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ARTICOLE SI NOTE MATEMATICE
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A refinement of the Modulus inequality

˘
Dorin MARGHIDANU 1

Among the properties of the modulus, there is one - intensively used - that
compares the modulus of the sum of real numbers with the sum of the modules of
the respective numbers, also called the Modulus inequality:

|x 1 + x 2 + . . . + x n | ≤ |x 1 | + |x 2 | + . . . + |x n | , ∀x 1 , x 2 , . . . , x n ∈ R, n ∈ N, n ≥ 2.

For the case n = 2, due to its geometric interpretation, the inequality is also
called the Triangle inequality.
Next we are interested to obtain a refinement of the Modulus inequality by
interposing a numerical expression between its two sides. For this we will first
look for a refinement of the Triangle inequality.

Lemma 1 (A refinement of Triangle inequality, [1], a)). For any x, y ∈ R
with x 6= y, the following inequalities hold

x |x| − y |y|
|x + y| ≤ ≤ |x| + |y| . (1)
x − y
Proof. For the real numbers x and y with x 6= y, we have the following possibilities.

I. If x ≥ 0, y ≥ 0, x 6= y, then the given inequalities become

2
x − y 2
x + y ≤ ≤ x + y,
x − y
true, even with equalities.
II. If x ≤ 0, y ≤ 0, x 6= y, then the given inequalities become

2
x − y 2
−x − y ≤ − ≤ −x − y,
x − y
true, even with equalities.
1
Profesor dr., Colegiul Nat , ional ,,Al. I. Cuza”, Corabia, d.marghidanu@gmail.com

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