Logarithmic mean

 In mathematics, the logarithmic mean is a function of two non-negative numbers which is equal to their difference divided by the logarithm of their quotient. This calculation is applicable in engineering problems involving heat and mass transfer.

Three-dimensional plot showing the values of the logarithmic mean.

DefinitionEdit

The logarithmic mean is defined as:

${\displaystyle {\begin{aligned}M_{\text{lm}}(x,y)&=\lim _{(\xi ,\eta )\to (x,y)}{\frac {\eta -\xi }{\ln(\eta )-\ln(\xi )}}\\[6pt]&={\begin{cases}x&{\text{if }}x=y,\\{\frac {y-x}{\ln(y)-\ln(x)}}&{\text{otherwise,}}\end{cases}}\end{aligned}}}$

for the positive numbers ${\displaystyle x,y}$.

InequalitiesEdit

The logarithmic mean of two numbers is smaller than the arithmetic mean and the generalized mean with exponent one third but larger than the geometric mean, unless the numbers are the same, in which case all three means are equal to the numbers.

${\displaystyle {\sqrt {xy}}\leq M_{\text{lm}}(x,y)\leq \left({\frac {x^{1/3}+y^{1/3}}{2}}\right)^{3}\leq {\frac {x+y}{2}}\qquad {\text{ for all }}x>0{\text{ and }}y>0.}$^[1][2][3]

DerivationEdit

Mean value theorem of differential calculusEdit

From the mean value theorem, there exists a value ${\displaystyle \xi }$ in the interval between x and y where the derivative ${\displaystyle f'}$ equals the slope of the secant line:

${\displaystyle \exists \xi \in (x,y):\ f'(\xi )={\frac {f(x)-f(y)}{x-y}}}$

The logarithmic mean is obtained as the value of ${\displaystyle \xi }$ by substituting ${\displaystyle \ln }$ for ${\displaystyle f}$ and similarly for its corresponding derivative:

${\displaystyle {\frac {1}{\xi }}={\frac {\ln(x)-\ln(y)}{x-y}}}$

and solving for ${\displaystyle \xi }$:

${\displaystyle \xi ={\frac {x-y}{\ln(x)-\ln(y)}}}$

IntegrationEdit

The logarithmic mean can also be interpreted as the area under an exponential curve.

${\displaystyle {\begin{aligned}L(x,y)={}&\int _{0}^{1}x^{1-t}y^{t}\ \mathrm {d} t={}\int _{0}^{1}\left({\frac {y}{x}}\right)^{t}x\ \mathrm {d} t={}x\int _{0}^{1}\left({\frac {y}{x}}\right)^{t}\mathrm {d} t\\[3pt]={}&\left.{\frac {x}{\ln \left({\frac {y}{x}}\right)}}\left({\frac {y}{x}}\right)^{t}\right|_{t=0}^{1}={}{\frac {x}{\ln \left({\frac {y}{x}}\right)}}\left({\frac {y}{x}}-1\right)={}{\frac {y-x}{\ln \left({\frac {y}{x}}\right)}}\\[3pt]={}&{\frac {y-x}{\ln \left(y\right)-\ln \left(x\right)}}\end{aligned}}}$

The area interpretation allows the easy derivation of some basic properties of the logarithmic mean. Since the exponential function is monotonic, the integral over an interval of length 1 is bounded by ${\displaystyle x}$ and ${\displaystyle y}$. The homogeneity of the integral operator is transferred to the mean operator, that is ${\displaystyle L(cx,cy)=cL(x,y)}$.

Two other useful integral representations are

$${\displaystyle {1 \over L(x,y)}=\int _{0}^{1}{\operatorname {d} \!t \over tx+(1-t)y}}$$

and

$${\displaystyle {1 \over L(x,y)}=\int _{0}^{\infty }{\operatorname {d} \!t \over (t+x)\,(t+y)}.}$$

GeneralizationEdit

Mean value theorem of differential calculusEdit

One can generalize the mean to ${\displaystyle n+1}$ variables by considering the mean value theorem for divided differences for the ${\displaystyle n}$th derivative of the logarithm.

We obtain

${\displaystyle L_{\text{MV}}(x_{0},\,\dots ,\,x_{n})={\sqrt[{-n}]{(-1)^{(n+1)}n\ln \left(\left[x_{0},\,\dots ,\,x_{n}\right]\right)}}}$

where ${\displaystyle \ln \left(\left[x_{0},\,\dots ,\,x_{n}\right]\right)}$ denotes a divided difference of the logarithm.

For ${\displaystyle n=2}$ this leads to

${\displaystyle L_{\text{MV}}(x,y,z)={\sqrt {\frac {(x-y)\left(y-z\right)\left(z-x\right)}{2\left(\left(y-z\right)\ln \left(x\right)+\left(z-x\right)\ln \left(y\right)+\left(x-y\right)\ln \left(z\right)\right)}}}}$.

IntegralEdit

The integral interpretation can also be generalized to more variables, but it leads to a different result. Given the simplex ${\textstyle S}$ with ${\textstyle S=\{\left(\alpha _{0},\,\dots ,\,\alpha _{n}\right):\left(\alpha _{0}+\dots +\alpha _{n}=1\right)\land \left(\alpha _{0}\geq 0\right)\land \dots \land \left(\alpha _{n}\geq 0\right)\}}$ and an appropriate measure ${\textstyle \mathrm {d} \alpha }$ which assigns the simplex a volume of 1, we obtain

${\displaystyle L_{\text{I}}\left(x_{0},\,\dots ,\,x_{n}\right)=\int _{S}x_{0}^{\alpha _{0}}\cdot \,\cdots \,\cdot x_{n}^{\alpha _{n}}\ \mathrm {d} \alpha }$

This can be simplified using divided differences of the exponential function to

${\displaystyle L_{\text{I}}\left(x_{0},\,\dots ,\,x_{n}\right)=n!\exp \left[\ln \left(x_{0}\right),\,\dots ,\,\ln \left(x_{n}\right)\right]}$.

Example ${\textstyle n=2}$

${\displaystyle L_{\text{I}}(x,y,z)=-2{\frac {x\left(\ln \left(y\right)-\ln \left(z\right)\right)+y\left(\ln \left(z\right)-\ln \left(x\right)\right)+z\left(\ln \left(x\right)-\ln \left(y\right)\right)}{\left(\ln \left(x\right)-\ln \left(y\right)\right)\left(\ln \left(y\right)-\ln \left(z\right)\right)\left(\ln \left(z\right)-\ln \left(x\right)\right)}}}$.

Connection to other meansEdit

• Arithmetic mean: ${\displaystyle {\frac {L\left(x^{2},y^{2}\right)}{L(x,y)}}={\frac {x+y}{2}}}$

• Geometric mean: ${\displaystyle {\sqrt {\frac {L\left(x,y\right)}{L\left({\frac {1}{x}},{\frac {1}{y}}\right)}}}={\sqrt {xy}}}$

• Harmonic mean: ${\displaystyle {\frac {L\left({\frac {1}{x}},{\frac {1}{y}}\right)}{L\left({\frac {1}{x^{2}}},{\frac {1}{y^{2}}}\right)}}={\frac {2}{{\frac {1}{x}}+{\frac {1}{y}}}}}$

This article uses material from the Wikipedia article  Metasyntactic variable, which is released under the  Creative Commons Attribution-ShareAlike 3.0 Unported License.