Logarithmic mean

Not to be confused with the log-average formulation of the geometric mean or the mean in the log semiring.

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.

Definition

The logarithmic mean is defined by

${\displaystyle L(x,y)=\left\{{\begin{array}{l l}x,&{\text{if }}x=y,\\{\dfrac {x-y}{\ln x-\ln y}},&{\text{otherwise}},\end{array}}\right.}$

for ${\displaystyle x,y\in \mathbb {R} }$.

Inequalities

The logarithmic mean of two numbers is smaller than the arithmetic mean and the generalized mean with exponent greater than 1. However, it is larger than the geometric mean and the harmonic mean, respectively. The inequalities are strict unless both numbers are equal.^[1][2][3][4] More precisely, for ${\displaystyle x,y\in \mathbb {R} }$ with ${\displaystyle x\neq y}$, we have $${\displaystyle {\frac {2xy}{x+y}}\leq {\sqrt {xy}}\leq {\frac {x-y}{\ln x-\ln y}}\leq {\frac {x+y}{2}}\leq \left({\frac {x^{2}+y^{2}}{2}}\right)^{1/2}.}$$ Sharma^[5] showed that, for any whole number ${\displaystyle n}$ and ${\displaystyle x,y\in \mathbb {R} }$ with ${\displaystyle x\neq y}$, we have $${\displaystyle {\sqrt {xy}}\ \left(\ln {\sqrt {xy}}\right)^{n-1}\left(n+\ln {\sqrt {xy}}\right)\leq {\frac {x(\ln x)^{n}-y(\ln y)^{n}}{\ln x-\ln y}}\leq {\frac {x(\ln x)^{n-1}(n+\ln x)+y(\ln y)^{n-1}(n+\ln y)}{2}}.}$$ This generalizes the arithmetic-logarithmic-geometric mean inequality. To see this, consider the case where ${\displaystyle n=0}$.

Derivation

Mean value theorem of differential calculus

From the mean value theorem, there exists a value ξ in the interval between x and y where the derivative 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 ξ by substituting ln for f and similarly for its corresponding derivative:

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

and solving for ξ:

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

Integration

The logarithmic is also given by the integral $${\displaystyle L(x,y)=\int _{0}^{1}x^{1-t}y^{t}\,\mathrm {d} t.}$$

This interpretation allows the derivation of some properties of the logarithmic mean. Since the exponential function is monotonic, the integral over an interval of length 1 is bounded by x and 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$${\displaystyle {1 \over L(x,y)}=\int _{0}^{\infty }{\operatorname {d} \!t \over (t+x)\,(t+y)}.}$$

Generalization

Mean value theorem of differential calculus

One can generalize the mean to n + 1 variables by considering the mean value theorem for divided differences for the 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 n = 2 this leads to

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

Integral

The integral interpretation can also be generalized to more variables, but it leads to a different result. Given the simplex ${\textstyle S}$ with $${\displaystyle S=\{\left(\alpha _{0},\,\dots ,\,\alpha _{n}\right):\left(\alpha _{0}+\dots +\alpha _{n}=1\right)\land \left(\alpha _{0}\geq 0\right)\land \dots \land \left(\alpha _{n}\geq 0\right)\}}$$ 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 n = 2:

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

Connection to other means

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

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.
• Logarithmic mean temperature difference
• Log semiring