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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).
Since the square root is a strictly concave function, it follows from Jensen's inequality that the square root of the sample variance is an underestimate. The use of n − 1 instead of n in the formula for the sample variance is known as Bessel's correction , which corrects the bias in the estimation of the population variance, and some, but ...
Indeed, convex functions are exactly those that satisfies the hypothesis of Jensen's inequality. A first-order homogeneous function of two positive variables x {\displaystyle x} and y , {\displaystyle y,} (that is, a function satisfying f ( a x , a y ) = a f ( x , y ) {\displaystyle f(ax,ay)=af(x,y)} for all positive real a , x , y > 0 ...
Hölder's inequality; Jackson's inequality; Jensen's inequality; Khabibullin's conjecture on integral inequalities; Kantorovich inequality; Karamata's inequality; Korn's inequality; Ladyzhenskaya's inequality; Landau–Kolmogorov inequality; Lebedev–Milin inequality; Lieb–Thirring inequality; Littlewood's 4/3 inequality; Markov brothers ...
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.
A convex function of a martingale is a submartingale, by Jensen's inequality. For example, the square of the gambler's fortune in the fair coin game is a submartingale (which also follows from the fact that X n 2 − n is a martingale). Similarly, a concave function of a martingale is a supermartingale.
The operator version of Jensen's inequality is due to C. Davis. [ 17 ] A continuous, real function f {\displaystyle f} on an interval I {\displaystyle I} satisfies Jensen's Operator Inequality if the following holds
English. Read; Edit; View history; Tools. ... In mathematics, Jensen's theorem may refer to: Johan Jensen's inequality for convex functions;