Jensen’s con aplicaciones S. Sever Dragomir1 1School of Computational & Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa, Africa Abstract. I used Jensen's inequality for concave function: $$f(x)=\sqrt{x}$$ The expression is homogeneous, so WLOG I assume: $$\frac{2a}{3a+b}+\frac{2b}{3b+c}+\frac{2c}{3c+a}=1$$ Therefore: $$\frac{2a}{3a+b}{\sqrt{3a+b}}+\frac{2b}{3b+c}{\sqrt{3b+c}}+\frac{2c}{3c+a}{\sqrt{3c+a}}\le\sqrt{\frac{2a}{3a+b}(3a+b)+\frac{2b}{3b+c}(3b+c)+\frac{2c}{3c+a}(3c+a)}=\sqrt{2a+2b+2c}$$ And that is not true. Jensen’s inequality - Application to time-delay and sampled-data systems Corentin Briat Abstract—The Jensen’s inequality plays a crucial role in the analysis of time-delay and sampled-data systems. Remarks. f is strictly convex if . Jensen’s inequality states that for a convex function , . Skip to main content Accessibility help We use cookies to distinguish you from other users and to provide you with a better experience on our websites. J Inequal Appl 2011, 101 (2011). Josip Pečarić. Get this Article. In this study, finite-time stability of the linear continuous time-delay systems was investigated. Introduction Less familiar are the ramifications of Jensen's inequality for estimation. Ebrahimi Vishki Abstract. Modern investment theory and asset pricing take place in dynamic settings, which inherently involve compounding. In the solitary atom on the left, electrons are symmetrically distributed. Recently, I came across “On an upper bound for Jensen’s inequality” by Slavko Simic that provides a technique to upper bound , when is a convex function. Stochastic Analysis and Applications: Vol. We study Jensen's inequality for generalized Peng's -expectations and give four equivalent conditions on Jensen's inequality for generalized Peng's -expectations without the assumption that the generator is continuous with respect to .This result includes and extends some existing results. Jensen Heute bestellen, versandkostenfrei 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.It was proved by Jensen in 1906. 2. 2015, No. This result includes and extends some existing results. AN IMPROVEMENT OF JENSEN'S INEQUALITY WITH SOME APPLICATIONS. Jadranka Mici´ ´c Hot Jensen’s inequality and its converses MIA2010 9 / 88. Jensen's inequality is an inequality involving convexity of a function. Die jensensche Ungleichung ist eine elementare Ungleichung für konvexe und konkave Funktionen. If fis strictly convex and Xis not constant then the inequality is strict. Search for more papers by this author. In general though, our main experience of convexity will be through the medium of Jensen’s inequality. However, when two atoms are close together, chance asymmetry in one atom's electron cloud produces an electrical dipole, which in turn induces a complimentary dipole in the other atom. Let f : X → R be a convex mapping and X a Hilbert space. Some standard applications include derivation of the arithmetic-geometric mean inequality, non-negativity of Kullback and Leibler divergence, and the convergence property of the expectation-maximization algorithm (Dempster et al., 1977). Now, in described setting Mićić et. 2 AbstractandAppliedAnalysis whichwillbeusefulinthispaper;inSection3,wegiveour mainresultsincludingtheproofsandapplications. There … For example, applying Jensen’s inequality to the function on the interval one gets the following result. 2 The Exponential Family RASSIAS Abstract. In Sect. We show implementation and results of the bounds in applications (i.e. Mathematics of computing. The application of multi-dimensional Jensen’s inequality for G-martingale. INTRODUCTION Operator inequalities have been important tools in the study of operator alge bras. The theory and applications of dynamic derivatives on time scales has recently received considerable attention. One simplification is to lower bound via Jensen’s inequality or EM. Exponentiating both sides, we get the AM-GM inequality. Some applications related to well-known inequalities are also established. The main purpose of this work is to present essential extensions of results in [7] and [8], and apply them to some special situations. Provided that all expectations are well-defined, the following hold. A novel formulation of the Lyapunov-like function was used to develop a new sufficient delay-dependent condition for finite-time stability. DOI: 10.11650/TJM.14.2010.400 Corpus ID: 8059520. The self-adjoint operators on Hilbert spaces with their numerous applications play an im‐ portant part in the operator theory. 