Logarithmic 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. This calculation is applicable in engineering problems involving heat and mass transfer.

Definition

The logarithmic mean is defined as:

$\begin{array}{cc}{M}_{\text{lm}}(x,y)& =\underset{(\xi ,\eta )\to (x,y)}{\lim }\frac{\eta -\xi }{\ln (\eta )-\ln (\xi )}\\ & =\{\begin{array}{cc}x& \text{if }x=y,\\ \frac{y-x}{\ln y-\ln x}& \text{otherwise,}\end{array}\end{array}$

for the positive numbers Template:Mvar.

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.

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

Now, for Template:Math: $$\begin{array}{ccccc}\sqrt{xy}{(\ln \sqrt{xy})}^{-1}\ln \sqrt{xy}& \le & \frac{x-y}{\ln x-\ln y}& \le & \frac{x(\ln x{)}^{-1}\ln x+y(\ln y{)}^{-1}\ln y}{2}\\ \sqrt{xy}& \le & \frac{x-y}{\ln x-\ln y}& \le & \frac{x+y}{2}\end{array}$$

This is the arithmetic logarithmic geometric mean inequality. Similarly, one can also obtain results by putting different values of Template:Mvar as below

For Template:Math: $$\sqrt{xy}(1+\ln \sqrt{xy})\le \frac{x\ln x-y\ln y}{\ln x-\ln y}\le \frac{x(1+\ln x)+y(1+\ln y)}{2}$$

for the proof go through the bibliography.

Derivation

Mean value theorem of differential calculus

From the mean value theorem, there exists a value Template:Mvar in the interval between Template:Mvar and Template:Mvar where the derivative Template:Mvar equals the slope of the secant line:

$\mathrm{\exists }\xi \in (x,y):f^{\prime }(\xi )=\frac{f(x)-f(y)}{x-y}$

The logarithmic mean is obtained as the value of Template:Mvar by substituting Template:Math for Template:Mvar and similarly for its corresponding derivative:

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

and solving for Template:Mvar:

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

Integration

The logarithmic mean can also be interpreted as the area under an exponential curve. $$\begin{array}{cc}L(x,y)=& \underset{0}{\overset{1}{\int }}{x}^{1-t}{y}^t\mathrm{d}t=\underset{0}{\overset{1}{\int }}{\left(\frac{y}{x}\right)}^{t}x\mathrm{d}t=x\underset{0}{\overset{1}{\int }}{\left(\frac{y}{x}\right)}^t\mathrm{d}t\\ =& {\frac{x}{\ln \frac{y}{x}}{\left(\frac{y}{x}\right)}^t|}_{t=0}^1=\frac{x}{\ln \frac{y}{x}}(\frac{y}{x}-1)=\frac{y-x}{\ln \frac{y}{x}}\\ =& \frac{y-x}{\ln y-\ln x}\end{array}$$

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 Template:Mvar and Template:Mvar. 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$$\frac{1}{L(x,y)}=\underset{0}{\overset{1}{\int }}\frac{\mathrm{d}t}{tx+(1-t)y}$$and$$\frac{1}{L(x,y)}=\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\mathrm{d}t}{(t+x)(t+y)}.$$

Generalization

Mean value theorem of differential calculus

One can generalize the mean to Template:Math variables by considering the mean value theorem for divided differences for the Template:Mvar-th derivative of the logarithm.

We obtain

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

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

For Template:Math this leads to

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

Integral

The integral interpretation can also be generalized to more variables, but it leads to a different result. Given the simplex $S$ with $$S=\{({\alpha }_0,\dots ,{\alpha }_n):({\alpha }_0+\dots +{\alpha }_n=1)\wedge ({\alpha }_0\ge 0)\wedge \dots \wedge ({\alpha }_n\ge 0)\}$$ and an appropriate measure $\mathrm{d}\alpha$ which assigns the simplex a volume of 1, we obtain

$L_{\text{I}}(x_0,\dots ,x_n)=\underset{S}{\int }{x}_0^{{\alpha }_0}\cdot \cdots \cdot x_n^{{\alpha }_n}\mathrm{d}\alpha$

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

$L_{\text{I}}(x_0,\dots ,x_n)=n!\exp [\ln \left(x_0\right),\dots ,\ln \left(x_n\right)]$.

Example Template:Math:

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

Connection to other means

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

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

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

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

References

Citations

Bibliography

• Oilfield Glossary: Term 'logarithmic mean'
• Template:Mathworld
• Template:Cite journal
• Toyesh Prakash Sharma.: https://www.parabola.unsw.edu.au/files/articles/2020-2029/volume-58-2022/issue-2/vol58_no2_3.pdf "A generalisation of the Arithmetic-Logarithmic-Geometric Mean Inequality, Parabola Magazine, Vol. 58, No. 2, 2022, pp 1–5