Skip to main content
SpringerLink
Log in

The inequality of Ky Fan and related results

  • Published:
Acta Applicandae Mathematica Aims and scope Submit manuscript

Abstract

In this survey paper, we present refinements, extensions, and variants of the inequality

$$\mathop \Pi \limits_{i = 1}^n (x_i (1 - x_i ))^{1/n}< \sum\limits_{i = 1}^n {x_i } /\sum\limits_{i = 1}^n {(1 - x_i ),}$$
((*))

valid for all real numbersx i ε (0,1/2] (i=1,...,n) which are not all equal. Inequality (*) was published for the first time in 1961 and is due to Ky Fan.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alzer, H.: Ein symmetrisches Mittel in zwei Veränderlichen,Bull. Greek Math. Soc. 28 (1987), 73–80.

    Google Scholar 

  2. Alzer, H.: Verschärfung einer Ungleichung von Ky Fan,Aequationes Math. 36 (1988), 246–250.

    Google Scholar 

  3. Alzer, H.: Ungleichungen für geometrische und arithmetische Mittelwerte,Proc. Kon. Nederl. Akad. Wetensch. 91 (1988), 365–374.

    Google Scholar 

  4. Alzer, H.: On an inequality of Ky Fan,J. Math. Anal. Appl. 137 (1989), 168–172.

    Google Scholar 

  5. Alzer, H.: A new proof of Ky Fan's inequality,Internat. J. Math. Ed. Sci. Tech. 20 (1989), 486–489.

    Google Scholar 

  6. Alzer, H.: Die Ungleichung von Ky Fan: Ein neuer Beweis,Mitt. Math. Ges. Hamburg 11 (1989), 699–701.

    Google Scholar 

  7. Alzer, H.: Rado-type inequalities for geometric and harmonic means,J. Pure Appl. Math. Sci. 24 (1989), 125–130.

    Google Scholar 

  8. Alzer, H.: On weighted geometric means,Canad. Math. Bull. 32 (1989), 199–206.

    Google Scholar 

  9. Alzer, H.: A converse of Ky Fan's inequality,C.R. Math. Rep. Acad. Sci. Canada 11 (1989), 1–3

    Google Scholar 

  10. Alzer, H.: Über gewichtete geometrische und arithmetische Mittelwerte,Anz. Österreich. Akad. Wiss. Math-Natur. Kl. 127 (1990), 33–36.

    Google Scholar 

  11. Alzer, H.: An inequality of W.-L. Wang and P.-F. Wang,Internat. J. Math. Math. Sci. 13 (1990), 295–298.

    Google Scholar 

  12. Alzer, H.: An extension and a converse of Ky Fan's inequality,Quaestiones Math. 13 (1990), 67–75.

    Google Scholar 

  13. Alzer, H.: Inequalities for arithmetic, geometric and harmonic means,Bull. London Math. Soc. 22 (1990), 362–366.

    Google Scholar 

  14. Alzer, H.: Über eine zweiparametrige Familie von Mittelwerten,Acta Math. Hungar. 56 (1990), 205–209.

    Google Scholar 

  15. Alzer, H.: A Popoviciu-type inequality for weighted geometric and harmonic means,Utilitas Math. 38 (1990), 189–192.

    Google Scholar 

  16. Alzer, H.: Eine komplementäre Ungleichung,Facta Univ. Ser. Math. Inform. 5 (1990), 51–54.

    Google Scholar 

  17. Alzer, H.: Ky Fan's inequality and related results,Punime Math. 5 (1990), 49–55.

    Google Scholar 

  18. Alzer, H.: A short proof of Ky Fan's inequality,Arch. Math. (Brno) 27 (1991), 199–200.

    Google Scholar 

  19. Alzer, H.: Inequalities for pseudo arithmetic and geometric means, in W. Walter (ed.),General Inequalities 6, Birkhäuser, Basel, 1992, pp. 5–16.

    Google Scholar 

  20. Alzer, H.: Refinements of Ky Fan's inequality,Proc. Amer. Math. Soc. 117 (1993), 159–165.

    Google Scholar 

  21. Alzer, H.: On weighted arithmetic, geometric and harmonic mean values,Glasnik Mat. 25 (1990), 279–284.

    Google Scholar 

  22. Alzer, H.: An inequality for arithmetic and harmonic means,Aequationes Math. 46 (1993), 257–263.

    Google Scholar 

  23. Alzer, H., Ando, T. and Nakamura, Y.: The inequalities of W. Sierpinski and Ky Fan,J. Math. Anal. Appl. 149 (1990), 497–512.

    Google Scholar 

  24. Alzer, H. and Pečarić, J. E.: On Ky Fan's inequalityPann. Math. (in print).

  25. Beckenbach, E. F. and Bellman, R.:Inequalities, Springer-Verlag, Berlin, 1961.

    Google Scholar 

  26. Bullen, P. S.: An inequality of N. Levinson,Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 412–460 (1973), 109–112.

