Summary
In this paper we prove the following:
IfA n ,G n andH n (resp.A′ n ,G′ n andH′ n ) denote the arithmetic, geometric and harmonic means ofa 1,⋯, a n (resp. 1 −a 1,⋯, 1 −a n ) and ifa i ∈ (0, 1/2],i = 1,⋯,n, then(G n /G′ n )n ⩽ (A n /A′ n )n-1 H n /H′ n , (*) with equality holding forn = 1,2. Forn ⩾ 3 equality holds if and only ifa 1 =⋯ =a n . The inequality (*) sharpens the well-known inequality of Ky Fan:G n /G′ n ⩽ A n /A′ n .
Similar content being viewed by others
Literatur
Alzer, H.,Über die Ungleichung zwischen dem geometrischen und dem arithmetischen Mittel. Quaestiones Math.10 (1987), 351–356.
Alzer, H.,On an inequality of Ky Fan. J. Math. Anal. Appl. (erscheint demnächst).
Bauer, H.,A class of means and related inequalities. Manuscripta Math.55 (1986), 199–211.
Beckenbach, E. F., andBellman, R.,Inequalities. Springer-Verlag, Berlin, 1983.
Bullen, P. S.,An inequality of N. Levinson. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz.412–460 (1973), 109–112.
Levinson, N.,Generalization of an inequality of Ky Fan. J. Math. Anal. Appl.8 (1964), 133–134.
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.
Popoviciu, T.,Sur une inegalité de N. Levinson. Mathematica (Cluj)6 (1964), 301–306.
Sierpinski, W.,Sur une inegalité pour la moyenne arithmétique, géométrique et harmonique (Polish). Warszawa Sitzungsber.2 (1909), 354–357.
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.
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.
Wang, C.-L.,Functional equation approach to inequalities II. J. Math. Anal. Appl.78 (1980), 522–530.
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).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Alzer, H. Verschärfung einer Ungleichung von Ky Fan. Aeq. Math. 36, 246–250 (1988). https://doi.org/10.1007/BF01836094
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01836094