aequationes mathematicae

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

Verschärfung einer Ungleichung von Ky Fan

  • Horst Alzer
Research Papers

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 26D15 Secondary 26D99 

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Copyright information

© Birkhäuser Verlag 1988

Authors and Affiliations

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

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