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On Certain Functional Equations and Mean Value Theorems

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Functional Equations, Inequalities and Applications

Abstract

In this paper we prove certain new characterizations of mean values in the spirit of Gauss type functional equations.

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References

  1. J. Aczèl: Lectures on Functional Equations and their Applications’, Academic Press, New York—London, 1966. (pp. 234–244 )

    MATH  Google Scholar 

  2. J. Aczèl: The notion of mean values’, Norske Vid. Selsk. Forh. Trondheim 19 (1946), 83–86.

    Google Scholar 

  3. G. Almkvist and B. Berndt: Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, 7r and the Ladies Diary’, Amer. Math. Monthly 95 (1988), 585–607.

    Article  MathSciNet  MATH  Google Scholar 

  4. J. Arazy, T. Cleasson, S. Janson and J. Peetre: Means and their iterations’, pages 191–217 In: Proceedings of the Nineteenth Nordic Congress of Mathematicians, Reykjavik, 1984.

    Google Scholar 

  5. E.F. Beckenbach: A class of mean value functions’, Amer. Math. Monthly 57 (1950), 1–6.

    Article  MathSciNet  MATH  Google Scholar 

  6. J.M. Borwein and P.B. Borwein: Pi and the AGM’, Wiley, New York, 1987.

    MATH  Google Scholar 

  7. P.S. Bullen, D.S. Mitrinovic and P. M. Vasic: Mean and Their Inequalities’, Reidel, Dordrecht, 1988.

    Google Scholar 

  8. D.A. Cox: The arithmetic-geometric mean of Gauss’, L’Enseign. Math. 30 (1984), 275–330.

    MATH  Google Scholar 

  9. D.M.E. Foster and G.M. Phillips: A generalization of the archimedean double sequence’, J. Math. Anal. Appl. 101 (1984), 575–581.

    Article  MathSciNet  MATH  Google Scholar 

  10. C.F. Gauss: ’Werke’,Göttingen—Leipzig, 1868–1927.

    Google Scholar 

  11. G.H. Hardy: A Course of Pure Mathematics’, Cambridge Univ. Press, 1958.

    Google Scholar 

  12. H. Haruki: New characterizations of the arithmetic—geometric mean of Gauss and other well-known values’, Publ. Math. Debrecen 38 (1991), 323–332.

    MathSciNet  MATH  Google Scholar 

  13. H. Haruki and Th.M. Rassias: New characterizations of some mean-values’, J. Math. Anal. Appl. 202 (1996), 333–348.

    Article  MathSciNet  MATH  Google Scholar 

  14. H. Haruki and Th.M. Rassias: A new analogue of Gauss’ functional equation’, Int. J. Math. Math. Sci. 18 (1995), 749–756.

    Article  MathSciNet  MATH  Google Scholar 

  15. D.H. Hyers, G. Isac and Th.M. Rassias: Stability of Functional Equations in Several Variables’, Birkhäuser, Boston, 1998.

    Book  MATH  Google Scholar 

  16. D.H. Hyers, G. Isac and Th.M. Rassias: Topics in Nonlinear Analysis and Applications’, World Scientific Publ., New Jersey—London. 1998

    Google Scholar 

  17. P. Kahlig and J. Matkowski: On the composition of homogeneous quasi-arithmetic means’, J. Math. Anal. Appl. 216 (1997), 69–85.

    Article  MathSciNet  MATH  Google Scholar 

  18. Y.H. Kim. New characterizations of well-known mean-values’, Far East J. Math. Sci. 6 (6) (1998), 939–947.

    Google Scholar 

  19. Y.H. Kim. On some further extensions of the characterizations of mean values by H. Haruki and Th. M. Rassias’, J. Math. Anal. Appl. 235 (1999), 598–607.

    Article  MathSciNet  MATH  Google Scholar 

  20. Y.H. Kim and J.S. Ume: New characterizations of well-known mean-values II’, Far East J. Math. Sci. 2 (3) (2000), 453–461.

    MathSciNet  MATH  Google Scholar 

  21. Y.H. Kim and Th.M. Rassias: On some mean value theorems and functional equations’Appl. Math. Corn. (to appear)

    Google Scholar 

  22. D.H. Lehmer: On the compounding of certain means’, J. Math. Anal. Appl. 36 (1971), 183–200.

    Article  MathSciNet  MATH  Google Scholar 

  23. P.K. Sahoo and T. Riedel: Mean Value Theorems and Functional Equations’, World Scientific Publ., Singapore—New Jersey—London. 1998

    Book  MATH  Google Scholar 

  24. Gh. Toader: Some mean values related to the arithmetic—geometric mean’, J. Math. Anal. Appl. 218 (1998), 358–368.

    Article  MathSciNet  MATH  Google Scholar 

  25. Gh. Toader and Th.M. Rassias: New properties of some mean values’, J. Math. Anal. Appl. 232 (1999), 376–383.

    Article  MathSciNet  MATH  Google Scholar 

  26. S. Toader, Th.M. Rassias and G. Toader: A Gauss type functional equation’, Int. J. Math. Math. Sci. 25 (9) (2001), 565–569.

    Google Scholar 

  27. J.S. Urne and Y.H. Kim: Some mean values related to the quasi-arithmetic mean’, J. Math. Anal. Appl. 252 (2000), 167–176.

    Article  MathSciNet  Google Scholar 

  28. E.T. Whittaker and G.N. Watson: A Course of Modern Analysis’, Cambridge Univ. Press, Cambridge, 1927.

    Google Scholar 

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Rassias, T.M., Kim, YH. (2003). On Certain Functional Equations and Mean Value Theorems. In: Rassias, T.M. (eds) Functional Equations, Inequalities and Applications. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0225-6_10

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  • DOI: https://doi.org/10.1007/978-94-017-0225-6_10

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-6406-6

  • Online ISBN: 978-94-017-0225-6

  • eBook Packages: Springer Book Archive

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