Pdf a visual explanation of jensens inequality researchgate. The value of this security is the same as the value of the simple european derivative security with. Perlman university of chicago, chicago, illinois, and university of minnesota, minneapolis, minnesota communicated by m. The proof of jensens inequality does not address the specification of the cases of equality. Jensens inequality predicts effects of environmental. Feb 18, 2018 there are many, many applications, but one i quite like is an alternative proof of the amgm inequality. 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. For instance, x x k is the payoff function for an american call. Generalizations and improvements of converse jensens inequality for convex hulls in. Pdf extension of jensens inequality to affine combinations. There are many, many applications, but one i quite like is an alternative proof of the amgm inequality. 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.
It states that if a1, 0an and 0 jensens inequality konstantinos g. Two is equalities with same direction can be added always and multiplied if both sides of the. Globalization, inequality and welfare harvard university. Question about the last step of this amgm inequality proof. Firstly, we give a general formulation of jensens operator inequality for a unital field of positive linear mappings see.
A story about pedro strong law of large numbers jensens inequality. Extension of jensens inequality to affine combinations article pdf available in journal of inequalities and applications 20141. Jensens inequality states that, for any convex function g, we have egx. Jensens inequality can be proved in several ways, and three different proofs corresponding to the different statements above will be offered. Tradeintegration socialwelfare inequality costlyredistribution abstract this paper studies the welfare implications of trade opening in a world in which trade raises aggregate. You can make it concave either by considering its inverse the exponential function or by working with lnx. 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. Jensen inequality basics this is a revision of material that appeared earlier in sep 37 and reprinted in pvi. 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 function gx jxjp is convex, because away from 0. An introductory look at jensens inequality for a nontechnical audience.
I if the segment between any two points taken on its graph. Recent research on jensens inequality for oparators. In a given inequality, termscoefficients from one side to other side can be transferred as in the case of an equality. 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. So jensens inequality gives that log 1 n xn k1 a k logex e logx 1 n xn k1 loga k. Hence, riskaversionisequivalent to the concavity of a bernoulli utility function ux. 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. If we use the generalized jensens inequality instead, we can get the weighted amgm inequality. Jensens inequality for a convex vectorvalued function on an. The function gx x2 is an example of convex function. Journal of multivariate analysis 4, 5265 1974 jensen s inequality for a convex vectorvalued function on an infinitedimensional space michael d.
The proof of jensen s inequality does not address the specification of the cases of equality. Now, for this expectation, we already know that this is equal to the variance of x plus the square of the expected value. Intuitively a function is convex if it is upward bending. 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. The conformable fractional derivative of a function f. And since the variance is always nonnegative, we obtain this. Jensens inequality plays a central role in the derivation of the expectation maximization algorithm 1 and the proof of consistency of maximum likelihood estimators. If f is a convex function on i, then the wellknown jensens inequality 1, 4 asserts that. Remember that variance of every random variable x is a positive value, i. We can find a general formulation in the mesure theoretic article in wikipedia. You can make it concave either by considering its inverse the. Pdf in this paper, we present more proofs of the new steffensens inequality for convex functions. Discretized jensens inequality archive ouverte hal.
Recent research on jensens inequality for oparators intechopen. Overview of jensens inequality overview of the kantorovich inequality mondpecari. Thanks for contributing an answer to mathematics stack exchange. Jensens inequality jensens inequality applies to convex functions. 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. Jensen s inequality is an inequality involving convexity of a function. Norms a norm is a function that measures the lengths of vectors in a vector space.
Asking for help, clarification, or responding to other answers. Proof of jensens inequality why is this progression valid. On a new converse of jensens inequality slavko simic. Generalization of jensens inequality to multivariate functions.
Jensen s inequality is going to tell us something about the expected value of x squared. I r defined on some open inter val and a selfadjoint element a. Rao jensen s inequality f ex 6 ef x for the expectation of a convex function of a random variable is extended to a. Jensens inequality in r matija vidmar november 4, 20 proposition 1 jensens inequality. Let f be a function with a positive second derivative. Oct 14, 20 proof of jensen s inequality for convex functions. Pdf on oct 1, 1993, tristan needham and others published a visual explanation of jensens inequality find, read and cite all the research. Consider the american derivative security with payoff g s k in period k. Apr 22, 2010 an introductory look at jensen s inequality for a nontechnical audience. On can addsubtract the same real number on both sides of an inequality, the direction of inequality does not change. Jensens inequality for a convex vectorvalued function on.
Strong law of large numbers and jensens inequality scott she eld mit 18. The special case p q 2 is called cauchys inequality, and justifies calling the angle. If we define gx x2, we can write the above inequality as egx. Convexity of f is defined with respect to closed cone partial orderings, or more general binary relations, on the range of f.
Lecture 32 strong law of large numbers and jensens. Jensens inequality convex functions and a proof for. And since the variance is always nonnegative, we obtain this inequality. Jensens inequality is an inequality involving convexity of a function.
354 1162 648 313 1097 1176 1270 1110 1433 247 551 1183 907 496 900 852 625 909 565 594 130 612 376 486 1495 1033 1325 818 261 1085 139 841 1303 193 208 1267 451 1496 548 1167 1217 444