Bernoulli's inequality

In mathematics, Bernoulli's inequality (named after Jacob Bernoulli) is an inequality that approximates exponentiations of

It is often employed in real analysis.

It has several useful variants:[1] Jacob Bernoulli first published the inequality in his treatise "Positiones Arithmeticae de Seriebus Infinitis" (Basel, 1689), where he used the inequality often.

[3] According to Joseph E. Hofmann, Über die Exercitatio Geometrica des M. A. Ricci (1963), p. 177, the inequality is actually due to Sluse in his Mesolabum (1668 edition), Chapter IV "De maximis & minimis".

[3] The first case has a simple inductive proof: Suppose the statement is true for

: Then it follows that Bernoulli's inequality can be proved for case 2, in which

By the modified induction we conclude the statement is true for every non-negative integer

is negative gives case 3.

can be generalized to an arbitrary real number as follows: if

This generalization can be proved by comparing derivatives.

The strict versions of these inequalities require

the inequality holds also in the form

Bernoulli's inequality is a special case when

This generalized inequality can be proved by mathematical induction.

In the second step we assume validity of the inequality for

After multiplying both sides with a positive number

So the quantity on the right-hand side can be bounded as follows:

The following theorem presents a strengthened version of the Bernoulli inequality, incorporating additional terms to refine the estimate under specific conditions.

) by using the formula for geometric series: (using

respectively, we get Note that and so our inequality is equivalent to After substituting

(bearing in mind that this implies

Bernoulli's inequality is equivalent to and by the formula for geometric series (using y = 1 + x) we get which leads to Now if

then by monotony of the powers each summand

Since the product of two non-positive numbers is non-negative, we get again (4).

One can prove Bernoulli's inequality for x ≥ 0 using the binomial theorem.

It is true trivially for r = 0, so suppose r is a positive integer.

and the reversed inequality is valid for

Another way of using convexity is to re-cast the desired inequality to

This inequality can be proved using the fact that the

function is concave, and then using Jensen's inequality in the form