aequationes mathematicae

, Volume 36, Issue 2–3, pp 246–250 | Cite as

Verschärfung einer Ungleichung von Ky Fan

  • Horst Alzer
Research Papers


In this paper we prove the following:

IfAn,Gn andHn (resp.A′n,G′n andH′n) denote the arithmetic, geometric and harmonic means ofa1,⋯, an (resp. 1 −a1,⋯, 1 −an) and ifai ∈ (0, 1/2],i = 1,⋯,n, then(Gn/G′n)n ⩽ (An/A′n)n-1Hn/H′n, (*) with equality holding forn = 1,2. Forn ⩾ 3 equality holds if and only ifa1 = =an. The inequality (*) sharpens the well-known inequality of Ky Fan:Gn/G′n⩽ An/A′n.

AMS (1980) subject classification

Primary 26D15 Secondary 26D99 


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  1. [1]
    Alzer, H.,Über die Ungleichung zwischen dem geometrischen und dem arithmetischen Mittel. Quaestiones Math.10 (1987), 351–356.Google Scholar
  2. [2]
    Alzer, H.,On an inequality of Ky Fan. J. Math. Anal. Appl. (erscheint demnächst).Google Scholar
  3. [3]
    Bauer, H.,A class of means and related inequalities. Manuscripta Math.55 (1986), 199–211.Google Scholar
  4. [4]
    Beckenbach, E. F., andBellman, R.,Inequalities. Springer-Verlag, Berlin, 1983.Google Scholar
  5. [5]
    Bullen, P. S.,An inequality of N. Levinson. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz.412–460 (1973), 109–112.Google Scholar
  6. [6]
    Levinson, N.,Generalization of an inequality of Ky Fan. J. Math. Anal. Appl.8 (1964), 133–134.Google Scholar
  7. [7]
    Mitrinović, D. S., andVasić, P. M.,On a theorem of W. Sierpinski concerning means. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz.544–576 (1976), 113–114.Google Scholar
  8. [8]
    Popoviciu, T.,Sur une inegalité de N. Levinson. Mathematica (Cluj)6 (1964), 301–306.Google Scholar
  9. [9]
    Sierpinski, W.,Sur une inegalité pour la moyenne arithmétique, géométrique et harmonique (Polish). Warszawa Sitzungsber.2 (1909), 354–357.Google Scholar
  10. [10]
    Wang, C.-L.,An extension of two sequences of inequalities of Mitrinović and Vasić. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz.634–677 (1979), 94–96.Google Scholar
  11. [11]
    Wang, C.-L.,On a Ky Fan inequality of the complementary A-G type and its invariants. J. Math. Anal. Appl.73 (1980), 501–505.Google Scholar
  12. [12]
    Wang, C.-L.,Functional equation approach to inequalities II. J. Math. Anal. Appl.78 (1980), 522–530.Google Scholar
  13. [13]
    Wang, W. andWang, P.,A class of inequalities for the symmetric functions. (Chinese). Acta Math. Sinica27 (1984), 485–497 (s. Zentralblatt. f. Math.561 (1985), 26013).Google Scholar

Copyright information

© Birkhäuser Verlag 1988

Authors and Affiliations

  • Horst Alzer
    • 1
  1. 1.WaldbrölFederal Republic of Germany

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