aequationes mathematicae

, Volume 36, Issue 2, pp 246–250

Verschärfung einer Ungleichung von Ky Fan

Authors

  • Horst Alzer
Research Papers

DOI: 10.1007/BF01836094

Cite this article as:
Alzer, H. Aeq. Math. (1988) 36: 246. doi:10.1007/BF01836094

Summary

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 26D15Secondary 26D99

Copyright information

© Birkhäuser Verlag 1988