499-508. Sabir Hussain. and. Versions of Jensen's inequality play in important role in several parts of mathe-matics and applications. The reverse upper bound is then derived. Read "On a Converse of Jensen's Discrete Inequality, Journal of Inequalities and Applications" on DeepDyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. For some applications of this result see for example the recent papers [3] and M-The main aim of this paper is to point out a refinement of the reverse of Jensen's inequality stated in (1.1). and. Weprove a pair of general companion inequalities to Jensen’s inequality in its discrete Jensen’s inequality is one of the fundamental inequalities in mathematics, and it underlies many vital statistical concepts and proofs. Let Y = jXjs and apply Jensen’s inequality to g(y) = jyjr, giving (E[jYj])r • E[jYjr]. e work is focused on applications and generalizations of this important result. Some inequalities of Jensen type and connected results are given for quasiconvex functions on convex sets in real linear spaces. 5, we show a Liouville’s theorem for harmonic maps described by Markov chains into \(2\)-uniformly convex space under (B) as an application of our Jensen’s The proper application of Jensen's inequality to the decision turns finance intuition on its head: Multi-period investments with negative risk premia can be profitable, risk-averse investors can have infinite demand for risky securities, settings exist in which risk-averse investors should not diversify, and demand for mutual funds with negative alphas may be rational. 2 Answers2. Classical Jensen’s inequality. Then it follows by Jensen's inequality that $$\mathbb E[\psi(X)]~~\le~~ \mathbb E[\psi(Y)]$$ and... Stack Exchange Network Stack Exchange network consists of 176 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 138, 137-156 (1989) Functional Differential Equations and Jensen's Inequality LEIGH C. DECKER Department of Mathematics, Christian Brothers College, 650 E. Parkway South, Memphis, Tennessee 38104-5581 AND T. A. BURTON* AND SHUNIAN ZHANGt Department of Mathematics, Southern Illinois University, … Stochastic Analysis and Applications: Vol. The idea of the video is taken from a article A Visual Explanation of Jensen's Inequality by Tristan Needham - Abstract. Full Access. This is a fact that can be used to prove that variance is non-negative. Jensen’s inequality is a well known inequality and has diverse applications in communication theory. Let / : / Ç R - >Ròea differentiable convex function on I, Xi G /, Pi > 0 (i = 1, . They represents a refinement of these inequalities in the case of convex/concave functions with numerous applications in Theory of Means and … A re nement and a new sharp reverse of Jensen’s inequality for convex functions in terms of divided diferences is obtained. 1. But you are right, Jensen's inequality tells us that the expected squared payoff is greater than the squared expected payoff. M. Akkouchi, “Inequalities for real random variables connected with Jensen’s inequality and applications”, Rivista di matematica della Universitá di … Applications of linearity of expectations: The coupon collector's problem, randomised quicksort and Jensen's inequality ; Tail bounds: Markov's inequality, Chebyshev's inequality and Chernoff bounds; Application of tail bounds to set balancing and load balancing via balls and bins; Markov chain algorithms for 2SAT, 3SAT and random walks mous Jensen inequality is one of the most usef ul inequalities in probability and statistics, which applies to convex functions. THE RESULTS The following inequality holds. a useful tool in mathematics, specifically in applied fields such as probability and statistics. Applications for norm inequalities in normed linear spaces and f -divergences in information theory are provided as well. Then. Suppose Ω is a measurable subset of the real line and f(x) is a non-negative function such that. In Sect. Of particular interest is the refinement of the integral Jensen inequality for vector valued integrable functions. Its conservatism is studied through the use of the Gru¨ss Inequality. FUNCTIONAL DIFFERENTIAL EQUATIONS AND JENSEN’S INEQUALITY Leigh C. Becker Department of Mathematics Christian Brothers College 650 E. Parkway South Memphis, Tennessee 38104–5581 T.A. For example, applying Jensen’s inequality to the function on the interval one