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Refinements of Bernoulli’s inequality

˘
Dorin MARGHIDANU 1

Bernoulli’s inequality [1], named after its author, the Swiss mathematician Ja-
cob Bernoulli (1654–1705), is one of the most important mathematical inequalities.
It has been known and used since the 17th century, has a very simple expression
and has many applications. For the general case - of real powers, we have the
following statement:

Proposition 1 (Bernoulli’s inequality). If x > −1 and r ∈ R, r ≥ 1, then

r
(1 + x) ≥ 1 + rx. (B)
Equality occurs if r = 1 or x = 0.

Remark 1. Inequality (B) also occurs if r ≤ 0. If r ∈ (0, 1), the inequality (B) is
reversed.

Bernoulli’s inequality underlies many famous inequalities, including: means
inequality, Young’s inequality, H¨older’s inequality, Minkowski’s inequality, Radon’s
inequality, and so on. In fact (B) is equivalent to each of the inequalities listed.
For further information on Bernoulli’s inequality, see [2], [3], [4].

Next we will interpose between the members of the inequality (B) certain
expressions - that is to say we will refine the inequality of Bernoulli (sometimes
with slight modifications for the exponent r). Interpolating expressions are also
generated by Bernoulli inequality. It could be said that the obtained refinements
of the Bernoulli inequality are in fact self-refinements!

Proposition 2 (Refinement of Bernoulli’s inequality). If x > −1 and r ∈ R,
r ≥ 2, then
r
(1 + x) ≥ (1 + x) [1 + (r − 1)x] ≥ 1 + rx. (B1)
The first inequality becomes equality if r = 2 or x = 0, and the second inequality
becomes equality if x = 0.
Proof. Applying the inequality (B) for the case r − 1, we obtain very simply:
(B)
r
(1 + x) = (1 + x) · (1 + x) r−1 ≥ (1 + x) · [1 + (r − 1)x], and then:
2
(1 + x) [1 + (r − 1)x] = 1 + rx + (r − 1)x ≥ 1 + rx.
1
Profesor dr., Colegiul Nat , ional ,,Al. I. Cuza”, Corabia, d.marghidanu@gmail.com

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