# Is the centroidal mean a generalized mean?

The centroidal mean of two numbers $a,b$ is the number $\dfrac{2(a^2+ab+b^2)}{3(a+b)}.$ In a trapezoid whose bases have lengths $a$ and $b$, it is the length of the line segment parallel to the bases which passes through the centroid.

For $p\in \Bbb R\setminus 0,$ a generalized $p$-mean of two numbers $a,b$ is the number $\left(\dfrac{a^p+b^p}2\right)^{1/p}.$

Is the centroidal mean a generalized $p$-mean for some $p$? I suspect it isn't, but I have no idea how to prove it. Just trying to equate the two expressions has led me nowhere.

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The $2$ in the numerator and denominator could be cancelled, y'know... also, your formula doesn't match the known formula for the length of the median. – J. M. Jul 16 '12 at 11:40
@J.M. Sorry, one of them should have been $3$. – Bartek Jul 16 '12 at 11:41

No, its not: Consider two different situations(e.g. $a=1$, $b=0$ and $a=2$, $b=1$) solve for $p$ and observe that this gives different values for $p$.
$\dfrac{2(a^2+ab+b^2)}{3(a+b)}$ is not a Hölder mean, but it is a special case of the "extended mean" discussed by Leach and Scholander in their paper. The Leach-Scholander extended mean is defined by
$$E(r,s;a,b)=\sqrt[s-r]{\frac{r(a^s-b^s)}{s(a^r-b^r)}}$$
and your mean in terms of the Leach-Scholander extended mean is denoted as $E(2,3;x,y)$.