In real analysis, Bernoulli's inequality (named after Jacob Bernoulli) is an inequality that approximates exponentiations of 1 + x.
The inequality states that
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
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,
is equivalent to 1 ≥ 1 which is true as required.
Now suppose the statement is true for r = k:
Then it follows that
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.
The exponent r can be generalized to an arbitrary real number as follows: if x > −1, then
for r ≤ 0 or r ≥ 1, and
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.
The following inequality estimates the r-th power of 1 + x from the other side. For any real numbers x, r > 0, one has
where e = 2.718.... This may be proved using the inequality (1 + 1/k)k < e.
- 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.
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