Is there a name for this "mean"?

Question

We all know these means:

$$GM = \sqrt[3]{xyz} $$ $$AM = \frac{x + y + z}{3}$$ $$QM = \sqrt{\frac{x^2 + y^2 + z^2}{3}} $$

Of course:

$$GM \le AM \le QM $$

What about this one:

$$XM = \sqrt{\frac{xy + yz + zx}{3}} $$

Does it have its name? Are there inequalities connecting it to other means?

I did some basic search, but was surprised not to find anything.

It can be rewritten as $\sqrt{\frac{xyz}{HM}}$ where $HM$ is the harmonic mean. Not sure how helpful that is. This in turn can be re-written as $$GM\sqrt{\frac{GM}{HM}}.$$ — Thomas Andrews, Aug 31 '15 at 18:59
What does QM stand for? I assume something with Euclidean... — Chris, Aug 26 '16 at 04:37

score 8 · Accepted Answer · edited Apr 13 '17 at 12:21

The quantity $XM$ lies between the arithmetic and geometric means, that is $$AM\geq XM\geq GM.$$ Notice that $$3AM^2=2XM^2+QM^2,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$$ and so since $AM\leq QM$, it follows that $$3AM^2=2XM^2+QM^2\geq 2XM^2 +AM^2\Rightarrow AM\geq XM.$$ The AM-GM inequality implies that $XM\geq GM$. From $(1)$ we may write $$XM=\sqrt{\frac{3AM^2-QM^2}{2}},$$ but a nicer way to express $XM$ is given in Thomas Andrews comment: $$XM=GM\sqrt{\frac{GM}{HM}}.$$

See Also: Newton's inequalities. More generally quantities such as $XM$ are often referred to as Elementary Symmetric Means.

Is there a name for this "mean"?

1 Answers1