(noun) We generalize and sharpen Aczél's inequality and Popoviciu's inequality by means of two classical inequalities, a unified improvement of Aczél's inequality and Popoviciu's inequality is given. Popoviciu, S"ophE Found insideNote that in all cases, (Var τ)/δ2≤ 1/4 by Popoviciu's inequality (Popoviciu, 1935); (Var τ)/δ2 = 1/12 for uniformly distributed τ. in the scenario ... In other words, we have to check the sign of {\displaystyle \mathbb {R} } Anyway, I flew back in time to attend the last day of the camp held at Oundle School to select the UK team for this . In Section 4, the obtained result will be used to establish an integral inequality of Aczel-Popoviciu type. As application, an integral inequality of Aczél-Popoviciu type is established. be sequences of positive real numbers such that for .Then with equality if and only if all the sequences are proportional. ⊆ 1.1 Some Inequalities Involving Convex Functions The first chapter contains: introductionto convexfunctions,variousinequalities involving convex functions, refinement of these inequalities given by various researchers in recent Let such that and let . Found inside – Page 60... f ( ai ) + Σ " ] 160 ) n 2 ( n − 1 ) n i = 1 From a generalization of Popoviciu's Inequality proved by the proposer of this problem and presented in ... We establish a general criterion for the validity of inequalities of the following form: A certain convex combination of the values of a convex function at n points and of its value at a weighted mean of these n points is always greater or equal to a convex combination of the . in some cases this inequality can be a powerful tool for proving other inequalities where Jensen’s inequality does not work. A particular inequality about convex functions, similar to Jensen's inequality . Then Popoviciu's inequality states: \text{variance} \le \frac14 (M - m)^2. Already have an account? NICULESCU Abstract. 2 As consequences, several interesting integral inequalities of Acz´el-Popoviciu-Bellman type are obtained. 3 0 obj << , and any n points He was 69 years old when he died. Popoviciu's inequality can give us an upper bound to the sample variance. It is similar to Jensen's inequality and was found in 1965 by Tiberiu Popoviciu, a Romanian mathematician. Then Popoviciu's inequality states: ().This equality holds precisely when half of the probability is concentrated at . Jensen's inequality, Popoviciu's inequality, 2D-convex function. 1 thought on " Popoviciu's Inequality " matthewhr on June 22, 2014 at 8:15 pm said: Thank you for the reference to Grinberg's paper. Sign up, Existing user? The most prominent example of such kind of inequalities, Popoviciu's inequality in its most general form, follows from the general criterion. See also . Now, in 1965, a similarly styled inequality was found by the Romanian Tiberiu Popoviciu: Theorem 2a, the Popoviciu inequality. We investigate the bounds for the identities related to the generalization of the Popoviciu inequality using inequalities for the Čebyšev functional. Constantin P. Niculescu; Find more articles with the same subject classes. 469 3 3 silver badges 8 8 bronze badges COMBINATORIAL EXTENSIONS OF POPOVICIU'S INEQUALITY 61 For two point right focal problem, the Abel-Gontscharoff theorem (see [1])isgivenas Theorem 1.4. Found inside – Page 97(3) Inequality (3) was first proved, in the case of equal weights by Popoviciu [2] , and is known as Popoviciu's inequality. It was implied in an earlier ... Then for any x,y,z∈[a,b]x, y, z \in [a, b]x,y,z∈[a,b] we have. In convex analysis, Popoviciu's inequality is an inequality about convex functions. x If f is convex, then for any three points x, y, z in I, If a function f is continuous, then it is convex if and only if the above inequality holds for all x, y, z from Several applications are included. If we knew the distribution where the samples stem from, then it would be easy to answer this question. Found inside – Page 53Given Popoviciu's inequality on variances , the variance of y . , is bounded by var ( ski ) < 1 / ( 462 ) . ( 5.12 ) Since each element in y ... {\displaystyle \mathbb {R} } Introduction The Pólya-Szegö's inequality can be stated as follows ([1] or ([2], p. 62)). Log in. □. >> n I've just returned to the UK after an excellent stay at the University of British Columbia. Let f be a continuous function from an interval In this paper, we show several new generalized and sharpened versions of Aczél's inequality and Popoviciu's inequality, our results contain as special cases the improvement of certain known results on Aczél's inequality and Popoviciu's inequality. Primary; 26A51; Secondary; 26D15; Find more articles with the same keywords. www.ssmrmh.ro 1 RMM-HORNICH-HLAWKA-POPOVICIU'S INEQUALITIES REVISITED HORNICH-HLAWKA-POPOVICIU'S INEQUALITIES REVISITED By Florică Anastase-Romania ABSTRACT: In this paper are presented proofs, generalizations, equivalent forms and connection between famous Hornich-Hlawka and Popoviciu's inequalities and applications. 3. Found inside – Page 1076( English summary ) Improvement of Aczél's inequality and Popoviciu's inequality . Univ . Beograd . Publ . Elektrotehn . Fak . Ser . Mat . Although maximums and minimums can be found using methods from calculus, the application of a classical inequality is often a simpler approach. Functions with Non-decreasing Increments and Popoviciu Type Identities and Inequalities for Sums and Integrals. Follow asked 9 mins ago. Problems illustrating important mathematical techniques with solutions and accompanying essays. Found inside – Page 502The inequality Eq . 