Bernoulli's inequality

In real analysis, Bernoulli's inequality (named after Jacob Bernoulli) 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.

Proof of the inequality

For r = 0,

\( (1+x)^0 \ge 1+0x \, \)

is equivalent to 1 ≥ 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

\( \begin{align} & {} \qquad (1+x)(1+x)^k \ge (1+x)(1+kx)\quad\text{(by hypothesis, since }(1+x)\ge 0) \\ & \iff (1+x)^{k+1} \ge 1+kx+x+kx^2, \\ & \iff (1+x)^{k+1} \ge 1+(k+1)x+kx^2. \end{align} \)

However, as 1 + (k + 1)x + kx² ≥ 1 + (k + 1)x (since kx² ≥ 0), it follows that (1 + x)^k + 1 ≥ 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 ≥ 0.By induction we conclude the statement is true for all r ≥ 0.

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.
Related inequalities

The following inequality estimates the r-th power of 1 + x from the other side. For any real numbers x, r > 0, one has

\( (1 + x)^r \le e^{rx},\! \)

where e = 2.718.... This may be proved using the inequality (1 + 1/k)^k < e.

Alternate Form

An alternate form of Bernoulli's inequality for \( t\geq 1 \) and \( 0\le x\le 1 \) is:

\( (1-x)^t \ge 1-xt \)

This can be proved (for integer t) by using the formula for geometric series: (using y=1-x)

\( t=1+1+\dots+1 \ge 1+y+y^2+\ldots+y^{t-1}=\frac{1-y^t}{1-y} or equivalently xt \ge 1-(1-x)^t \)

References

Carothers, N. (2000). Real Analysis. Cambridge: Cambridge University Press. p. 9. ISBN 978-0-521-49756-5.
Bullen, P.S. (1987). Handbook of Means and Their Inequalities. Berlin: Springer. p. 4. ISBN 978-1-4020-1522-9.
Zaidman, Samuel (1997). Advanced Calculus. City: World Scientific Publishing Company. p. 32. ISBN 978-981-02-2704-3.

External links

Weisstein, Eric W., "Bernoulli Inequality" from MathWorld.
Bernoulli Inequality by Chris Boucher, Wolfram Demonstrations Project.
Arthur Lohwater (1982). "Introduction to Inequalities". Online e-book in PDF format.

Mathematics Encyclopedia