**AM-GM**for

**arithmetic-mean/geometric-mean**is frequently used while referring to the

**AM-GM inequality**. The “means” are defined as

**and**

*A*= (_{n}*x*+...+_{1}*x*)/_{n}*n*

*G*= (_{n}*x*⋅...⋅_{1}*x*)_{n}^{1/n}for all

**. The AM-GM inequality states that**

*x*≥ 0_{k}**.**

*A*≥_{n}*G*_{n}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:

*x*≥ 1 +^{n}*n*(*x*- 1)for any natural number

**and**

*n***.**

*x*> 0Our acronym expands to AM-GM-HM, when the harmonic mean (HM or

*H*) is included. In the

_{n}**AM-GM-HM inequalitiy**the “means”are related as follows [2]:

*A*≥_{n}*G*_{n}**≥**.

*H*_{n}*: mathematics, statements, relationships, inequalities, equivalence.*

**Keywords****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**: www.physicsforums.com/library.php?do=view_item&itemid=14.

*: the*

**Note****harmonic mean**is defined as

*H*=_{n}

*n/***(1/**

*x*_{1}**+**

**...**

**+**

**1/**.

*x*)_{n}
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