Bernoulli's inequality

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 References
5 External links

[ 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 $r\ge 0.$

[ 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 thee inequality (1 + 1/k)^k < e.

[ References

Carothers, N. (2000). Real Analysis. Cambridge: Cambridge University Press. pp. 9. ISBN 0521497566.
Bullen, P.S. (1987). Handbook of Means and Their Inequalities. Berlin: Springer. pp. 4. ISBN 1402015224.
Zaidman, Samuel (1997). Advanced Calculus. City: World Scientific Publishing Company. pp. 32. ISBN 9810227043.

[ External links

Eric W. Weisstein, Bernoulli Inequality at MathWorld.
Bernoulli Inequality by Chris Boucher, Wolfram Demonstrations Project.

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Category: Inequalities

Bernoulli inequality