gets the following result. It has been reported in the literature that fragmentation (or partitioning) (2019). We are always given a convex function f defined on an interval I= [a,b], and , and weights which sum to 1. Professor Ng discusses unsupervised learning in the context of clustering, Jensen's inequality, mixture of Gaussians, and expectation-maximization. Finally, a strict algebraic proof is given to validate the reverse-bound. As applications, we obtain an interpolation of the weighted arithmetic- geometric mean inequality for the Karcher mean of positive invertible operators on a Hilbert space. In many applications the population consists of positive members only, so the function f(p) need have a positive second derivative only for positive values of p. The function f(p) = 1/p yields a Jensen inequality for the harmonic mean: H = Xw i p i − 1 P w ip i ≥ 0 (5) A more important case is the geometric inequality. Y k : 7.2 Optimal Exercise of an American Call This follows from Jensen’s inequality. This book offers a concise introduction to mathematical inequalities for graduate students and researchers in the fields of engineering and applied mathematics. Jensen inequality. Jensen computes simple lower bounds on otherwise intractable quantities such as products of sums and latent log-likelihoods. Jensen's Inequality is one of the most used inequality in different exams like #ISI , #CMI , different #olympiads . Keywords and phrases: convex functions, Jensen’s inequality, norms, mean f -deviation, f … The bounds research for self-adjoint operators is a very useful area of this theory. 1. conditional maximum likelihood (CML)). (1)The expectation EX2C (2)If f: C!R is convex then f(EX) Ef(X). models which are intractable. Josip Pečarić. Jensen’s inequality can also be used to give easier proofs of certain other trigonometric inequalities whose direct proofs are either difficult or clumsy. The specifics are as follows: Stack Exchange Network. Jensen’s inequality - Application to time-delay and sampled-data systems Corentin Briat Abstract—The Jensen’s inequality plays a crucial role in the analysis of time-delay and sampled-data systems. Taking , which is convex (because and ), and , we obtain. Mathematics Subject Classification: 26D07, 26D1 . (A) The London dispersion force. Sign in. . Department of Mathematics, Faculty of Science, Islamic Azad University, Central Tehran Branch,Tehran, Iran. In this paper we introduce a new mapping connected with the classical inequality due to Jensen and point out its main properties. Applications for norm inequalities in normed linear spaces and f -divergences in information theory are provided as well. Virtually every practical problem studied in finance requires some nonlinear estimation. Lecture by Professor Andrew Ng for Machine Learning (CS 229) in the Stanford Computer Science department. Mathematical analysis. This paper is organized as follows: in Section2,we introducesomenotations,assumptions,notions,andlemmas Department of Mathematics, National Central University, Taiwan; Department of Information Management, Lunghwa University of Science and Technology, Taiwan. I will give two applications of the method – USAMO 2017, problem 6 and CGMO 2007, p3. One simplification is to lower bound via Jensen’s inequality or EM. conditional maximum likelihood (CML)). ∙ Association for Computing Machinery ∙ 0 ∙ share . In this video we prove two versions of Jensen's Inequality. ON JENSEN’S MULTIPLICATIVE INEQUALITY FOR POSITIVE CONVEX FUNCTIONS OF SELFADJOINT OPERATORS IN HILBERT SPACES SILVESTRU SEVER DRAGOMIR1 Communicated by H.R. Some Equivalent Forms of Bernoulli’s Inequality: A Survey() Yuan-Chuan Li, Cheh-Chih Yeh. To use Jensen's inequality, we need to determine if a function g is convex. Similarly, arithmetic mean - geometric mean inequality ( AM-GM) can be obtained from Jensen's inequality by considering . 