On the inequalities of Ky Fan, Wang-Wang and Alzer (Q1614715)

Let \(A\), \(G\) and \(H\) (or \(A(a)\), \(G(a)\) and \(H(a)\)) be, respectively, the unweighted arithmetic, geometric and harmonic means of the real numbers \(a_1,\dots, a_n\) (\(a_i>0\), \(i= 1,\dots, n\)). Let \(1= (1,\dots, 1)\) and \(A^+= A(1+ a)\), \(G^+= G(1+a)\), \(H^+= H(1+ a)\), then \[ {1\over H^+}- {1\over H}\leq {1\over G^+}- {1\over G}\leq {1\over A^+}- {1\over A} \] with equalities iff \(a_1=\cdots= a_n\). New proofs of Ky Fan, Wang-Wang, Alzer and El-Neweihi-Proschan inequalities are also given.