Abstract :
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.
