Gianni Bellocchi

2014-02-01T20:00:32Z

88.138.71.53: /* Biography */

{{Dablink|Not to be confused "[[Geometric mean#Log-average|log-average]]" which is a [[geometric mean]]}}

[[Image:Logarithmic mean 3D plot from 0 to 100.png|thumb|300px|Three dimensional plot showing the values of the logarithmic mean.]]

{{Refimprove|date=April 2009}}
In [[mathematics]], the '''logarithmic mean''' is a [[function (mathematics)|function]] of two non-negative [[number]]s which is equal to their [[Difference (mathematics)|difference]] divided by the [[logarithm]] of their [[quotient]]. In symbols:

:<math>
\begin{array}{ll}
M_{\text{lm}}(x,y)
&=
\lim_{(\xi,\eta)\to(x,y)} \frac{\eta - \xi}{\ln \eta - \ln \xi},
\\
&=
\begin{cases}
0 & \text{if }x=0 \text{ or } y=0 ,\\
x & \text{if }x=y ,\\
\frac{y - x}{\ln y - \ln x} & \text{otherwise,}
\end{cases}
\end{array}
</math>

for the positive numbers <math>x, y</math>.
This calculation is applicable in [[engineering]] problems involving [[heat transfer|heat]] and [[mass transfer]].

== Inequalities ==

The logarithmic mean of two numbers is smaller than the [[arithmetic mean]] but larger than the [[geometric mean]] (unless the numbers are the same, in which case all three means are equal to the numbers):

: <math> \sqrt{x\cdot y} < M_{\text{lm}}(x,y) < \frac{x+y}{2} \qquad \text{ for all } x>0 \text{ and } y>0.</math>

== Derivation of the mean ==
=== Mean value theorem of differential calculus ===

From the [[mean value theorem]]
:<math> \exists \xi\in[x,y] : \ f'(\xi) = \frac{f(x)-f(y)}{x-y} </math>
the logarithmic mean is obtained as the value of <math>\xi</math>
by substituting <math>\ln</math> for <math>f</math>
:<math> \frac{1}{\xi} = \frac{\ln x - \ln y}{x-y} </math>
and solving for <math>\xi</math>.
:<math> \xi = \frac{x-y}{\ln x - \ln y} </math>

=== Integration ===

The logarithmic mean can also be interpreted as the [[area]] under an [[exponential function|exponential curve]].
:<math>L(x,y) = \int_0^1 x^{1-t} y^t\ \mathrm{d}t</math>
<math>\begin{array}{rcl}
\int_0^1 x^{1-t} y^t\ \mathrm{d}t
&=& \int_0^1 \left(\frac{y}{x}\right)^t x\ \mathrm{d}t \\
&=& x \int_0^1 \left(\frac{y}{x}\right)^t \mathrm{d}t \\
&=& \frac{x}{\ln \frac{y}{x}} \left(\frac{y}{x}\right)^t|_{t=0}^{1}\\
&=& \frac{x}{\ln \frac{y}{x}} \left(\frac{y}{x}-1\right)\\
&=& \frac{y-x}{\ln y - \ln x}

\end{array}</math>

The area interpretation allows the easy derivation of some basic properties of the logarithmic mean.
Since the exponential function is [[monotonic function|monotonic]],
the integral over an interval of length 1 is bounded by <math>x</math> and <math>y</math>.
The [[Homogeneous function|Homogeneity]] of the integral operator is transferred to the mean operator,
that is <math>L(c\cdot x, c\cdot y) = c\cdot L(x,y)</math>.

== Generalization ==
=== Mean value theorem of differential calculus ===

You can generalize the mean to <math>n+1</math> variables by considering the [[mean value theorem (divided differences)|mean value theorem for divided differences]] for the <math>n</math>th [[derivative]] of the logarithm.
You obtain
:<math>L_{\mathrm{MV}}(x_0,\dots,x_n) = \sqrt[-n]{(-1)^{(n+1)}\cdot n \cdot \ln[x_0,\dots,x_n]}</math>
where <math>\ln[x_0,\dots,x_n]</math> denotes a [[divided difference]] of the logarithm.

For <math>n=2</math> this leads to
:<math>L_{\mathrm{MV}}(x,y,z) = \sqrt{\frac{(x-y)\cdot(y-z)\cdot(z-x)}{2\cdot((y-z)\cdot\ln x + (z-x)\cdot\ln y + (x-y)\cdot\ln z)}}</math>.

=== Integral ===

The integral interpretation can also be generalized to more variables,
but it leads to a different result.
Given the [[simplex]] <math>S</math>
with <math>S = \{(\alpha_0,\dots,\alpha_n) : \alpha_0+\dots+\alpha_n=1\ \land\ \alpha_0\ge0\ \land\ \dots\ \land\ \alpha_n\ge0\}</math> and an appropriate measure <math>\mathrm{d}\alpha</math> which assigns the simplex a volume of 1, we obtain
:<math>L_{\mathrm{I}}(x_0,\dots,x_n) = \int_S x_0^{\alpha_0}\cdot\dots\cdot x_n^{\alpha_n}\ \mathrm{d}\alpha</math>
This can be simplified using divided differences of the exponential function to
:<math>L_{\mathrm{I}}(x_0,\dots,x_n) = n!\cdot\exp[\ln x_0, \dots, \ln x_n]</math>.

Example <math>n=2</math>
:<math>L_{\mathrm{I}}(x,y,z) = -2\cdot\frac{x\cdot(\ln y-\ln z) + y\cdot(\ln z-\ln x) + z\cdot(\ln x-\ln y)}{(\ln x-\ln y)\cdot(\ln y-\ln z)\cdot(\ln z-\ln x)}</math>.


== Connection to other means ==

* <math>\frac{L(x^2,y^2)}{L(x,y)} = \frac{x+y}{2}</math> ([[Arithmetic mean]])

== See also ==

* A different mean which is related to logarithms is the [[geometric mean]].
* The logarithmic mean is a special case of the [[Stolarsky mean]].

== References ==
*[http://www.everything2.com/index.pl?node_id=801020 Logarithmic mean @ Everything2.com]
*[http://jipam-old.vu.edu.au/v4n4/088_03.html Oilfield Glossary: Term 'logarithmic mean']
* {{mathworld|Arithmetic-Logarithmic-GeometricMeanInequality|Arithmetic-Logarithmic-Geometric-Mean Inequality}}
* Stolarsky, Kenneth B.: ''[http://links.jstor.org/sici?sici=0025-570X%28197503%2948%3A2%3C87%3AGOTLM%3E2.0.CO%3B2-6 Generalizations of the logarithmic mean]'', Mathematics Magazine, Vol. 48, No. 2, Mar., 1975, pp 87–92

[[Category:Means]]

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Gianni Bellocchi