Page 25 - MATINF Nr. 11-12

Refinements of classical means inequalities 25

Definition 2. For any strictly positive real numbers a i , i = 1, n, and any p i ∈ [0, 1], i = 1, n,
such that p 1 + p 2 + . . . + p n = 1, we will denote and define:

Ã
! ! !
n n n
X X X
M p 1 ,p 2 ,...,p n = n p i a i · p i a i+1 · . . . · p i a i+n−1 , (5)
a 1 ,a 2 ,...,a n
i=1 i=1 i=1
1
p 1 ,p 2 ,...,p n
M , (6)
a 1 ,a 2 ,...,a n = Å ã Å ã Å ã
n n n
P P P
n p i · p i · . . . · p i
a i a i+1 a i+n−1
i=1 i=1 i=1
where a n+i = a i , for all i = 1, n − 1.
For the expressions (5) and (6), we will also have the following chain with the classical means
([3]).

Proposition 2 (A refinement of the AM-GM-HM Inequality, n-ary version). For any
strictly positive real numbers a i , i = 1, n and any p i ∈ [0, 1], i = 1, n, such that p 1 +p 2 +. . .+p n =
1, the following inequalities hold:

A n [a] ≥ M p 1 ,p 2 ,...,p n ≥ G n [a] ≥ M p 1 ,p 2 ,...,p n ≥ H n [a].
a 1 ,a 2 ,...,a n a 1 ,a 2 ,...,a n
Proof. For the first inequality, by the GM-AM Inequality we have

Ã
! ! !
n n n
X X X
M p 1 ,p 2 ,...,p n = n p i a i · p i a i+1 · . . . · p i a i+n−1
a 1 ,p 2 ,...,a n
i=1 i=1 i=1
n n n
P P P
p i a i + p i a i+1 + . . . + p i a i+n−1
≤ i=1 i=1 i=1
n
n n n
P P P
a 1 · p i + a 2 · p i + . . . + a n · p i
i=1 i=1 i=1
=
n
a 1 + a 2 + . . . + a n
= = A n [a],
n
with equality when a 1 = a 2 = . . . = a n .
For the second inequality, by the weighted AM − GM Inequality, applied n times, we have

Ã
! ! !
n n n
X X X
M p 1 ,p 2 ,...,p n = n p i a i · p i a i+1 · . . . · p i a i+n−1
a 1 ,a 2 ,...,a n
i=1 i=1 i=1
Ã
! ! !
n n n
Y Y Y
≥ n a p i · a p i · . . . · a p i
i i+1 i+n−1
i=1 i=1 i=1
»
p 1 +p 2 +...+p n
= n a p 1 +p 2 +...+p n · a p 1 +p 2 +...+p n · . . . · a n
2
1
√
= n a 1 · a 2 · . . . · a n = G n [a],
with equality when a 1 = a 2 = . . . = a n .