Verschärfung einer Ungleichung von Ky Fan. (German) Zbl 0655.26011
In this paper we prove the following: If \(A_ n\), \(G_ n\) and \(H_ n\) (resp. \(A_ n'\), \(G_ n'\) and \(H_ n')\) denote the arithmetic, geometric and harmonic means of \(a_ 1,...,a_ n\) (resp. \(1-a_ 1,...,1-a_ n\)) and if \(a_ i\in (0,1/2]\), \(i=1,...,n,\) then
\[
(*)\quad (G_ n/G_ n')^ n\leq (A_ n/A_ n')^{n-1}H_ n/H_ n',
\]
with equality holding for \(n=1,2\). For \(n\geq 3\) equality holds if and only if \(a_ 1=...=a_ n\). The inequality (*) sharpens the well-known inequality of Ky Fan: \(G_ n/G_ n'\leq A_ n/A_ n'\).
MSC:
| 26D15 | Inequalities for sums, series and integrals |
References:
| [1] | Alzer, H.,Über die Ungleichung zwischen dem geometrischen und dem arithmetischen Mittel. Quaestiones Math.10 (1987), 351–356. · Zbl 0632.26009 · doi:10.1080/16073606.1987.9632134 |
| [2] | Alzer, H.,On an inequality of Ky Fan. J. Math. Anal. Appl. (erscheint demnächst). · Zbl 0668.26012 |
| [3] | Bauer, H.,A class of means and related inequalities. Manuscripta Math.55 (1986), 199–211. · Zbl 0592.26019 · doi:10.1007/BF01168685 |
| [4] | Beckenbach, E. F., andBellman, R.,Inequalities. Springer-Verlag, Berlin, 1983. |
| [5] | Bullen, P. S.,An inequality of N. Levinson. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz.412–460 (1973), 109–112. · Zbl 0268.26014 |
| [6] | Levinson, N.,Generalization of an inequality of Ky Fan. J. Math. Anal. Appl.8 (1964), 133–134. · Zbl 0128.05803 · doi:10.1016/0022-247X(64)90089-7 |
| [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. |
| [8] | Popoviciu, T.,Sur une inegalité de N. Levinson. Mathematica (Cluj)6 (1964), 301–306. |
| [9] | Sierpinski, W.,Sur une inegalité pour la moyenne arithmétique, géométrique et harmonique (Polish). Warszawa Sitzungsber.2 (1909), 354–357. |
| [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. |
| [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. · Zbl 0437.26010 · doi:10.1016/0022-247X(80)90294-2 |
| [12] | Wang, C.-L.,Functional equation approach to inequalities II. J. Math. Anal. Appl.78 (1980), 522–530. · Zbl 0461.26010 · doi:10.1016/0022-247X(80)90164-X |
| [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). |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
