality between the weighted arithmetic mean and the weighted geometric mean stated above. Since an x k with weight w k = 0 has no influence on the inequality, we may assume in the following that all weights are positive.
There are three inequalities between means to prove. There are various methods to prove the inequalities, including mathematical induction, the Cauchy–Schwarz inequality, Lagrange multipliers, and Jensen's inequality. For several proofs that GM ≤ AM, see Inequality of arithmetic and geometric means.
The proof of this inequality follows from the above combined with Klein's inequality. ... satisfies Jensen's Operator Inequality if the following holds ...
Jensen's formula can be used to estimate the number of zeros of an analytic function in a circle. Namely, if is a function analytic in a disk of radius centered at and if | | is bounded by on the boundary of that disk, then the number of zeros of in a circle of radius < centered at the same point does not exceed
Jan Jensen had numerous head coaching opportunities in her 24 years as an assistant on Lisa Bluder’s coaching staff at Iowa. A few of the offers, Jensen said, were good enough to get serious ...
The result can alternatively be proved using Jensen's inequality, the log sum inequality, or the fact that the Kullback-Leibler divergence is a form of Bregman divergence. Proof by Jensen's inequality
CES 2025 has once again showcased a remarkable array of health and fitness technology that promises to transform our well-being. From artificial intelligence-powered sleep aids to smart ear ...