    Google Scholar 

  27. Bullen, P. S., Mitrinović, D. S. and Vasić, P. M.:Means and Their Inequalities, D. Reidel, Dordrecht, 1988.

    Google Scholar 

  28. Chan, F., Goldberg, D. and Gonek, S.: On extensions of an inequality among means,Proc. Amer. Math. Soc. 42 (1974), 202–207.

    Google Scholar 

  29. Dinghas, A.: Some identities between arithmetic means and the other elementary functions ofn numbers,Math. Ann. 120 (1948), 154–157.

    Google Scholar 

  30. Dinghas, A.: Zum Beweis der Ungleichung zwischen dem arithmetischen und geometrischen Mittel vonn Zahlen,Math. Phys. Semesterber. 9 (1962/63), 157–163.

    Google Scholar 

  31. Dinghas, A.: Superadditive Funktionale, Identitäten und Ungleichungen der elementaren Analysis,Math. Ann. 178 (1968), 315–334.

    Google Scholar 

  32. El-Neweihi, E. and Proschan, F.: Unified treatment of inequalities of the Weierstrass product type,Amer. Math. Monthly 86 (1979), 206–208.

    Google Scholar 

  33. Hardy, G. H., Littlewood, J. E. and Pólya, G.:Inequalities, Cambridge Univ. Press, Cambridge, 1952.

    Google Scholar 

  34. Klamkin, M. S. and Newman, D. J.: Extensions of the Weierstrass product inequalities,Math. Mag. 43 (1970), 137–140.

    Google Scholar 

  35. Lawrence, S. and Segalman, D.: A generalization of two inequalities involving means,Proc. Amer. Math. Soc. 35 (1972), 96–100.

    Google Scholar 

  36. Levinson, N.: Generalization of an inequality of Ky Fan,J. Math. Anal. Appl. 8 (1964), 133–134.

    Google Scholar 

  37. Marshall, A. W. and Olkin, I.:Inequalities: Theory of Majorization and Its Applications, Academic Press, New York, 1979.

    Google Scholar 

  38. Mitrinović, D. S.:Analytic Inequalities, Springer-Verlag, New York, 1970.

    Google Scholar 

  39. Pečarić, J. E.: On an inequality of N. Levinson,Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 678–715 (1980), 71–74.

    Google Scholar 

  40. Pečarić, J. E.: An inequality for 3-convex functions,J. Math. Anal. Appl. 90 (1982), 213–218.

    Google Scholar 

  41. Pečarić, J. E.: On Levinson's inequality,Real Anal. Exchange 15 (1989–90), 710–712.

    Google Scholar 

  42. Pečarić J. E. and Klamkin, M. S.: Extensions of the Weierstrass product inequalities, III,SEA Bull. Math. 11 (1988), 123–126.

    Google Scholar 

  43. Popoviciu, T.: Sur une inégalité de N. Levinson,Mathematica (Cluj) 6 (1964), 301–306.

    Google Scholar 

  44. Roberts, A. W. and Varberg, D. E.:Convex Functions, Academic Press, New York, 1973.

    Google Scholar 

  45. Sandor, J.: On an inequality of Ky Fan,Babes-Bolyai Univ., Fac. Math. Phys., Res. Semin. 1990, No. 7, 29–34.

    Google Scholar 

  46. Sandor, J.: On an inequality of Ky Fan, II,Internat. J. Math. Ed. Sci. Tech. 22 (1991), 326–328.

    Google Scholar 

  47. Vasić, P. M. and Janić, R. R.: On an inequality of N. Levinson,Publ. Inst. Math. (Belgrade) 10 (1970), 155–157.

    Google Scholar 

  48. Wang, C.-L.: On a Ky Fan inequality of the complementary A-G type and its variants,J. Math. Anal. Appl. 73 (1980), 501–505.

    Google Scholar 

  49. Wang, C.-L.: Functional equation approach to inequalities, II,J. Math. Anal. Appl. 78 (1980), 522–530.

    Google Scholar 

  50. Wang, C.-L.: Inequalities of the Rado-Popoviciu type for functions and their applications,J. Math. Anal. Appl. 100 (1984), 436–446.

    Google Scholar 

  51. Wang, C.-L.: On development of a Ky Fan inequality of the complementary A-G type,J. Math. Res. Exp. 8 (1988), 513–519.

    Google Scholar 

  52. Wang, W.-L. and Wang, P.-F.: A class of inequalities for the symmetric functions (in Chinese),Acta Math. Sinica 27 (1984), 485–497.

    Google Scholar 

  53. Wang, Z. and Chen, J.: A generalization of Ky Fan inequality,Math. Balkanica 5 (1991), 373–380.

    Google Scholar 

  54. Wu, C., Wang, W. and Fu, L.: Inequalities for symmetric functions and their applications (in Chinese),J. Chengdu Univ. Sci. Tech., No. 1 (1982), 103–108.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Morsbacher Str. 10, 51545, Waldbröl, Germany

    Horst Alzer

Authors
  1. Horst Alzer
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Alzer, H. The inequality of Ky Fan and related results. Acta Appl Math 38, 305–354 (1995). https://doi.org/10.1007/BF00996150

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00996150

Mathematics subject classification (1991)

Key words

Search

Navigation