Bernoulli's inequality
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In mathematics, Bernoulli's inequality states that
- <math>(1+x)^n\geq 1+nx\,<math>
for every integer n ≥ 0 and every real number x ≥ −1. If n ≥ 0 is even, then the inequality is valid for all real numbers x. The strict version of the inequality reads
- <math>(1+x)^n>1+nx\,<math>
for every integer n ≥ 2 and every real number x ≥ −1 with x ≠ 0.
The inequality is often used as the crucial step in the proof of other inequalities. It can be proven using mathematical induction.
The following generalizations for real exponents can be proved by comparing derivatives: if x > −1, then
- <math>(1+x)^r\geq 1+rx\,<math>
for r ≤ 0 or r ≥ 1 and
- <math>(1+x)^r\leq 1+rx\,<math>
for 0 ≤ r ≤ 1.
Related inequalities
The following inequality estimates the r-th power of 1+x from the other side. For any <math>x, r > 0<math> one has
- <math>(1+x)^r < e^{rx}.\,<math>pl:nierówność Bernoulliego