A good explanation of jensens inequality should get the oh, duh reaction. At the end of this paper, let us discuss an application of jensen s inequality for monetary utility functions. The article provides the generalization of jensens inequality for convex functions on the line segments. Feb 10, 2015 proof of the cauchyschwarz inequality vectors and spaces linear algebra khan academy duration. A common description of this theorem would be that the function of the expectation is always less than. Jensens inequality holds that a function is convex in the interval if and only if the following inequality is satisfied for all in and for all with. Overview of jensens inequality overview of the kantorovich inequality mondpecari.
Jensens inequality theorem for any concave function f, efx fex proof. Jensens inequality provides us with an important tool for deciding whether noise will be beneficial or not. This is an extension of the definition of convexity on a finite number of points. Sandor, some inequalities for uniformly convex spaces,mathematica cluj,34 1992, 38. Jensen s inequality is an inequality involving convexity of a function.
Featured on meta the q1 2020 community roadmap is on the blog. May 31, 2012 the intuition behind jensen s inequality duration. Aug 19, 2014 the article provides the generalization of jensens inequality for convex functions on the line segments. The notion of convexity is very close to jensens inequality. A sequence of inequalities which include mcshanes generalization of jensens inequality for isotonic positive linear functionals and convex fu.
The resulting quasiarithmetic means are used to extend the two wellknown inequalities. The function gx x2 is an example of convex function. Properties of some functionals related to jensens inequality. Utility and risk preferences part 1 utility function duration.
Inequalities and equations solver math 101 wolfram alpha. Now, for this expectation, we already know that this is equal to the variance of x plus the square of the expected value. There are a lot of refinements of the discrete jensens inequality, and this problem has been studied by many authors. 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 is an inequality involving convexity of a function. Since the graph of every convex function lies above its tangent line at every point, we can compare the function with the linear function, whose graph is tangent to the graph of at the point.
Jensen s inequality can be proved in several ways, and three different proofs corresponding to the different statements above will be offered. In addition to establishing jensens inequality which will be used in the next post, the theorem above nicely connects the notion of a convex set with that of a convex function. Some inequalities of jensen type and connected results are given for quasiconvex functions on convex sets in real linear spaces. Jensen s inequality 2 can be generalized by taking instead a probability measure on a algebra in a set, a bounded realvalued function in and a convex function on the range of. Jensens inequality states that, for any convex function g, we have egx. Download wolfram player this demonstration allows you to input a system of inequalities that will be graphed for you. Convex sets and jensens inequality university college dublin. Examples of jensen inequalities the most familiar example of a jensen inequality occurs when the weights are all equal to 1n and the convex function is fx x2. Pdf some integral inequalities are proved in probability spaces, which go back to some. Dragomir, an improvement of jensens inequality,bull. The following discrete jensens inequality is also well known see 5.
Namely, if f is a function analytic in a disk of radius r centered at z 0 and if f is bounded by m on the boundary of that disk, then the number of zeros of f in a circle of radius r feb 18, 2018 there are many, many applications, but one i quite like is an alternative proof of the amgm inequality. Jensens inequality is one of the most basic problem solving tools. Concave function, convex function, jensens formula references. To embed a widget in your blogs sidebar, install the wolframalpha widget sidebar plugin, and copy and paste the widget id below into the id field. Pdf jensens inequality for operators and l wners theorem. If all the p j are the same, then both of the two terms in s are the same, and s vanishes. 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. Jensen s inequality holds that a function is convex in the interval if and only if the following inequality is satisfied for all in and for all with. This is accomplished by the introduction of genuine noncommutative convex combinations of operators, as opposed to the contractions used in earlier versions of the theory.
What can we say about the expected value of, by using jensen s inequality. Extension of jensens inequality to affine combinations. Jensens inequality can be proved in several ways, and three different proofs corresponding to the different statements above will be offered. The purpose of this paper is to give some refinements of the classical jensens inequality. If its that obvious, itll be easy to extrapolate to other related circumstances. Jensens inequality predicts effects of environmental variation. The function s defined by is like the entropy defined in thermodynamics.
Using a locator you can then choose test points for each inequality and shade on the side of the locator. Functions on linear spaces, mathematica moravica vol. The main and preliminary inequalities are expressed in discrete form using affine combinations that can be reduced to convex combinations. Solving linear inequalities wolfram demonstrations project. The natural logarithm is a strictly concave function, because its second derivative is strictly negative on its domain of definition. So jensens inequality gives that log 1 n xn k1 a k logex e logx 1 n xn k1 loga k. Hence, minimizing s is like urging all the p j to be identical. The special case of equal with the concave function gives.
