Bernoulli inequality
Bernoulli's inequality in real analysis states that
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
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 is non-zero, then
for r ≤ 0 or r ≥ 1 and
for 0 ≤ r ≤ 1.
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List of mathematical topics | List of mathematical topics (A-C) | List of mathematics topics | List of real analysis topics | Real analysis
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