A note on multivariate Ostrowski type inequalities *.1. Introduction Throughout, R and [R.sup.n] denote de·note tr.v. de·not·ed, de·not·ing, de·notes 1. To mark; indicate: a frown that denoted increasing impatience. 2. the set of real numbers and the n--dimensional Euclidean space Euclidean space In geometry, a two- or three-dimensional space in which the axioms and postulates of Euclidean geometry apply; also, a space in any finite number of dimensions, in which points are designated by coordinates (one for each dimension) and the distance between , respectively. Let D = {([x.sub.1], ..., [x.sub.n])[member of] [R.sup.n]: [a.sub.i] < [x.sub.i] < [b.sub.i] (i = 1, ..., n)} and [bar.D] be the closure of D. For a function u (x): [R.sup.n][right arrow] R, we denote the first order partial derivative partial derivative In differential calculus, the derivative of a function of several variables with respect to change in just one of its variables. Partial derivatives are useful in analyzing surfaces for maximum and minimum points and give rise to partial differential by [[partial derivative]u(x)/[partial derivative][x.sub.i]] (i = 1, ..., n) and [[integral].sub.D] u (x) dx the n--fold integral [[integral].sub.a1.sup.b1] ... [[integral].sub.an.sup.bn] u ([x.sub.1], ..., [x.sub.n]) [dx.sub.1] ... [dx.sub.n]. In 1938, Ostrowski [3, p.468] established the following integral inequality inequality, in mathematics, statement that a mathematical expression is less than or greater than some other expression; an inequality is not as specific as an equation, but it does contain information about the expressions involved. : Let f: [a, b] [right arrow] R be continuous on [a, b] and differentiable dif·fer·en·tia·ble adj. 1. That can be differentiated: differentiable species. 2. Mathematics Possessing a derivative. on (a, b). If the derivative derivative: see calculus. derivative In mathematics, a fundamental concept of differential calculus representing the instantaneous rate of change of a function. f' is bounded on (a, b), that is, [MATHEMATICAL EXPRESSION A group of characters or symbols representing a quantity or an operation. See arithmetic expression. NOT REPRODUCIBLE re·pro·duce v. re·pro·duced, re·pro·duc·ing, re·pro·duc·es v.tr. 1. To produce a counterpart, image, or copy of. 2. Biology To generate (offspring) by sexual or asexual means. IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. ] then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1) for all x [member of] [a; b]. The inequality (1:1) is known in the literature as the Ostrowski inequality. For some recent results which generalize generalize /gen·er·al·ize/ (-iz) 1. to spread throughout the body, as when local disease becomes systemic. 2. to form a general principle; to reason inductively. , improve and extend the inequality (1:1); see [1--8]. In [8] Pachpatte established the following two theorems This is a list of theorems, by Wikipedia page. See also
Theorem theorem, in mathematics and logic, statement in words or symbols that can be established by means of deductive logic; it differs from an axiom in that a proof is required for its acceptance. A. Let f; g: [R.sup.n] [right arrow] R be functions continuous on [bar.D], differentiable on D and whose derivatives [[partial derivative]f(x)/[partial derivative][x.sub.i]] and [[partial derivative]g(x)/[partial derivative][s.sub.i]] (i = 1, ...,n) are buonded,i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all i = 1, ...,n. Let the function w (x) be nonnegative non·neg·a·tive adj. Of, relating to, or being a quantity that is either positive or zero. Adj. 1. nonnegative - either positive or zero , integrable for every x [member of] D and [[integral].sub.D] w (y) dy > 0. Then for every x [member of] [bar.D], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3) where M=mesD=[n[proudct](i=1)]([b.sub.i]-[a.sub.i]), dy=[dy.sub.1] ... [dy.sub.n] and [E.sub.i](x)=[[integral].sub.D][absolute value of [x.sub.i]-[y.sub.i]]dy. Theorem