An inequality for arithmetic and harmonic means
The following result is proved: If \(A_ n\) and \(H_ n\) (resp. \(A_ n'\) and \(H_ n')\) denote the arithmetic and harmonic means of \(x_ 1,\ldots,x_ n\) (resp. \(1-x_ 1,\ldots,1-x_ n)\) with \(x_ i\in(0,1/2]\) \((i=1,\ldots,n)\), then \[ H_ n-H_ n'\leq A_ n-A_ n', \] with equality if and only if \(x_ 1=\cdots=x_ n\).
scientific article; zbMATH DE number 493227