Open Access

On Opial-Type Integral Inequalities

Journal of Inequalities and Applications20072007:038347

DOI: 10.1155/2007/38347

Received: 22 January 2007

Accepted: 4 April 2007

Published: 13 May 2007


We establish some new Opial-type inequalities involving functions of two and many independent variables. Our results in special cases yield some of the recent results on Opial's inequality and also provide new estimates on inequalities of this type.


Authors’ Affiliations

Department of Mathematics, The University of Hong Kong
Department of Information and Mathematics Sciences, College of Science, China Jiliang University


  1. Opial Z: Sur une inégalité. Annales Polonici Mathematici 1960, 8: 29–32.MathSciNetMATHGoogle Scholar
  2. Agarwal RP, Pang PYH: Opial Inequalities with Applications in Differential and Difference Equations, Mathematics and Its Applications. Volume 320. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1995:x+393.View ArticleGoogle Scholar
  3. Agarwal RP, Lakshmikantham V: Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations, Series in Real Analysis. Volume 6. World Scientific, River Edge, NJ, USA; 1993:xii+312.View ArticleGoogle Scholar
  4. Baĭnov D, Simeonov P: Integral Inequalities and Applications, Mathematics and Its Applications (East European Series). Volume 57. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1992:xii+245.Google Scholar
  5. Li JD: Opial-type integral inequalities involving several higher order derivatives. Journal of Mathematical Analysis and Applications 1992,167(1):98–110. 10.1016/0022-247X(92)90238-9MathSciNetView ArticleMATHGoogle Scholar
  6. Mitrinović DS, Pečarić JE, Fink AM: Inequalities Involving Functions and Their Integrals and Derivatives, Mathematics and Its Applications (East European Series). Volume 53. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1991:xvi+587.View ArticleGoogle Scholar
  7. Cheung W-S: On Opial-type inequalities in two variables. Aequationes Mathematicae 1989,38(2–3):236–244. 10.1007/BF01840008MathSciNetView ArticleMATHGoogle Scholar
  8. Cheung W-S: Some new Opial-type inequalities. Mathematika 1990,37(1):136–142. 10.1112/S0025579300012869MathSciNetView ArticleMATHGoogle Scholar
  9. Cheung W-S: Some generalized Opial-type inequalities. Journal of Mathematical Analysis and Applications 1991,162(2):317–321. 10.1016/0022-247X(91)90152-PMathSciNetView ArticleMATHGoogle Scholar
  10. Cheung W-S: Opial-type inequalities withfunctions invariables. Mathematika 1992,39(2):319–326. 10.1112/S0025579300015047MathSciNetView ArticleMATHGoogle Scholar
  11. Cheung W-S, Dandan Z, Pečarić JE: Opial-type inequalities for differential operators. Nonlinear Analysis: Theory, Methods & Applications 2007,66(9):2028–2039. 10.1016/ ArticleMATHGoogle Scholar
  12. Godunova EK, Levin VI: An inequality of Maroni. Matematicheskie Zametki 1967, 2: 221–224.MathSciNetGoogle Scholar
  13. Mitrinović DS: Analytic Inequalities, Die Grundlehren der mathematischen Wisenschaften. Volume 1965. Springer, New York, NY, USA; 1970:xii+400.Google Scholar
  14. Pachpatte BG: On integral inequalities similar to Opial's inequality. Demonstratio Mathematica 1989,22(1):21–27.MathSciNetMATHGoogle Scholar
  15. Pachpatte BG: On inequalities of the Opial type. Demonstratio Mathematica 1992, 25: 35–45.MathSciNetMATHGoogle Scholar
  16. Pachpatte BG: Some inequalities similar to Opial's inequality. Demonstratio Mathematica 1993,26(3–4):643–647.MathSciNetMATHGoogle Scholar
  17. Pachpatte BG: A note on generalized Opial-type inequalities. Tamkang Journal of Mathematics 1993,24(2):229–235.MathSciNetMATHGoogle Scholar
  18. Pečarić JE: An integral inequality. In Analysis, Geometry and Groups: A Riemann Legacy Volume—Part II, Hadronic Press Collect. Orig. Artic.. Edited by: Srivastava HM, Rassias ThM. Hadronic Press, Palm Harbor, Fla, USA; 1993:471–478.Google Scholar
  19. Pečarić JE, Brnetić I: Note on generalization of Godunova-Levin-Opial inequality. Demonstratio Mathematica 1997,30(3):545–549.MathSciNetMATHGoogle Scholar
  20. Pečarić JE, Brnetić I: Note on the generalization of the Godunova-Levin-Opial inequality in several independent variables. Journal of Mathematical Analysis and Applications 1997,215(1):274–282. 10.1006/jmaa.1997.5529MathSciNetView ArticleMATHGoogle Scholar
  21. Rozanova GI: Integral inequalities with derivatives and with arbitrary convex functions. Moskovskiĭ Gosudarstvennyĭ Pedagogicheskiĭ Institut imeni V. I. Lenina. Uchenye Zapiski 1972, 460: 58–65.MathSciNetGoogle Scholar
  22. Yang GS: Inequality of Opial-type in two variables. Tamkang Journal of Mathematics 1982,13(2):255–259.MathSciNetMATHGoogle Scholar


© W.-S. Cheung and C.-J. Zhao. 2007

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