, Volume 36, Issue 2-3, pp 246-250

Verschärfung einer Ungleichung von Ky Fan

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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 .