Page 23 - MATINF Nr. 11-12

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ARTICOLE SI NOTE DE MATEMATICA
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Refinements of classical means inequalities

Dorin M˘arghidanu 1

In this short note, some refinements of the classical inequalities between the arithmetic mean,
the geometric mean and the harmonic mean are presented. The expressions that achieve these
refinements are also highlighted. Furthermore, these expressions are means too.

We recall that a (binary) mean in S ⊂ R is a function M : S × S −→ S with the property
of internality:
min{a, b} ≤ M(a, b) ≤ max{a, b}, (∀) a, b ∈ S.

The following properties are specific to many types of means (see e.g. [1], [2]):

• symmetry: M(a, b) = M(b, a), (∀) a, b ∈ S;

∗
• homogeneity: M(ka, kb) = kM(a, b), (∀) a, b ∈ S, (∀) k ∈ R .
+
The following two expressions have been introduced in [2]:

»
2
2
M a,b (x) = a sin x + b cos x · a cos x + b sin x ,
2
2
ab
.
M a,b (x) = »
2
2
a sin x + b cos x · a cos x + b sin x
2
2
If a and b are strictly positive real numbers, we denote by
a + b √ 2
A 2 (a, b) = , G 2 (a, b) = ab and H 2 (a, b) =
2 1 + 1
a b
the arithmetic mean, the geometric mean, and the harmonic mean of numbers a and b, respectively.
For all these quantities, the following multiple inequality is established in [2]:

A 2 (a, b) ≥ M a,b (x) ≥ G 2 (a, b) ≥ M a,b (x) ≥ H 2 (a, b), (1)

from which we deduce that the expressions M a,b (x) and M a,b (x) are in turn means of the
numbers a and b, and (1) refines the classical means inequality.

2
2
Like as in these expressions, where the coefficients of a and b are sin x and cos x, which can
be regarded as weights (thanks to the fundamental formula of trigonometry), it makes sense to
consider the following general weighted expressions.
1
Profesor dr., Colegiul Nat , ional ,,Al. I. Cuza”, Corabia, d.marghidanu@gmail.com

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