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aequationes mathematicae

, Volume 40, Issue 1, pp 261–270 | Cite as

On the identric and logarithmic means

  • J. Sándor
Research Papers

Summary

Leta, b > 0 be positive real numbers. The identric meanI(a, b) of a andb is defined byI = I(a, b) = (1/e)(b b /a a ) 1/(b−a) , fora ≠ b, I(a, a) = a; while the logarithmic meanL(a, b) ofa andb isL = L(a, b) = (b − a)/(logb − loga), fora ≠ b, L(a, a) = a. Let us denote the arithmetic mean ofa andb byA = A(a, b) = (a + b)/2 and the geometric mean byG =G(a, b) =\(\sqrt {ab}\). In this paper we obtain some improvements of known results and new inequalities containing the identric and logarithmic means. The material is divided into six parts. Section 1 contains a review of the most important results which are known for the above means. In Section 2 we prove an inequality which leads to some improvements of known inequalities. Section 3 gives an application of monotonic functions having a logarithmically convex (or concave) inverse function. Section 4 works with the logarithm ofI(a, b), while Section 5 is based on the integral representation of means and related integral inequalities. Finally, Section 6 suggests a new mean and certain generalizations of the identric and logarithmic means.

AMS (1980) subject classification

Primary 26D99 Secondary 26D15, 26D20 

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References

  1. [1]
    Alzer, H.,Two inequalities for means. C.R. Math. Rep. Acad. Sci. Canada.9 (1987), 11–16.Google Scholar
  2. [2]
    Alzer, H.,Ungleichungen für Mittelwerte. Arch. Math. (Basel)47 (1986), 422–426.Google Scholar
  3. [3]
    Alzer, H.,On an inequality of Ky Fan. J. Math. Anal. Appl.137 (1989), 168–172.CrossRefGoogle Scholar
  4. [4]
    Beckenbach, E. F. andBellman, R.,Inequalities. Springer, New York, 1965.Google Scholar
  5. [5]
    Carlson, B. C.,Some inequalities for hypergeometric functions. Proc. Amer. Math. Soc.17 (1966), 32–39.Google Scholar
  6. [6]
    Carlson, B. C.,The logarithmic mean. Amer. Math. Monthly79 (1972), 615–618.Google Scholar
  7. [7]
    Hardy, G. H., Littlewood, J. E. andPolya, G.,Inequalities. Cambridge Univ. Press, Cambridge—New York, 1988.Google Scholar
  8. [8]
    Leach, E. B. andSholander, M. C.,Extended mean values II. J. Math. Anal. Appl.92 (1983), 207–223.CrossRefGoogle Scholar
  9. [9]
    Lin, T. P.,The power mean and the logarithmic mean. Amer. Math. Monthly81 (1974), 879–883.Google Scholar
  10. [10]
    Mitrinovic, D. S. (in cooperation withP. M. Vasic),Analytic Inequalities. Springer, Berlin—Heidelberg—New York, 1970.Google Scholar
  11. [11]
    Ostle, B. andTerwilliger, H. L.,A comparison of two means. Proc. Montana Acad. Sci.17 (1957), 69–70.Google Scholar
  12. [12]
    Rüthing, D.,Eine allgemeine logarithmische Ungleichung. Elem. Math.41 (1986), 14–16.Google Scholar
  13. [13]
    Sándor, J.,Some integral inequalities. Elem. Math.43 (1988), 177–180.Google Scholar
  14. [14]
    Sándor, J.,An application of the Jensen — Hadamard inequality. To appear in Nieuw Arch. Wisk. (4)8 (1990).Google Scholar
  15. [15]
    Sándor, J.,On an inequality of Ky Fan. To appear in Sem. Math. Anal., Babes—Bolyai Univ.Google Scholar
  16. [16]
    Seiffert, H.-J.,Eine Integralungleichung für streng monotone Funktionen mit logarithmische konvexer Umkehrfunktion. Elem. Math.44 (1989), 16–17.Google Scholar
  17. [17]
    Stolarsky, K. B.,Generalizations of the logarithmic mean. Math. Mag.48 (1975), 87–92.Google Scholar
  18. [18]
    Stolarsky, K. B.,The power and generalized logarithmic means. Amer. Math. Monthly87 (1980), 545–548.Google Scholar
  19. [19]
    Zaiming, Z.,Problem E 3142. Amer. Math. Monthly93 (1986), 299.Google Scholar

Copyright information

© Birkhäuser Verlag 1990

Authors and Affiliations

  • J. Sándor
    • 1
  1. 1.Jud. HarghitaRomania

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