Page 24 - MATINF Nr. 11-12

24 D. M˘arghidanu

Definition 1. If a, b > 0 and p, q ≥ 0, with p + q = 1, we will denote and define:

»
p,q
M a,b = (pa + qb) · (qa + pb) , (2)

p,q ab
M . (3)
a,b = p
(pa + qb) · (qa + pb)
For these expressions we will also have the framing with the classical means.

Proposition 1 (A refinement of the AM-GM-HM Inequality, binary version). If
a, b > 0 and p, q ≥ 0, with p + q = 1, then the following chain of inequalities holds:

a + b p,q √ p,q 2
≥ M (x) ≥ ab ≥ M (x) ≥ . (4)
2 a,b a,b 1 + 1
a b
Proof. For the first inequality, by the GM-AM Inequality we have

» (GM−AM) (pa + qb) + (qa + pb) a(p + q) + b(p + q) a + b
M p,q = (pa + qb) · (qa + pb) ≤ = = .
a,b
2 2 2
For the second inequality, by the weighted AM-GM Inequality we have
√ √
» (w AM−GM) »
p,q
p
q
q
p
M = (pa + qb) · (qa + pb) ≥ (a · b ) · (a · b ) = a p+q · b p+q = a · b.
a,b
p,q
The last two inequalities from enounce, those regarding the expression M a,b , are obtained from
1 1
the first two inequalities by the substitutions a → , b → .
a b
p,q p,q
Remark 1. From (4), it follows that the quantities M and M are means too.
a,b a,b
Remark 2. Mean pairs M a,b (x), M a,b (x) and M p,q , M p,q are respectively equivalent, in the sense
a,b a,b
that they can be obtained from each other.
p,q
2
2
Indeed, by taking p = sin x and q = cos x in M a,b , one obtain M a,b (x).
√ p,q
Conversely, by taking x = arcsin p in M a,b (x), one obtain M .
a,b
p,q
A similar equivalence can be obtained between M a,b (x) and M a,b .
p,q
Although the two pairs of means are equivalent, we prefer the pair M a,b , M p,q , because it
a,b
also allows a beautiful generalization, which we will detail as follows.
For any strictly positive real numbers a 1 , a 2 , . . . , a n , we will consider the following (classical)
means of these numbers:
a 1 + a 2 + . . . + a n
A n [a] = − the arithmetic mean,
√ n
G n [a] = n a 1 · a 2 · . . . · a n − the geometric mean,
n
H n [a] = − the harmonic mean.
1 1 1
+ + . . . +
a 1 a 2 a n

As in the binary case, we will consider the following general version of the expressions from
the relations (2) and (3).