2000 Mathematics subject classification: primary 26D15; secondary 94A17. The relation to variance is incidental in this example. The proposed new bound is simple and insightful, is broadly applicable by imposing minimum assumptions, and provides fairly accurate results in spite of its simple form. Probability Inequalities Jensen’s Inequality has many applications. Fractional Jensen’s Inequality Serkan ASLIYÜCE and A. Feza GÜVEN˙IL ˙IR Communicated by Jose Luis Lopez-Bonilla MSC 2010 Classifications: Primary 26A33, ... For more information and applications on conformable fractional derivative and integral, we refer [1, 4, 11, 17]. Jensen’s inequality is an extension of 1 . Jensen’s inequality states that for a convex function , . Tightness of Jensen’s Bounds and Applications to MIMO Communications Abstract: Due to the difficulty in manipulating the distribution of Wishart random matrices, the performance analysis of multiple-input-multiple-output (MIMO) channels has mainly focused on deriving capacity bounds via Jensen's inequality. Introduction A Refinement of Jensen's Inequality with Applications for f-Divergence Measures @article{Dragomir2010ARO, title={A Refinement of Jensen's Inequality with Applications for f-Divergence Measures}, author={S. Dragomir}, journal={Taiwanese Journal of Mathematics}, year={2010}, volume={14}, pages={153-164} } Jensen's inequality generalizes the statement that the secant line of a convex function lies above the graph of the function, which is Jensen's inequality for two points: the secant line consists of weighted means of the convex function, t f + f, {\displaystyle tf+f,} while the graph of the function is the convex function of the weighted means, f. {\displaystyle f\left.} In a true mathematical sense, It is superior to most inequalities as those others are just a special case of this one. It begins by reviewing essential facts from algebra and calculus and proceeds with a presentation of the central inequalities of applied If fis strictly concave and Xis Get this Article. For other refinements and applications related to Ky Fan’s inequality, the arith-metic mean-geometric mean inequality, the generalized triangle inequality, the f-Divergence measure etc., see [3]-[12]. We study Jensen's inequality for generalized Peng's g -expectations and give four equivalent conditions on Jensen's inequality for generalized Peng's g -expectations without the assumption that the generator g is continuous with respect to t . Check if you have access through your login credentials or your institution to get full access on this article. For more information and applications on conformable fractional derivative and integral, we refer [1, 4, 11, 17]. The bounds research for self-adjoint operators is a very useful area of this theory. There is no better inequality in bounds examination than Jensen's inequality. ON JENSEN’S MULTIPLICATIVE INEQUALITY FOR POSITIVE CONVEX FUNCTIONS OF SELFADJOINT OPERATORS IN HILBERT SPACES SILVESTRU SEVER DRAGOMIR1 Communicated by H.R. This is a fact that can be used to prove that variance is non-negative. But we can apply the following trick. ity, Jensen’s inequality for convex functions and Fenchel’s dual inequality In this section, we review three basic inequalities that are Cauchy-Schwarz inequality, Jensen’s inequality for convex functions and Fenchel’s inequality for duality. If is concave then for any probability measure on we have Jensen’s inequality holds for all measurable real valued functions provided that is measurable.. Probabilist would write Jensen’s inequality as where is Borel measurable real valued random variable.. Jensen’s inequality with several variables has series of applications. A new form of Jensen's inequality and its application to statistical experiments - Volume 36 Issue 4. ... which has applications in the Calculus of Variations. Department of Applied Mathematics, National Chung-Hsing University, Taiwan. In this chapter we give cyclic refinements of Jensen’s inequality and their applications. Jensen’s inequality is a well known inequality and has diverse applications in communication theory. A list of 10 econometric theorems was circulated on Twitter citing what Jeffrey Wooldridge claims you just need to apply repeatedly in order to do econometrics. an assertion about how convex functions interact with expected values of random variables, Jensen’s Inequality: Let f : ... Jensen’s Inequality has variety of applications. Note that the proof holds for any finite dimensions as long as we know that the subgradient exists. Jensen’s inequality can also be used to give easier proofs of certain other trigonometric inequalities whose direct proofs are either difficult or clumsy. Ebrahimi Vishki Abstract. Accepted: 31 October 2011. Received: 17 April 2011. FURTHER REVERSE RESULTS FOR JENSEN’S DISCRETE INEQUALITY AND APPLICATIONS IN INFORMATION THEORY I. BUDIMIR, S.S. DRAGOMIR, AND J. PECARIˇ C´ Abstract. 