# Bernoulli's inequality

 related topics {math, number, function}

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.

## Contents

### 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.

### Generalization

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.

### Related inequalities

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

where e = 2.718.... This may be proved using the 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.