B. Let f,g,[[partial derivative]f(x)]/[[partial derivative][x.sub.i]] and [[partial derivative]g(x)]/[[partial derivative][x.sub.i]]] (i=1, ...,n) be defined as in Theorem A. Then for every x [member of] [bar.D], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5) where M, dy and [E.sub.i] (x) are defined as in Theorem A. The main purpose of the present paper is to establish some generalizations of Theorems A and B. 2. Main Results Theorem 1. Let f,g,w,M dy, [[[partial derivative]f(x)]/[[partial derivative][x.sub.i]] [[partial derivative]g(x)]/[[partial derivative][x.sub.i]]] and [E.sub.i] (x) (i = 1, ..., n) be defined as in Theorem A and [alpha], [beta], [member of] R with [alpha] + [beta] = 1. Then for every x [member of] [bar.D]. we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2) Proof. Let x = ([x.sub.1], ..., [x.sub.n]) and y = ([y.sub.1], ..., [y.sub.n]) (x [member of] [bar.D], y [member of] D). By the assumptions on f; g and the n--dimensional version of the mean value theorem In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal to the "average" derivative of the section. , we have (see [4, p. 121] or [9, p. 174]) f(x) - f(y) = [n summation summation n. the final argument of an attorney at the close of a trial in which he/she attempts to convince the judge and/or jury of the virtues of the client's case. (See: closing argument) over(i=1)][[partial derivative]f(c)]/[[partial derivative][x.sub.i]]]([x.sub.i] - [y.sub.i]) (2.3) and g(x) - g(y) = [n summation over(i=1)] [[[partial derivative]g(d)]/[[partial derivative][x.sub.i]]]([x.sub.i] - [y.sub.i]),(2.4) where c = ([y.sub.1] + [gamma]([x.sub.1] - [y.sub.1]), ...,[y.sub.n] + [gamma]([x.sub.n] - [y.sub.n])) and d = ([y.sub.1] + [rho]([x.sub.1] - [y.sub.1]), ...,[y.sub.n] + [rho] ([x.sub.n] - [y.sub.n])) for some [gamma],[rho] [member of] (0,1). Multiplying mul·ti·ply 1 v. mul·ti·plied, mul·ti·ply·ing, mul·ti·plies v.tr. 1. To increase the amount, number, or degree of. 2. Mathematics To perform multiplication on. both sides of (2:3) and (2:4) by [[beta]g](x) and [[alpha]f](x) respectively and adding the resulting identities, we get f(x)g(x) - [alpha]f](x)g(y) - [beta]g(x)f(y) = [alpha]f(x)[n summation over(i=1)]([[[partial derivative]g(d)]/[[partial derivative][x.sub.i]]]([x.sub.i] - [y.sub.i]) + [beta]g(x)[n summation over(i=1)][[[partial derivative]f(c)]/[[partial derivative][x.sub.i]]]([x.sub.i] - [y.sub.i]). (2.5) Integrating both sides of (2:5) with respect to y over D and using M = mesD = [n.[proudct](i=1)]([b.sub.i] - [a.sub.i]), we obtain f(x)g(x) - [[[alpha]f(x) [[infinity infinity, in mathematics, that which is not finite. A sequence of numbers, a1, a2, a3, … , is said to "approach infinity" if the numbers eventually become arbitrarily large, i.e. ].sub.D]g(y)dy + [beta]g(x)[[infinity].sub.D]f(y)dy]/M] = [alpha]f(x)[[infinity].sub.D][n summation over(i=1)] ([[[partial derivative]g(d)]/[[partial derivative][x.sub.i]]]([x.sub.i] - [y.sub.i]) + [beta]g(x)[n summation over(i=1)] [[[partial derivative]f(c)]/[[partial derivative][x.sub.i]]]([x.sub.i] - [y.sub.i])dy)]/M]. (2.6) By (2:6) and the properties of modulus See modulo. , we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which is the inequality (2:1). Multiplying both sides of (2:5) by w (y) and integrating the resulting identity with respect to y on D and following the proof of inequality (2:1), we obtain the inequality (2:2). This completes the proof. Remark 1. If we choose n = 1. [alpha] = 0, [beta] = 1 and g (x) [equivalent to] 1 in Theorem 1, then (2.1) reduces to (1.1). Remark 2. If we choose [alpha] = 1/2 and [beta] = 1/2 in Theorem 1, then Theorem 1 reduces to Theorem A. Theorem 2. Let [f.sub.1], [f.sub.2], ...,[f.sub.m]: [R.sup.n] [right arrow] R be functions continuous on [bar.D], differentiable on D and whose derivatives [[partial derivative]f.sub.k](x)]/ [[partial derivative]x.sub.i]] (i = 1, ...,n, k = 1, ...,m) are bounded. Further,let [alpha].sub.1],[alpha].sub.2] ...,[alpha].sub.m] in R with [m.