An = (x1 +...+ xn)/n and
Gn = (x1 ⋅...⋅ xn)1/n
for all xk ≥ 0. The AM-GM inequality states that
An ≥ Gn .
The AM-GM equality is sometimes called the Cauchy inequality. In a recent note, Lech Maligranda shows that the AM-GM inequality is equivalent to what now is called the Bernoulli inequality . The latter was proved in the 17th century by Isaac Barrow and Jacob Bernoulli:
for any natural number n and x > 0.
Our acronym expands to AM-GM-HM, when the harmonic mean (HM or Hn) is included. In the AM-GM-HM inequalitiy the “means”are related as follows :
An ≥ Gn ≥ Hn .
Keywords: mathematics, statements, relationships, inequalities, equivalence.
References and details
 L. Maligranda: The AM-GM Inequality is Equivalent to the Bernoulli Inequality. The Mathematical Intelligencer 2012, 34 (1), page 1. DOI: 10.1007/s00283-011-9266-8.
 Physics Forums > Mathematics > Calculus/Analysis > Inequalities > AM-GM-HM inequality: www.physicsforums.com/library.php?do=view_item&itemid=14.
Note: the harmonic mean is defined as