Logarithmic mean

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

Three-dimensional plot showing the values of the 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. In symbols:

\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}

for the positive numbers $x,y$ . This calculation is applicable in engineering problems involving 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):

\sqrt{x\cdot y} \leq M_{\text{lm}}(x,y) \leq \frac{x+y}{2} \qquad \text{ for all } x\geq 0 \text{ and } y\geq 0.

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Derivation of the mean

Mean value theorem of differential calculus

From the mean value theorem

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

the logarithmic mean is obtained as the value of $\xi$ by substituting $\ln$ for $f$

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

and solving for $\xi$ .

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

Integration

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

L(x,y) = \int_0^1 x^{1-t} y^t\ \mathrm{d}t

$\begin{array}{rcl} \int_0^1 x^{1-t} y^t\ \mathrm{d}t &=& \int_0^1 \left(\frac{y}{x}\right)^t x\ \mathrm{d}t \\ &=& x \int_0^1 \left(\frac{y}{x}\right)^t \mathrm{d}t \\ &=& \frac{x}{\ln \frac{y}{x}} \left(\frac{y}{x}\right)^t|_{t=0}^{1}\\ &=& \frac{x}{\ln \frac{y}{x}} \left(\frac{y}{x}-1\right)\\ &=& \frac{y-x}{\ln y - \ln x} \end{array}$

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 $x$ and $y$ . The Homogeneity of the integral operator is transferred to the mean operator, that is $L(c\cdot x, c\cdot y) = c\cdot L(x,y)$ .

Generalization

Mean value theorem of differential calculus

You 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. You obtain

L_{\mathrm{MV}}(x_0,\dots,x_n) = \sqrt[-n]{(-1)^{(n+1)}\cdot n \cdot \ln[x_0,\dots,x_n]}

where $\ln[x_0,\dots,x_n]$ denotes a divided difference of the logarithm.

For $n=2$ this leads to

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)}}

Integral

The integral interpretation can also be generalized to more variables, but it leads to a different result. Given the simplex $S$ with $S = \{(\alpha_0,\dots,\alpha_n) : \alpha_0+\dots+\alpha_n=1\ \land\ \alpha_0\ge0\ \land\ \dots\ \land\ \alpha_n\ge0\}$ and an appropriate measure $\mathrm{d}\alpha$ which assigns the simplex a volume of 1, we obtain

L_{\mathrm{I}}(x_0,\dots,x_n) = \int_S x_0^{\alpha_0}\cdot\dots\cdot x_n^{\alpha_n}\ \mathrm{d}\alpha

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

L_{\mathrm{I}}(x_0,\dots,x_n) = n!\cdot\exp[\ln x_0, \dots, \ln x_n]

Example $n=2$

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)}

Connection to other means

$\frac{L(x^2,y^2)}{L(x,y)} = \frac{x+y}{2}$ (Arithmetic mean)

References

↑ B. C. Carlson (1966). "Some inequalities for hypergeometric functions". Proc. Amer. Math. Soc. 17: pp. 32–39. doi:10.1090/s0002-9939-1966-0188497-6. CS1 maint: Extra text (link)
↑ B. Ostle and H. L. Terwilliger (1957). "A comparison of two means". Proc. Montana Acad. Sci. 17: pp. 69–70.

Stolarsky, Kenneth B.: Generalizations of the logarithmic mean, Mathematics Magazine, Vol. 48, No. 2, Mar., 1975, pp 87–92

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Logarithmic mean

Inequalities

Derivation of the mean

Mean value theorem of differential calculus

Integration

Generalization

Mean value theorem of differential calculus

Integral

Connection to other means

See also

References