I r defined on some open inter val and a selfadjoint element a. Now, for this expectation, we already know that this is equal to the variance of x plus the square of the expected value. Introduction overview of jensens inequality let t be a locally compact hausdorff space and let a be a calgebra of operators on some hilbert space h. Rao jensen s inequality f ex 6 ef x for the expectation of a convex function of a random variable is extended to a. Intuitively a function is convex if it is upward bending. Jensens inequality provides a fundamental tool for understanding and predicting consequences of variance, but it is only just beginning to be explicitly acknowledged in the primary literature 25, and we can find no mention of jensens inequality in any biology or biometry textbook. So jensens inequality gives that log 1 n xn k1 a k logex e logx 1 n xn k1 loga k. Globalization, inequality and welfare harvard university. Jensen s inequality is one of the most basic problem solving tools. Jensens inequality convex functions and a proof for. Jensens inequality for a convex vectorvalued function on. On a new converse of jensens inequality slavko simic. Jensen inequality basics this is a revision of material that appeared earlier in sep 37 and reprinted in pvi.
Pdf in this paper, we present more proofs of the new steffensens inequality for convex functions. Recent research on jensens inequality for oparators. 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. You can make it concave either by considering its inverse the exponential function or by working with lnx.
There are many, many applications, but one i quite like is an alternative proof of the amgm inequality. If f is a convex function on i, then the wellknown jensens inequality 1, 4 asserts that. Norms a norm is a function that measures the lengths of vectors in a vector space. Efx for the expectation of a convex function of a random variable is extended to a generalized class of convex functions f whose domain and range are subsets of possibly infinitedimensional linear topological spaces. In a given inequality, termscoefficients from one side to other side can be transferred as in the case of an equality. And since the variance is always nonnegative, we obtain this inequality.
Consider the american derivative security with payoff g s k in period k. Remember that variance of every random variable x is a positive value, i. Let xbe a random variable, ejxj jensen s inequality can be proved in several ways, and three different proofs corresponding to the different statements above will be offered. Jensens inequality predicts effects of environmental. If we define gx x2, we can write the above inequality as egx. Goal of the lecture 2 main results generalization of jensens inequality generalization of converses of jensens inequality 3 quasiarithmetic means monotonicity difference and ratio type inequalities 4 power functions ratio type order. The value of this security is the same as the value of the simple european derivative security with. On can addsubtract the same real number on both sides of an inequality, the direction of inequality does not change. This is an extension of the definition of convexity on a finite number of points. Jensens inequality is one of the most basic problem solving tools. Generalizations and improvements of converse jensens. Here are some special cases of the power mean inequality. Jensens inequality plays a central role in the derivation of the expectation maximization algorithm 1 and the proof of consistency of maximum likelihood estimators. Jensens inequality in r matija vidmar november 4, 20 proposition 1 jensens inequality.
Proof of jensens inequality why is this progression valid. Jensens inequality plays a central role in the derivation of the expectation maximization algorithm 1 and the proof of consistency of maximum. Hence, riskaversionisequivalent to the concavity of a bernoulli utility function ux. Jensens inequality jensens inequality applies to convex functions.
An introductory look at jensens inequality for a nontechnical audience. You can make it concave either by considering its inverse the. Strong law of large numbers and jensens inequality scott she eld mit 18. For instance, x x k is the payoff function for an american call.
Let f be a function with a positive second derivative. Extension of jensens inequality to affine combinations article pdf available in journal of inequalities and applications 20141. We can find a general formulation in the mesure theoretic article in wikipedia. Thanks for contributing an answer to mathematics stack exchange. Discretized jensens inequality archive ouverte hal. Overview of jensens inequality overview of the kantorovich inequality mondpecari. Feb 18, 2018 there are many, many applications, but one i quite like is an alternative proof of the amgm inequality. Jensens inequality for a convex vectorvalued function on an. Jensens inequality theorem for any concave function f, efx fex proof. Journal of multivariate analysis 4, 5265 1974 jensen s inequality for a convex vectorvalued function on an infinitedimensional space michael d. It can be shown that strict inequality exists unless all of the are equal or is linear on an interval containing all of the. Jensens inequality states that, for any convex function g, we have egx. Question about the last step of this amgm inequality proof. The conformable fractional derivative of a function f.
Oct 14, 20 proof of jensen s inequality for convex functions. Two is equalities with same direction can be added always and multiplied if both sides of the. Pdf on oct 1, 1993, tristan needham and others published a visual explanation of jensens inequality find, read and cite all the research. Jensens inequality can be proved in several ways, and three different proofs corresponding to the different statements above will be offered. The function gx jxjp is convex, because away from 0. Jensen s inequality is an inequality involving convexity of a function.
Convexity, inequalities, and norms 9 applying the same reasoning using the integral version of jensens inequality gives p q z x fpd 1p x fqd 1q for any l1 function f. Tradeintegration socialwelfare inequality costlyredistribution abstract this paper studies the welfare implications of trade opening in a world in which trade raises aggregate. Recent research on jensens inequality for oparators intechopen. And since the variance is always nonnegative, we obtain this. Jensen s inequality is going to tell us something about the expected value of x squared. Convexity of f is defined with respect to closed cone partial orderings, or more general binary relations, on the range of f. To prove this, apply jensens inequality to the convex function xq with weights w j uj p and arguments xj ujvjwj. Exponentiating both sides, we get the amgm inequality. Integral, discrete and functional variants of jensens inequality. The special case p q 2 is called cauchys inequality, and justifies calling the angle.
If we use the generalized jensens inequality instead, we can get the weighted amgm inequality. Firstly, we give a general formulation of jensens operator inequality for a unital field of positive linear mappings see. The proof of jensen s inequality does not address the specification of the cases of equality. In mathematics, jensens inequality, named after the danish mathematician johan jensen. Asking for help, clarification, or responding to other answers. Generalizations and improvements of converse jensens inequality for convex hulls in. Perlman university of chicago, chicago, illinois, and university of minnesota, minneapolis, minnesota communicated by m. Pdf a visual explanation of jensens inequality researchgate. Apr 22, 2010 an introductory look at jensen s inequality for a nontechnical audience. The function gx x2 is an example of convex function. Jensens inequality predicts effects of environmental variation. I if the segment between any two points taken on its graph.
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