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Jensen's inequality generalizes the statement that a secant line of a convex function lies above its graph. Visualizing convexity and Jensen's inequality. 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.
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.
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 ...
Example of Frost diagram for the manganese species. A Frost diagram or Frost–Ebsworth diagram is a type of graph used by inorganic chemists in electrochemistry to illustrate the relative stability of a number of different oxidation states of a particular substance. The graph illustrates the free energy vs oxidation state of a chemical species.
Jensen's inequality: If : is a convex function, then (()) (()). Conditional variance : Using the conditional expectation we can define, by analogy with the definition of the variance as the mean square deviation from the average, the conditional variance
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 ...
Note that the operator ... for example. Löwner–Heinz theorem ... satisfies Jensen's Operator Inequality if the following holds ...