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.

Definition
The logarithmic mean is defined as:


 * $$\begin{align}

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 ,\\[2pt] \dfrac{y - x}{\ln y - \ln x} & \text{otherwise,} \end{cases} \end{align}$$

for the positive numbers $x, y$.

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.

$$ \frac{2}{\displaystyle \frac1{x} + \frac1{y}} \leq \sqrt{x y} \leq \frac{x - y}{\ln x - \ln y} \leq \frac{x + y}{2} \leq \left(\frac{x^2+y^2}2\right)^{1/2} \qquad \text{ for all } x > 0 \text{ and } y > 0.$$ Toyesh Prakash Sharma generalizes the arithmetic logarithmic geometric mean inequality for any $n$ belongs to the whole number as $$\sqrt{xy}\ \left( \ln \sqrt{xy} \right)^{n-1} \left(n + \ln \sqrt{xy}\right) \leq \frac{x(\ln x)^n - y(\ln y)^n}{\ln x - \ln y} \leq \frac{x(\ln x)^{n-1} (n + \ln x) + y(\ln y)^{n-1} (n + \ln y)} {2}$$

Now, for $n = 0$: $$\begin{array}{ccccc} \sqrt{xy} \left(\ln \sqrt{xy} \right)^{-1} \ln\sqrt{xy} & \leq & \dfrac{x-y}{\ln x - \ln y} & \leq & \dfrac{x(\ln x)^{-1} \ln x + y(\ln y)^{-1} \ln y}{2} \\[4pt] \sqrt{xy} & \leq & \dfrac{x-y}{\ln x - \ln y} & \leq & \dfrac{x + y}{2} \end{array}$$

This is the arithmetic logarithmic geometric mean inequality. similarly, one can also obtain results by putting different values of $n$ as below

For $n = 1$: $$\sqrt{xy} \left(1 + \ln \sqrt{xy} \right) \leq \frac{x\ln x - y\ln y}{\ln x - \ln y} \leq \frac{x (1 + \ln x) + y(1 + \ln y)}{2}$$

for the proof go through the bibliography.

Mean value theorem of differential calculus
From the mean value theorem, there exists a value $&xi;$ in the interval between $x$ and $y$ where the derivative $f&thinsp;&prime;$ equals the slope of the secant line:
 * $$\exists \xi \in (x, y): \ f'(\xi) = \frac{f(x) - f(y)}{x - y}$$

The logarithmic mean is obtained as the value of $&xi;$ by substituting $ln$ for $f$ and similarly for its corresponding derivative:
 * $$\frac{1}{\xi} = \frac{\ln x - \ln y}{x-y}$$

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. $$\begin{align} L(x, y) ={} & \int_0^1 x^{1-t} y^t\ \mathrm{d}t ={}  \int_0^1 \left(\frac{y}{x}\right)^t x\ \mathrm{d}t ={}  x \int_0^1 \left(\frac y x\right)^t \mathrm{d}t \\[3pt] ={} & \left.\frac{x}{\ln\frac y x} \left(\frac{y}{x}\right)^t\right|_{t=0}^1 ={}  \frac{x}{\ln\frac y x} \left(\frac{y}{x} - 1\right) ={}  \frac{y - x}{\ln\frac y x} \\[3pt] ={} & \frac{y - x}{\ln y - \ln x} \end{align}$$

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 $$L(cx, cy) = cL(x, y)$$.

Two other useful integral representations are$${1 \over L(x,y)} = \int_0^1 {\operatorname{d}\!t \over t x + (1-t)y}$$and$${1 \over L(x,y)} = \int_0^\infty {\operatorname{d}\!t \over (t+x)\,(t+y)}.$$

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
 * $$L_\text{MV}(x_0,\, \dots,\, x_n) = \sqrt[-n]{(-1)^{n+1} n \ln\left(\left[x_0,\, \dots,\, x_n\right]\right)}$$

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

For $n = 2$ this leads to
 * $$L_\text{MV}(x, y, z) = \sqrt{\frac{(x-y)(y-z)(z-x)}{2 \bigl((y-z) \ln x + (z-x) \ln y + (x-y) \ln z \bigr)}}.$$

Integral
The integral interpretation can also be generalized to more variables, but it leads to a different result. Given the simplex $S$ with $$S = \{\left(\alpha_0,\, \dots,\, \alpha_n\right) : \left(\alpha_0 + \dots + \alpha_n = 1\right) \land \left(\alpha_0 \ge 0\right) \land \dots \land \left(\alpha_n \ge 0\right)\}$$ and an appropriate measure $\mathrm{d}\alpha$  which assigns the simplex a volume of 1, we obtain
 * $$L_\text{I}\left(x_0,\, \dots,\, x_n\right) = \int_S x_0^{\alpha_0} \cdot \,\cdots\, \cdot x_n^{\alpha_n}\ \mathrm{d}\alpha$$

This can be simplified using divided differences of the exponential function to
 * $$L_\text{I}\left(x_0,\, \dots,\, x_n\right) = n! \exp\left[\ln\left(x_0\right),\, \dots,\, \ln\left(x_n\right)\right]$$.

Example $n = 2$:
 * $$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: $$\frac{L\left(x^2, y^2\right)}{L(x, y)} = \frac{x + y}{2}$$


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


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