On the inequalities of Ky Fan, Wang–Wang and Alzer

We denote by A, B and H (or A(a), G(a) and H(a)) the unweighted arithmetic, geometric and harmonic means of the real numbers a1, . . . ,an with a = (a1, . . . ,an) ∈ Rn +. Let 1 = (1, . . . , 1), A − = A(1 − a), G − = G(1 − a), H − = H(1 − a) with 0 < ai < 1, i = 1, . . .,n, and A + = A(1 + a), G + = G(1 + a), H + = H(1+ a) with ai > 0, i = 1, . . . ,n. The fundamental inequalities H G A (1) can be considered as the separation of arithmetic means from harmonic by geometric means. The same goes for H H− G G− A A− , (2) H H+ G G+ A A+ , (3) and 1 H− − 1 H 1 G− − 1 G 1 A− − 1 A , (4) with ai ∈ (0, 1/2], i = 1, . . . ,n.