Bernoulli's inequality

From Wikipedia, the free encyclopedia

This article does not cite any references or sources. (April 2008)
Please help improve this article by adding citations to reliable sources. Unverifiable material may be challenged and removed.

An illustration of Bernoulli's inequality, with the graphs of

y = (1 + x) r

and

y = 1 + r x

shown in red and blue respectively. Here,

r = 3.

In real analysis, Bernoulli's inequality is an inequality that approximates exponentiations of 1 + x.

The inequality states that

$(1 + x)^r \geq 1 + rx\!$

for every integer r ≥ 0 and every real number x > −1. If the exponent r is even, then the inequality is valid for all real numbers x. The strict version of the inequality reads

$(1 + x)^r > 1 + rx\!$

for every integer r ≥ 2 and every real number x ≥ −1 with x ≠ 0.

Bernoulli's inequality is often used as the crucial step in the proof of other inequalities. It can itself be proved using mathematical induction, as shown below.

1 Proof of the inequality
2 Generalization
3 Related inequalities
4 External links

[edit] Proof of the inequality

For $r = 0,$ $(1+x)^0 \ge 1+0x$ is equivalent to $1\ge 1$ which is true as required.

Now suppose the statement is true for $r = k$ :

$(1+x)^k \ge 1+kx.$

Then it follows that

$(1+x)(1+x)^k \ge (1+x)(1+kx)$ (by hypothesis, since $(1+x)\ge 0$ )

$\begin{matrix} & \iff & (1+x)^{k+1} \ge 1+kx+x+kx^2 \\ & \iff & (1+x)^{k+1} \ge 1+(k+1)x+kx^2 \end{matrix}.$

However, as $1+(k+1)x + kx^2 \ge 1+(k+1)x$ (since $kx^2 \ge 0$ ), it follows that $(1+x)^{k+1} \ge 1+(k+1)x$ , which means the statement is true for $r = k + 1$ as required.

By induction we conclude the statement is true for all $r\ge 0.$

[edit] Generalization

The exponent r can be generalized to an arbitrary real number as follows: if x > −1, then

$(1 + x)^r \geq 1 + rx\!$

for r ≤ 0 or r ≥ 1, and

$(1 + x)^r \leq 1 + rx\!$

for 0 ≤ r ≤ 1. This generalization can be proved by comparing derivatives. Again, the strict versions of these inequalities require x ≠ 0 and r ≠ 0, 1.