4 reduces to an equality for f = 0 and thus we may , without ... Now , let us consider Popoviciu's inequality 3 . w ; dil .... in 2 I w ... As application, we establish a Minkowski inequality, which in special case yields the well-known dual Minkowski inequality for volumes difference. What does popoviciu-s-inequality mean? Popoviciu's Inequality. We generalize and sharpen Aczél's inequality and Popoviciu's inequality by means of two classical inequalities, a unified improvement of Aczél's inequality and Popoviciu's inequality is given. In 1965, T. Popoviciu [8] (see also [4], [7, p.171]) proved the following characteri- zation of convex functions: Theorem 1. (convex analysis) A particular inequality about convex functions, similar to Jensen's inequality. x [5][6][7], Popoviciu's paper has been published in Romanian language, but the interested reader can find his results in the review, https://en.wikipedia.org/w/index.php?title=Popoviciu%27s_inequality&oldid=1037374807, Creative Commons Attribution-ShareAlike License, This page was last edited on 6 August 2021, at 05:30. Found inside – Page 964Alg. Appl., Vol. 142, pp. 43–54, 1990. (Cited on pages 394 and 395.) S. Wu, “A Further Generalization of Aczel's Inequality and Popoviciu's Inequality,” ... Found inside – Page 93... a Jensen type functional equation that arises from Popoviciu's inequality. ... satisfies the inequality f ( x + y 2 ) ≤ f(x) + 2 f(y) (7.1) for all x, ... Found inside – Page 7Authors improve a number of algebraic inequalities and obtain some new ones . ... In the end , authors has given an analogue of Popoviciu's inequality . In probability theory, Popoviciu's inequality, named after Tiberiu Popoviciu, is an upper bound on the variance σ 2 of any bounded probability distribution. Found inside – Page 220R Rado's Geometric - Arithmetic Mean Inequality Extension If n > 2 , then Wn ... Popoviciu's GeometricArithmetic Mean Inequality Extension COMMENTS ( ii ) . from I, Popoviciu's inequality can also be generalized to a weighted inequality. Found inside – Page 400POPOVICIU-type inequalities 94, 98 – 102. POPOvICIU's inequality 58: positive sequence XII. — vector XII. pseudo-means 338. quadratic form, ... in some cases this inequality can be a powerful tool for proving other inequalities where Jensen's inequality does not work. Key Words: Convex function, right derivative, Leibniz- Newton for-mula f(s+ . I Some of the proofs I've seen have first shown … THE INTEGRAL VERSION OFPOPOVICIU'S INEQUALITY 325 By Lemma1above,the proofofthe integralversionof Popoviciu'sinequalitycan be reducedto the case of functionsof the form f(x)=(x−c)+, where c is a real parameter. inequality jensen-inequality. Without loss of generality, we assume that x≤y≤z.x \le y \le z.x≤y≤z. In convex analysis, Popoviciu's inequality is an inequality about convex functions. Found inside – Page 1733.4.6 Rado–Popoviciu inequality For x = (x1,...,xn) e IR'', let A(x) = *. Gn(X) = VII* i=1 The inequalities n(An(x) - Gn(x)) > (n − 1)(An_1(x) – Gh 1(x)) ... %PDF-1.4 Read full biography. Found inside – Page 226Rado's inequality | 49 3.2 . Popoviciu's inequality | 50 3.3 . Extensions of the inequalities of Rado and Popoviciu | 52 3.4 . The results of Everitt | 54 ... It is similar to Jensen's inequality and was found in 1965 by Tiberiu Popoviciu,[1][2] a Romanian mathematician. The main aim of this paper is to initiate a more consequent study of classical inequalities in this more general frame. \end{aligned}f(2x+z)≤s⋅f(3x+y+z)+(1−s)⋅f(z)f(2y+z)≤t⋅f(3x+y+z)+(1−t)⋅f(z). Finally, we establish the time scales version of Beckenbach-type inequality. THE INTEGRAL VERSION OF POPOVICIU'S INEQUALITY CONSTANTINP. Comments: The subject class "functional analysis" is an approximation; I would describe the field as "elementary inequalities for convex . Generalizations of Popoviciu's inequality Grinberg, Darij; Abstract. Found inside – Page 505... 123 Planck's constant 414 plank problem 473 polynomial interpolation 73–90 Popoviciu's inequality 8, 9, 10 pseudo arithmetic and geometric means 7, ... Found inside – Page 362DARST, R. and H. POLLARD, An inequality for the Riemann-Stieltjes integral, Proc. Amer. ... PECARIC, J. E., On an inequality of T. Popoviciu, Bul. Sti. The upper bound derived from Popoviciu's inequality is 1 4(b . 2. . [ a, b]. MathSciNet MATH Google Scholar 14. 3x+y+z≤2x+z≤z and 3x+y+z≤2y+z≤z. Popoviciu's Inequality For N Convex Functions Josip Pecaric, Muralism without Walls: Rivera, Orozco, and Siqueiros in the United States, 1927-1940 (Pitt Illuminations)|Anna Indych-Lopez, Coalbed Methane And Coal Geology (Geological Society of London Special Publications)|R. POPOVICIU'S INEQUALITY REVISITED. POPOVICIU'S INEQUALITY REVISITED. I NICULESCU Abstract. As application, an integral inequality of Aczél-Popoviciu type is established The integral version of Popoviciu's inequality. Tiberiu Popoviciu was a Romanian mathematician and the namesake of Popoviciu's inequality andPopoviciu's inequality on variances. Wu S, Debnath L: Generalizations of Aczél's inequality and Popoviciu's inequality. Are being queued ) improvement of Aczél 's inequality proposed by Cezar Lupu, University of British Columbia using! Exampleintegral versionof Popoviciu & # x27 ; s inequality will be used in the same keywords 2005,36 ( 2:49-62. 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New results provide new estimates on inequalities, such as Jensen ’ s inequality, 2D-convex function ( 2:49-62...
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