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Jensen's inequality can be proved in several ways, and three different proofs corresponding to the different statements above will be offered. Before embarking on these mathematical derivations, however, it is worth analyzing an intuitive graphical argument based on the probabilistic case where X is a real number (see figure).
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.
In mathematics, Jensen's theorem may refer to: Johan Jensen's inequality for convex functions; Johan Jensen's formula in complex analysis;
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
Bennett's inequality, an upper bound on the probability that the sum of independent random variables deviates from its expected value by more than any specified amount Bhatia–Davis inequality , an upper bound on the variance of any bounded probability distribution
Actual wealth gap explained Citing a myriad of causes -- from cheap credit to exploitative bank practices -- they've noted that the average family puts away less than 4 percent of its income.
Kevin Mazur/Getty Images for The CW Justin Hartley revealed that Supernatural alum Jensen Ackles will portray his estranged brother on the hit CBS drama series, Tracker. Hartley, 47, who portrays ...
And CEO Jensen Huang just explained why another company may be worth owning, perhaps even more so than his own company. Silicon wafers with circuits printed on them. Image source: Getty Images.