Bernoulli's inequality

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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. 

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