Jensen inequality an overview sciencedirect topics. For finite sets of probability measures, sufficiency is character ized by means of certain positively homogeneous convex functions. Recall that the range of possible outcomes is 1,2,3,4,5,6 and the mean of the possible outcome is 3. It contains 175 solved problems in the form of exercises and. In this paper, we obtain jensens inequality for ggconvex functions.
Exponential convexity for jensens inequality for norms. This is jensen s inequality, which were going to develop. This work is about inequalities which play an important role in mathematical olympiads. Jensens inequality and its converses pdf book manual. Download pdf an introduction to the theory of functional equations and inequalities second edition cauchy s equation and jensen s inequality book full free. Download englishus transcript pdf let x be a random variable, and let g be a function.
The last part is dedicated to the proof of the well known jensens inequality using the new inequality. Pdf jensens inequality for convexconcave antisymmetric. If f is a convex function on i, then the wellknown jensens inequality 1, 4 asserts that. Hence, riskaversionisequivalent to the concavity of a bernoulli utility function ux. Expected utility and jensen s inequality risk aversion. Expected utility and jensen s inequa lity free download as powerpoint presentation. Consequently we have equality in 1 if, and only if.
Also, we get in equalities alike to hermitehadamard inequality for ggconvex functions. By jensen type inequalities in 8 we have that for every contraction x. Pdf the weighted jensen inequality for convexconcave antisymmetric functions is proved and some applications are given. Expected utility and jensen s inequality free download as powerpoint presentation. Jensens inequality plays a central role in the derivation of the expectation maximization algorithm 1 and the proof of consistency of maximum likelihood estimators. The basic observation is the fact that f 0 and z x f d 0 imply f 0 almost everywhere. A variant of jensens inequality rims, kyoto university. Jensens inequality suggests that the mean of the payoffs will always be larger than or equal to the payoff of the mean outcome. Operator jensens, bohrs and popovicius type inequalities for. We study how good the jensen inequality is, that is, the discrepancy between.
The simplest form of jensens inequality is that if 4x is a convex function, and m is the arithmetic mean of xi. The following theorem is a consequence of the jensen inequality proved by pecaric and. On the equality in jensens inequality for operator convex functions. Sufficiency and jensens inequality for conditional expectations. We know that if g is linear, then the expected value of the function is the same as that linear function of the expected value. In this talk we pursue the concept further and derive a number of jensen type inequalities for quasiconvex functions. All books are in clear copy here, and all files are secure so dont worry about it. Jensens inequa lity for operators and l wners theorem.