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  • Research Article
  • Open Access

Improvement of Aczél's Inequality and Popoviciu's Inequality

Journal of Inequalities and Applications20072007:072173

https://doi.org/10.1155/2007/72173

  • Received: 30 December 2006
  • Accepted: 24 April 2007
  • Published:

Abstract

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. As application, an integral inequality of Aczél-Popoviciu type is established.

Keywords

  • Integral Inequality
  • Classical Inequality
  • Unify Improvement

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Authors’ Affiliations

(1)
Department of Mathematics, Longyan College, Longyan, Fujian, 364012, China

References

  1. Aczél J: Some general methods in the theory of functional equations in one variable. New applications of functional equations. Uspekhi Matematicheskikh Nauk (N.S.) 1956,11(3(69)):3–68.MATHGoogle Scholar
  2. Cho YJ, Matić M, Pečarić J: Improvements of some inequalities of Aczél's type. Journal of Mathematical Analysis and Applications 2001,259(1):226–240. 10.1006/jmaa.2000.7423MathSciNetView ArticleMATHGoogle Scholar
  3. Sun X-H: Aczél-Chebyshev type inequality for positive linear functions. Journal of Mathematical Analysis and Applications 2000,245(2):393–403. 10.1006/jmaa.2000.6754MathSciNetView ArticleMATHGoogle Scholar
  4. Losonczi L, Páles Z: Inequalities for indefinite forms. Journal of Mathematical Analysis and Applications 1997,205(1):148–156. 10.1006/jmaa.1996.5188MathSciNetView ArticleMATHGoogle Scholar
  5. Mercer AM: Extensions of Popoviciu's inequality using a general method. Journal of Inequalities in Pure and Applied Mathematics 2003,4(1, Article 11):4 pages.Google Scholar
  6. Mascioni V: A note on Aczél type inequalities. Journal of Inequalities in Pure and Applied Mathematics 2002,3(5, Article 69):5 pages.MathSciNetGoogle Scholar
  7. Dragomir SS, Mond B: Some inequalities of Aczél type for Gramians in inner product spaces. Nonlinear Functional Analysis and Applications 2001,6(3):411–424.MathSciNetMATHGoogle Scholar
  8. Bellman R: On an inequality concerning an indefinite form. The American Mathematical Monthly 1956,63(2):108–109. 10.2307/2306434MathSciNetView ArticleMATHGoogle Scholar
  9. Vasić PM, Pečarić JE: On the Jensen inequality for monotone functions. Analele Universităţii din Timişoara. Seria Matematică-Informatică 1979,17(1):95–104.MATHGoogle Scholar
  10. Kuang JC: Applied Inequalities. 2nd edition. Hunan Education Press, Changsha, China; 1993:xxvi+794.Google Scholar
  11. Mitrinović DS, Pečarić JE, Fink AM: Classical and New Inequalities in Analysis. Volume 61. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1993:xviii+740.View ArticleMATHGoogle Scholar
  12. Popoviciu T: On an inequality. Gazeta Matematica si Fizica. Seria A 1959, 11 (64): 451–461.MathSciNetGoogle Scholar
  13. Wu S, Debnath L: Generalizations of Aczél's inequality and Popoviciu's inequality. Indian Journal of Pure and Applied Mathematics 2005,36(2):49–62.MathSciNetMATHGoogle Scholar
  14. Wu S: A further generalization of Aczél's inequality and Popoviciu's inequality. Mathematical Inequalities and Application 2007.,10(3):Google Scholar
  15. Beckenbach EF, Bellman R: Inequalities. Springer, New York, NY, USA; 1983:xi+198.Google Scholar
  16. Hardy GH, Littlewood JE, Pólya G: Inequalities. 2nd edition. Cambridge University Press, Cambridge, UK; 1952:xii+324.MATHGoogle Scholar

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