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The harmonic mean of a finite set of positive real numbers $\{x_1, x_2, \ldots, x_n\}$ is defined to be $H(\{x_1, x_2, \ldots, x_n\}) = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n}}.$

The logarithmic mean of two distinct positive real numbers $a$ and $b$ is defined to be $L(a,b) = \frac{b - a}{\ln b - \ln a}.$

One of the first applications of integration that students often see is the extension of the arithmetic mean of a finite set of real numbers to the arithmetic mean of a function $f(x)$ on a continuous interval $[a,b]$ via $\frac{1}{b-a} \int_a^b f(x) dx.$

In the same way, you can extend the harmonic mean so that it applies to a positive function $f(x)$ over a continuous interval $[a,b]$. You get $\frac{b-a}{\int_a^b \frac{dx}{f(x)}} .$

Thus, if you take $f(x) = x$ you obtain the harmonic mean of the continuous interval $[a,b]$. This is $H([a,b]) = \frac{b-a}{\int_a^b \frac{1}{x} dx} = \frac{b-a}{\ln b - \ln a} = L(a,b).$

My question is this: Is there an intuitive reason why $H([a,b]) = L(a,b)$?

For comparison purposes, note that if $A$ denotes the arithmetic mean, then $A([a,b]) = A(a,b) = \frac{a+b}{2}$.

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    @Rahul : yes, I saw it in Amercian Mathemtical Monthly, $\displaystyle\lim_{n\to-1} \left( \int {x}^n dx = \frac {x^{n+1}}{n+1} \right)$2011-05-14

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The harmonic mean can be generalized to an arbitrary invertible function $w$: define the mean of $x_1,\ldots,x_n$ to be

$w^{-1}\left(\frac{w(x_1) + \cdots + w(x_n)}{n}\right)$

The logarithmic mean can also be generalized re its mean value interpretation for an arbitrary function $f$ such that $f'$ is invertible: define the mean of $x,y$ as

$(f')^{-1}\left(\frac{f(x) - f(y)}{x - y}\right)$

We can generalize your observation by taking $w = f'$. The corresponding mean, before taking $w^{-1}$, is

$\frac{1}{y-x}\int_x^y f'(t) \, dt = \frac{f(y)-f(x)}{y-x}$

and so the (former) mean of the interval with weight $f'$ is the same as the (latter) mean with respect to $f$.

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    I don't have a specific one in mind; I was just hoping that there was some way of connecting the concepts. For instance, an interpretation of the geometric mean is that "the geometric mean of growth over periods yields the equivalent constant growth rate that would yield the same final amount." The harmonic mean "is appropriate for situations when the average of rates is desired" or when calculating equivalent resistance for resistors in parallel. Something like that. (Quotes from Wikipedia.)2010-11-05