Friday, March 30, 2012

Acronym in mathematics: AM-GM for arithmetic-mean/geometric-mean

The mathematical acronym AM-GM for arithmetic-mean/geometric-mean is frequently used while referring to the AM-GM inequality. The “means” are defined as

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 [1]. The latter was proved in the 17th century by Isaac Barrow and Jacob Bernoulli:

xn 1 + n(x - 1)

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 [2]:

  An Gn  Hn .

Keywords: mathematics, statements, relationships, inequalities, equivalence.

References and details
[1] 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.
[2] Physics Forums > Mathematics > Calculus/Analysis > Inequalities > AM-GM-HM inequality:

Note: the harmonic mean is defined as

Hn = n/(1/x1 +...+ 1/xn).

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