On some inequalities of Ky Fan and Wang-Wang (Q1322318)

The main result of the paper states the following inequality: \[ H_ n\leq {n+\sum b_ i\over n+\sum 1/b_ i}\leq G_ n\leq {\sum b_ i/(1+ b_ i)\over \sum 1/(1+ b_ i)}\leq A_ n, \] where \(0< b_ i\leq 1\) and \(A_ n\), \(G_ n\), \(H_ n\) denote the arithmetic, geometric, and harmonic mean, respectively, of the numbers \(b_ 1,\dots,b_ n\). As an immediate corollary, the author obtains the well-known Ky Fan inequality and an inequality due to \textit{W.-L. Wang} and \textit{P.-F. Wang} [Acta Math. Sin. 27, 485-497 (1984; Zbl 0561.26013)].