[summation over(k=1)]] [alpha].sub.k] = 1. Then for every x [member of] [bar.D], we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.7) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.8) Proof. Let x = ([x.sub.1], ..., [x.sub.n]) and y = ([y.sub.1], ..., [y.sub.n]) (x [member of] [bar.D], y [member of] D). By the assumptions on [f.sub.k] (k = 1, ...,m) and the n-dimensional version of the mean value theorem, we have [f.sub.1](x)-[f.sub.1](y) = [n.[summation over(i=1)]] [[partial darivative][f.sub.1]([C.sub.1]/[partial darivative][x.sub.i]]([x.sub.i]-[y.sub.i]), (2.9) [f.sub.2](x)-[f.sub.2](y) = [n.[summation over(i=1)]] [[[partial darivative][f.sub.2]([C.sub.2])/[[partial darivative][x.sub.i]]]([x.sub.i]-[y.sub.i]), (2.10) [f.sub.m](x)-[f.sub.m](y) = [n.[summation over(i=1)]] [[partial darivative][f.sub.m]]([C.sub.m])/[partial darivative][x.sub.i]]]([x.sub.i]-[y.sub.i]), (2.11) where [c.sub.k] = ([y.sub.1] +[[gamma].sub.k] ([x.sub.1] - [y.sub.1]) ..., [y.sub.n] + [[gamma].sub,k] ([x.sub.n] - [y.sub.n]) (k = 1, ..., m) for some 0 < [gamma].sub.k] < 1 (k = 1, ..., m). Multiplying both sides of (2:9:k) by [[alpha].sub.k] [m.[product].l=1,l[not equal to]k] [f.sub.l] (x) (k = 1, ..., m) and adding the resulting identities, we get [m.[product].(k=1)][f.sub.k](x)-[m.summation over(k=1)] [[alpha].sub.k][m.[product](l=1,l[not equal to]k)][f.sub.1](y) = [m.summation over(k=1)] {[alpha].sub.k][m.[product](l=1,l[not equal to]k)][f.sub.1](x)][[n. summation over(i=1)][[partial derivative][f.sub.k]([c.sub.k])]/[partial derivative].[x.sub.i]]]([x.sub.i]-[y.sub.i])]}]. (2.12) Integrating both sides of (2:10) with respect to y over D and using M = mesD = [n.[product].i=1] ([b.sub.i] - [a.sub.i]), we obtain [m.[product] over(k=1)]-[[m.summation over(k=1)][[alpha].sub.k][m.[product] over(l=1,l[not equal to]k][f.sub.1](x)[[integral].sub.D][f.sub.k](y)dy]/M] = [[m.summation over(k=1)[{[[alpha].sub.k][m.[product] over(l=1,l[not equal to]k][f.sub.l](x)][[integral].sub.D][n.summation over(i=1)][[partial derivative][f.sub.k]([c.sub.k])/[partial.derivative][x.sub.i]]([x.sub.i]-[y.sub.i])dy]]/M]]. (2.13) By (2:11) and the properties of modulus, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.14) which is the inequality (2:7). Multiplying both sides of (2:11) by w (y) and integrating the resulting identity with respect to y on D and following the proof of inequality (2:7), we obtain the inequality (2:8). This completes the proof. Remark 3. If we choose n = 2 in Theorem 2, then Theorem 2 reduces to Theorem 1. Remark 4. If we choose n = 2 and [[alpha].sub.1] = [[alpha].sub.2] = 1/2 in Theorem 2, then Theorem 2 reduces to Theorem A. Theorem 3. Let f, g, w, dy, [[partial derivative]f(x)]/ [[partial derivative][x.sub.i]] and [[partial derivative]g(x)]/ [[partial derivative][x.sub.i]] (i = 1, ..., n) be defined as in Theorem A. Then for every x [member of] [bar.D], we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.15) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.16) Proof. From the hypotheses, as in the proof of Theorem 1, the identities (2:3) and (2:4) hold. Multiplying the left and right sides of (2:3) and (2:4), we have f(x)g(x)-f(x)g(y)-g(x)f(y)+f(y)g(y) = [n.summation over(i=1)][[partial derivative]f(c)/[partial derivative][x.sub.i]]([x.sub.i]-[y.sub.i])][[n.summation over.(i=1)][[partial derivative][x.sub.i]]]([x.sub.i]-[y.sub.i]). (2.17) Multiplying both sides of (2:15) by w (y) and integrating the resulting identity with respect to y on D, we get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which is the required inequality in (2:13): Multiplying both sides of (2:3) and (2:4) by w (y) and integrating the resulting identities with respect to y on D, we get f(x)-[[integral].sub.D]f(y)w(y)dy/[[integral].sub.D]w(y)dy] = [1/[[integral].sub.D]w(y)dy][[integral].sub.D][n.summation over(i=1)][[partial derivative]f(c)]/[[partial derivative][x.sub.i]]]([x.sub.i]-[y.sub.i])w(y)dy (2.18) and g(x)-[[integral].sub.D]g(y)w(y)dy/[[integral].sub.D]w(y)dy] = [1/[[integral].sub.D]w(y)dy][[integral].sub.D][n.summation over(i=1)][partial derivation derivation, in grammar: see inflection. ]g(d)/[partial derivation][x.sub.i]]([x.sub.i]-[y.sub.i])w(y)dy, (2.19) respectively. Multiplying the left and right sides of (2:16) and (2:17), we have f(x)g(x)-[f(x)[[integral].sub.D]g(y)w(y)dy/[[integral].sub.D]w(y)dy] -g(x)[[integral].sub.D]f(y)w(y)dy/[[integral].sub.D]w(y)dy]+[[[integral].sub.D]f(y)w(y)dy[[integral].sub.D]g(y)w(y)dy/([[integral].sub.D]g(y)w(y)dy/[([[integral].sub.D]w(y)dy]).sup.2]] =[1/([[integral].sub.D]w(y)dy).sub.2]][[[integral].sub.D][n.summation over(i=1)][[partial derivation]f(c)/[partial derivation][x.sub.i]]([x.sub.i]-[y.sub.i])w(y)dy]]x[[[integral].sub.D][[n]summation over(i=1)][[partial derivation]g(d)/[partial derivation][x.sub.i]]([x.sub.i]-[y.sub.i])w(y)dy]]. (2.20) By (2:18) and the properties of modulus, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which is the inequality (2:14). This completes the proof Remark 5. For w (y) [equivalent to] 1, Theorem 3 reduces to Theorem B. References (1) G. A. Anastassiou, Multivariate Ostrowski Type Inequalities This page lists Wikipedia articles about named mathematical inequalities. Pure mathematics
(2) N. S. Barnett and S. S. Dragomir, An Ostrowski Type Inequality for Double Integrals and Applications for Cubature cu·ba·ture n. 1. The determination of the volume of a solid. 2. Cubage. [cub(e) + (quadr)ature.] Formulae, RGMIA RGMIA Research Group in Mathematical Inequalities and Applications Res. Rep. Coll., 1(1)(1998), 13-22. (http://rgmia.vu.edu.au/v1n1.html) (3) S. S. Dragomir, N. S. Barnett and P. Cerone, An n-dimension Version of Ostrowski's Inequality for mappings of the Holder Type, RGMIA Res. Rep. Coll., 2(2)(1999), 169-180. (http://rgmia.vu.edu.au/v2n2.html) (4) G. V. Milovanovic, On Some Integral Inequalities, Univ. Beograd. Publ. Elek. Fak. Ser. Mat. Fiz., No. 496-No. 541(1975), 119-124. (5) D. S. Mitrinovic, J. E. Pecaric and A. M. Fink fink Slang n. 1. A contemptible person. 2. An informer. 3. A hired strikebreaker. intr.v. finked, fink·ing, finks 1. To inform against another person. , Inequalities for Functions and Their Integrals and Derivatives, Kluwer Academic Pulishers, Dordrecht, 1994. (6) B. G. Pachpatte, On An Inequality of Ostrowski Type in Three Independent Variables, J. Math. Anal anal (a´n'l) relating to the anus. a·nal adj. 1. Of, relating to, or near the anus. 2. . Appl., 249(2000), 583-591. (7) B. G. Pachpatte, On A New Ostrowski Type Inequality in Two Independent Variables, Tamkang J. Math. 32(1) (2001), 45-49. (8) B. G. Pachpatte, On Multivariate Ostrowski Type Inequalities, J. Inequal. Pure and Appl. Math., 3(4), Art. 58, 2002. (9) W. Rudin, Principles of Mathematical Analysis Analysis has its beginnings in the rigorous formulation of calculus. It is the branch of mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function. , McGraw-Hill Book Company Inc., 1953. * 2000 Mathematics subject classi_cation cation (kăt'ī`ən), atom or group of atoms carrying a positive charge. The charge results because there are more protons than electrons in the cation. : 26D15, 26D20. [dagger] E-mail: hsru@cc.chit chit 1 n. 1. A statement of an amount owed for food and drink; a check. 2. A short letter; a note. 3. .edu.tw [double dagger double dagger n. A reference mark (  ) used in printing and writing. Also called diesis.Noun 1. ] E-mail: Meniar.Haddad@fst.rnu.tn Shiow-Ru Hwangy [dagger] China Institute of Technology Nankang, Taipei, Taiwan 11522 Kuang-Tsan Cheng and Chung-Shin Wangz Department of Mathematics, Aletheia University Aletheia University (after Greek αλήθεια, truth) is a university in Tamsui, Taipei County, Taiwan founded by George Leslie Mackay as the Oxford (University) College. , Tamsui, Taiwan 25103 Received November 29, 2007, Accepted January 8, 2008. |
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