Jensen's inequality
Web12 nov 2024 · The Jensen inequality is a widely used tool in a multitude of fields, such as for example information theory and machine learning. It can be also used to derive other standard inequalities such as the inequality of arithmetic and geometric means or the Hölder inequality. In a probabilistic setting, the Jensen inequality describes the … Web6 lug 2010 · Many important inequalities depend upon convexity. In this chapter, we shall establish Jensen's inequality, the most fundamental of these inequalities, in various forms. A subset C of a real or complex vector space E is convex if whenever x and y are in C and 0 ≤ θ ≤ 1 then (1 − θ) x + θ y ∈ C. This says that the real line segment ...
Jensen's inequality
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http://users.mat.unimi.it/users/libor/AnConvessa/Jensen.pdf Web24 mar 2024 · Jensen's Inequality. If , ..., are positive numbers which sum to 1 and is a real continuous function that is convex, then. which can be exponentiated to give the arithmetic mean - geometric mean inequality. Here, equality holds iff .
Web12 nov 2024 · The Jensen inequality is a widely used tool in a multitude of fields, such as for example information theory and machine learning. It can be also used to derive other standard inequalities such as the inequality of arithmetic and geometric means or the … WebEvan Chen (April 30, 2014) A Brief Introduction to Olympiad Inequalities Example 2.7 (Japan) Prove P cyc (b+c a)2 a 2+(b+c) 3 5. Proof. Since the inequality is homogeneous, we may assume WLOG that a+ b+ c= 3. So the inequality we wish to prove is X cyc (3 2a)2 a2 + (3 a)2 3 5: With some computation, the tangent line trick gives away the magical ...
WebLet us return to the Jensen inequality. We can apply it to an image measure to obtain the following Theorem 0.7 (Second Jensen inequality). Let (; ; ) be a probability measure space, and g: !Rd a measurable mapping that is -integrable. Let CˆRd be a convex set such that g(!) 2Cfor -a.e. !2, and f: C!(1 ;+1] a l.s.c. convex function. Then: R gd 2C;
WebJensen's inequality applies to convex and concave functions. The properties of these functions that are relevant for understanding the proof of the inequality are: the tangents of a convex function lie entirely below its graph; the tangents of a concave function lie …
WebJensen’s Inequality is a statement about the relative size of the expectation of a function compared with the function over that expectation (with respect to some random variable). To understand the mechanics, I first define convex functions and then walkthrough the logic … club tennis ile bizardhttp://users.mat.unimi.it/users/libor/AnConvessa/Jensen.pdf club tennis hospitaletWebInégalité de Jensen. En mathématiques, et plus précisément en analyse, l’ inégalité de Jensen est une relation utile et très générale concernant les fonctions convexes, due au mathématicien danois Johan Jensen et dont il donna la preuve en 1906. On peut l'écrire de deux manières : discrète ou intégrale. Elle apparaît notamment ... cabled concrete block matsWeb22 feb 2015 · Letting c: = ∫Xfdμ, it follows that for every κ ∈ [λ, ρ] and t ∈ (a, b) we have. ϕ(t) ⩾ ϕ(c) + κ ⋅ (t − c) and hence. ϕ(f(x)) ⩾ ϕ(c) + κ ⋅ (f(x) − c) for every x ∈ X. Integrating (2) gives Jensen's inequality, and it follows that we have the equality. ∫Xϕ ∘ fdμ = ϕ(∫Xfdμ) if … cabled definition bibleWebLa disuguaglianza di Jensen (dal nome del matematico danese Johan Jensen) è una disuguaglianza che lega il valore di una funzione convessa al valore della medesima funzione calcolata nel valor medio del suo argomento. Essa è stata enunciata e … club tennis marly le roiWebThe integral form of Jensen's inequality can be phrased in terms of permuting a convex function $\varphi$ (say, with the prop... Stack Exchange Network Stack Exchange network consists of 181 Q&A communities including Stack Overflow , the largest, most trusted … cabled crewneck sweater crochet patternWeb17 mag 2024 · Abstract. We generalize Jensen’s integral inequality for real Stieltjes measure by using Montgomery identity under the effect of convex functions; also, we give different versions of Jensen’s discrete inequality along with its converses for real weights. As an application, we give generalized variants of Hermite–Hadamard inequality. club tennis els gorchs