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NEW INEQUALITIES OF ČEBYŠEV TYPE FOR DOUBLE INTEGRALS

NEW INEQUALITIES OF ČEBYŠEV TYPE FOR DOUBLE INTEGRALS DEMONSTRATIO MATHEMATICAVol. XLNo 12007B. G. PachpatteNEW INEQUALITIES OF CEBYSEV TYPEFOR DOUBLE INTEGRALSAbstract. In this paper, we establish new inequalities of Cebysev t y p e involvingfunctions of two independent variables by using certain integral identities.1. IntroductionIn 1882, P. L. Cebysev [2] proved the following classical inequality6(1.1)j f(x)g(x)dxa/ - . b-(^\$f(x)dxa/ibj f\ g(x)d:X) — a J"" '' \aprovided / , g are absolutely continuous functions defined on [a, 6] and / ' , g' €Loo [a, b].Since the publication of [2], a number of researchers have given variousgeneralizations, extensions and variants of the above inequality, see [4] andalso some of the recent papers appeared in RGMIA Research Report Collection. The main purpose of this paper is to establish new inequalities similarto the inequality (1.1), involving functions of two independent variables andtheir partial derivatives and double integrals. The analysis used in the proofsis based on the integral identities proved in [1] and [3].2. Statement of resultsLet R denotes the set of real numbers and A = [a, b] x [c, d], a, b, c, dG R. The partial derivatives of a function z(x, y) defined on A are denoted by Diz{x,y)= J^z(x,y),D2z(x,y)= J^z(x,y),D2Diz{x,y)=(x, y). We denote by C (A) the class http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

NEW INEQUALITIES OF ČEBYŠEV TYPE FOR DOUBLE INTEGRALS

Demonstratio Mathematica , Volume 40 (1): 8 – Jan 1, 2007

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References (6)

Publisher
de Gruyter
Copyright
© by B. G. Pachpatte
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-2007-0107
Publisher site
See Article on Publisher Site

Abstract

DEMONSTRATIO MATHEMATICAVol. XLNo 12007B. G. PachpatteNEW INEQUALITIES OF CEBYSEV TYPEFOR DOUBLE INTEGRALSAbstract. In this paper, we establish new inequalities of Cebysev t y p e involvingfunctions of two independent variables by using certain integral identities.1. IntroductionIn 1882, P. L. Cebysev [2] proved the following classical inequality6(1.1)j f(x)g(x)dxa/ - . b-(^\$f(x)dxa/ibj f\ g(x)d:X) — a J"" '' \aprovided / , g are absolutely continuous functions defined on [a, 6] and / ' , g' €Loo [a, b].Since the publication of [2], a number of researchers have given variousgeneralizations, extensions and variants of the above inequality, see [4] andalso some of the recent papers appeared in RGMIA Research Report Collection. The main purpose of this paper is to establish new inequalities similarto the inequality (1.1), involving functions of two independent variables andtheir partial derivatives and double integrals. The analysis used in the proofsis based on the integral identities proved in [1] and [3].2. Statement of resultsLet R denotes the set of real numbers and A = [a, b] x [c, d], a, b, c, dG R. The partial derivatives of a function z(x, y) defined on A are denoted by Diz{x,y)= J^z(x,y),D2z(x,y)= J^z(x,y),D2Diz{x,y)=(x, y). We denote by C (A) the class

Journal

Demonstratio Mathematicade Gruyter

Published: Jan 1, 2007

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