Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
    
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

2 Answers 2

up vote 1 down vote accepted

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 $p$.

share|improve this answer

$\dfrac{2(a^2+ab+b^2)}{3(a+b)}$ 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 $E(2,3;x,y)$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.