2. e work is focused on applications and generalizations of this important result. There is no better inequality in bounds examination than Jensen's inequality. This implies that (E[jXjs])t=s • E[jXjt] Taking the tth root of each side gives the result. Now, let be a Rademacher random variable. Actually, Jensen’s inequality usually is not true for all monetary utility functions even when the related convex or concave function is a linear function. Due to Jensen’s inequality, the EM algorithm is designed to maximizing its lower-bound iteratively. For example, there is a set of observations , latent variable and parameter . According to probability theory, we have: is optimal when is maximum, as well as is maximum. This article proposes a new sharpened version of Jensen's inequality. Corollary 2.25 Given a convex function g : 0;1 !IR where g It has numerous applications in abstract and applied sciences ; One of the simplest examples of Jensen's inequality is the quadratic mean - arithmetic mean inequality. Applications for Its applications therein include the EM algorithm, Bayesian estimation and Bayesian inference. If is not monotone, then we translate so that its minimum and express where with. A MAPPING ASSOCIATED WITH JENSEN’S INEQUALITY AND APPLICATIONS S.S. DRAGOMIR AND T.M. Login options. 2. We obtain some multiplicative refinements and reverses of Jensen's inequality for positive convex/concave functions of selfadjoint operators in Hilbert spaces. Furthermore, we give some applications of Jensen's inequality for generalized Peng's -expectations. An important one is the Inequality … For example, if g(x) = x2, then g ″ (x) = 2 ≥ 0, thus g(x) = x2 is convex over R. Let X be a random variable defined on an open interval with cdf fX(x), and let g(x … - Selection from Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications [Book] An extension of Jensen's discrete inequality to half convex functions. 09/21/2010 ∙ by Frank Nielsen, et al. Jensen computes simple lower bounds on otherwise intractable quantities such as products of sums and latent log-likelihoods. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): Jensen's inequality is a powerful mathematical tool and one of the workhorses in statistical learning. First we derive a refinement of integral Jensen’s inequality associated to two functionswhose sum is equalto unity. If we use the generalized Jensen’s inequality instead, we can get the weighted AM-GM inequality. F.5 JENSEN'S INEQUALITY Theorem F.10 (Jensen's inequality). 37, No. models which are intractable. Jensen’s Inequality: Let C Rdbe convex and suppose that X2C. AB - This article proposes a new sharpened version of Jensen's inequality. . DOI name: Click to select In this article we give some improvements and generalizations of the famous Jensen's and Jensen-Mercer inequalities for twice differentiable functions, where convexity property of the target function is not assumed in advance. Recently, I came across “On an upper bound for Jensen’s inequality” by Slavko Simic that provides a technique to upper bound , when is a convex function. Jensen’s inequality plays a crucial role to obtain inequalities for divergences between probability distributions, which have been introduced to measure the difference between them. Note that the proof holds for any finite dimensions as long as we know that the subgradient exists. (It takes a while.) 1 Weprove a pair of general companion inequalities to Jensen’s inequality in its discrete 3, pp. The reverse upper bound is then derived. Applications of Jensen's inequality. 37, No. Hence, taking expectation over equation ( 1) and using linearity of expectation, I obtain. 2 The Exponential Family Jensen's Inequality Under Nonlinear Expectation Generated by BSDE with Jumps: Na ZHANG 1, Guang-yan JIA 2: 1 School of Science, Tianjin University of Technology, Tianjin 300384, China; 2 Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan 250100, China 10, iss. This presentation can be incorporated in any calculus-based statistical course. ON AN UPPER BOUND FOR JENSEN’S INEQUALITY On An Upper Bound Slavko Simic SLAVKO SIMIC vol. crete Jensen’s inequalities. If is concave then for any probability measure on we have Jensen’s inequality holds for all measurable real valued functions provided that is measurable.. Probabilist would write Jensen’s inequality as where is Borel measurable real valued random variable.. Jensen’s inequality with several variables has series of applications.
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