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# order of six means

The size order of the six usual means of two positive numbers ($a$ and $b$) is from the least to the greatest one

1. 2. 3. 4. 5. 6.

i. e.

$\frac{2ab}{a\!+\!b}\;\leqq\;\sqrt{ab}\;\leqq\;\frac{a\!+\!\sqrt{ab}\!+\!b}{3}% \;\leqq\;\frac{a\!+\!b}{2}\;\leqq\;\sqrt{\frac{a^{2}\!+\!b^{2}}{2}}\;\leqq\;% \frac{a^{2}\!+\!b^{2}}{a\!+\!b}.$ |

Proof. If $x^{2}-y^{2}\geqq 0$ for nonnegative $x$ and $y$, then $x\geqq y$.

“$1\leqq 2$”:

$\displaystyle\left(\sqrt{ab}\right)^{2}-\left(\frac{a\!+\!b}{2}\right)^{2}=ab%
\!-\!\frac{4a^{2}b^{2}}{(a\!+\!b)^{2}}=ab\left(1\!-\!\frac{4ab}{(a\!+\!b)^{2}}%
\right)=ab\cdot\frac{(a\!+\!b)^{2}-4ab}{(a\!+\!b)^{2}}=\frac{ab(a\!-\!b)^{2}}{%
(a+b)^{2}}\geqq 0$

“$2\leqq 3$” and “$3\leqq 4$”: proven in Heronian mean is between geometric and arithmetic mean

“$4\leqq 5$”:

$\displaystyle\left(\sqrt{\frac{a^{2}\!+\!b^{2}}{2}}\right)^{2}-\left(\frac{a\!%
+\!b}{2}\right)^{2}=\frac{2a^{2}\!+\!2b^{2}\!-\!a^{2}\!-\!2ab\!-\!b^{2}}{4}=%
\left(\frac{a\!-\!b}{2}\right)^{2}\geqq 0$

“$5\leqq 6$”:

$\displaystyle\left(\frac{a^{2}\!+\!b^{2}}{a\!+\!b}\right)^{2}-\left(\sqrt{%
\frac{a^{2}\!+\!b^{2}}{2}}\right)^{2}=\frac{2(a^{2}\!+\!b^{2})^{2}-(a^{2}\!+\!%
b^{2})(a\!+\!b)^{2}}{2(a\!+\!b)^{2}}=\frac{(a^{2}\!+\!b^{2})(2a^{2}\!+\!2b^{2}%
\!-\!a^{2}\!-\!2ab\!-\!b^{2})}{2(a\!+\!b)^{2}}\\
=\frac{(a^{2}\!+\!b^{2})(a\!-\!b)^{2}}{2(a\!+\!b)^{2}}\geqq 0$

## Mathematics Subject Classification

06A05*no label found*26B35

*no label found*26D07

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## Comments

## Generalized mean

There are a formula of "generalized mean" (if I translated it to English correctly) which generalizes arithmetic, geometric, quadratic and other means.

It should be mentioned.

Sorry, now I don't have free time to write about it (even if I would remember the details).

--

Victor Porton - http://www.mathematics21.org

* Algebraic General Topology and Math Synthesis

## Re: Generalized mean

Do you mean http://en.wikipedia.org/wiki/Generalized_mean?

It does not cover e.g. the Heronian mean.

Jussi

## Re: Generalized mean

> Do you mean http://en.wikipedia.org/wiki/Generalized_mean?

Yes.

--

Victor Porton - http://www.mathematics21.org

* Algebraic General Topology and Math Synthesis

## Re: Generalized mean

Porton,

there are some such entries in PM (e.g. http://planetmath.org/encyclopedia/PowerMean.html.

Jussi

## Re: Generalized mean

Hey pahio and Porton,

However I'm just looking at Wiki as well as in PM pure *discrete* means.

peruchin

## Re: Generalized mean

Dear Peruchin,

I have never heard of discrete means! What are they?

Jussi

## Re: Generalized mean

Hi Jussi,

well my friend, the data to obtain any mean of that type, is a finite set of numbers. As an example of continuum mean you have

\bar{x} = [\int_a^b w(x).x.dx]/[\int_a^b w(x).dx].

Pedro