Jensens inequality describes how variance depresses the response variable in decelerating functions and elevates the response variable in accelerating functions. Let s, f, p be a probability space along with a sequence of subsigma. The demonstration solves linear inequalities of the form in terms of the parameter. Jensens inequality jensens inequality applies to convex functions. Indirectly, it also provides an indication of when the addition of noise will be most beneficialif ventilation is centred at or near the point at which the convexity is largest or where the second derivative, f. If is concave, then the inequality reverses, giving. Inequalities and equations solver math 101 added aug 1, 2010 by fawad in mathematics this widget gives the solutions of simple and compound inequalities and equations. In spite of this, few papers have been published dealing with this problem. Jensen s inequality is one of the most basic problem solving tools. The function gx jxjp is convex, because away from 0.
Since the inequality is of considerable utility, it seems worth while to. Jensens inequality for operators and l wners theorem article pdf available in mathematische annalen 2583 september 1982 with 368 reads how we measure reads. As a consequence, we no longer need to impose conditions on the. Periodica mathematica hungarica volume 58, pages91542009cite this article. Graphing systems of inequalities wolfram demonstrations. Jensens inequality jensens inequality is that its converse is also true. Jensens inequality convex functions and a proof for.
Jensens formula can be used to estimate the number of zeros of analytic function in a circle. Convexity, jensens inequality and benefits of noisy. Efz for any random variable z basic inequality is special case with discrete distribution prob z x. Pdf inequalities corresponding to the classical jensens 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. In mathematics, jensens inequa lity, named after the danish mathematician johan jensen, relates the value of a convex function of an integral to the integral of the convex function. Jensens inequality generalizes the statement that the secant line of a convex function lies above the graph of the function, which is jensens inequality for two. Whereas equations can often be solved in terms of numbers, even representing solution sets for inequalities is only made possible by the wolfram languages symbolic capabilities. Expected utility and jensen s inequality free download as powerpoint presentation. Proof of the cauchyschwarz inequality vectors and spaces linear algebra khan academy duration. The wolfram language uses a large number of original algorithms to provide automatic systemwide support for inequalities and inequality constraints. For three hypothetical predators, figs ac, energy gained from a prey item is a function of prey mass.
Horova, bernstein polynomials of convex functions, mathematica cluj 33, no. In addition to establishing jensen s inequality which will be used in the next post, the theorem above nicely connects the notion of a convex set with that of a convex function. 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. Concave function, convex function, jensen s formula references. It is also a natural problem to give analogous results for the classical jensens inequality. At the end of this paper, let us discuss an application of jensens inequality for monetary utility functions.
Jensens inequality for monetary utility functions springerlink. Jensens inequality 2 can be generalized by taking instead a probability measure on a algebra in a set, a bounded realvalued function in and a convex function on the range of. Browse other questions tagged inequality convexanalysis contestmath proofexplanation functionalinequalities or ask your own question. What can we say about the expected value of, by using jensens inequality. The simplest form of jensens inequality is that if 4x is a. Jensens inequa lity is going to tell us something about the expected value of x squared. Jensen s formula can be used to estimate the number of zeros of analytic function in a circle. If we define gx x2, we can write the above inequality as egx. Intuitively a function is convex if it is upward bending.
Equilibrium is when s is reduced to the smallest possible value which satisfies any constraints that may be applicable. To add a widget to a mediawiki site, the wiki must have the widgets extension installed, as well as the code for the wolframalpha widget. We apply jensens inequality to the convex function fz zp, writing. Expected utility and jensen s inequality risk aversion. Let xbe a random variable, ejxj jensens inequality basic inequality.
I if the segment between any two points taken on its graph. Remember that variance of every random variable x is a positive value, i. What is an intuitive explanation of jensens inequality. Given its generality, the inequality appears in many forms depending on the context, some of which are presented below. And since the variance is always nonnegative, we obtain this inequality.
Fix 2r such that t 0 jensens inequality in r matija vidmar november 4, 20 proposition 1 jensens inequality. Pdf jensens inequality for convex functions on ncoordinates. Namely, if f is a function analytic in a disk of radius r centered at z 0 and if f is bounded by m on the boundary of that disk, then the number of zeros of f in a circle of radius r jensens inequality gives that log 1 n xn k1 a k logex e logx 1 n xn k1 loga k. A refinement of the integral form of jensens inequality. We establish what we consider to be the definitive versions of jensens operator inequality and jensens trace inequality for functions defined on an interval.
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