Verschärfung einer Ungleichung von Ky Fan (Q1109156)

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: \(G_ n/G_ n'\leq A_ n/A_ n'\).