Bernoulli's inequality

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In mathematics, Bernoulli's inequality is an inequality that approximates exponentiations of 1 + x.

The inequality states that

<math>(1 + x)^r \geq 1 + rx\!</math>

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

<math>(1 + x)^r > 1 + rx\!</math>

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.

The exponent r can be generalized to an arbitrary real number as follows: 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. 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

<math>(1 + x)^r < e^{rx},\!</math>

where e = 2.718.... This may be proved using the inequality (1 + 1/k)^k < e.de:Bernoullische Ungleichung eo:Vikipedio:Projekto matematiko/Neegalaĵo de Bernoulli fr:Inégalité de Bernoulli it:Diseguaglianza di Bernoulli he:אי שוויון ברנולי pl:Nierówność Bernoulliego pt:Desigualdade de Bernoulli ru:Неравенство Бернулли zh:伯努利不等式