Logarithmic mean

The logarithmic mean is defined by

:$$

M_\text{lm}(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 $x,\; y\; \backslash in\; \backslash mathbb\{R\}$.

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.

{{cite journal

| author=B. C. Carlson

| title=Some inequalities for hypergeometric functions

| journal=Proc. Amer. Math. Soc.

| volume=17 | year=1966 | pages=32–39

| doi=10.1090/s0002-9939-1966-0188497-6

| doi-access=free

}}

{{cite journal

| author1=B. Ostle | author2=H. L. Terwilliger | name-list-style=amp

| title=A comparison of two means

| journal=Proc. Montana Acad. Sci.

| volume=17 | year=1957 | pages=69–70

}}

{{cite journal

| author1=Tung-Po Lin

| title=The Power Mean and the Logarithmic Mean

| journal=The American Mathematical Monthly

| year=1974

| volume=81| issue=8

| pages=879–883

| doi=10.1080/00029890.1974.11993684

}}

{{cite journal

| author1=Frank Burk

| title=The Geometric, Logarithmic, and Arithmetic Mean Inequality

| journal=The American Mathematical Monthly

| year=1987

| volume=94| issue=6

| pages=527–528

| doi=10.2307/2322844

| jstor=2322844

}}

More precisely, for $x,\; y\; \backslash in\; \backslash mathbb\{R\}$ with $x\; \backslash neq\; y$, we have

\frac{2xy}{x + 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}.

Sharma{{cite journal

| author=T. P. Sharma

| title=A generalisation of the Arithmetic-Logarithmic-Geometric Mean Inequality

| journal=Parabola Magazine

| volume=58 | issue=2 | year=2022 | pages=25–29

| doi=10.1090/s0002-9939-1966-0188497-6

| doi-access=free

}} showed that, for any whole number $n$ and $x,\; y\; \backslash in\; \backslash mathbb\{R\}$ with $x\; \backslash neq\; y$, we have

$$\backslash sqrt\{xy\}\backslash \; \backslash left(\; \backslash ln\; \backslash sqrt\{xy\}\; \backslash right)^\{n-1\}\; \backslash left(n\; +\; \backslash ln\; \backslash sqrt\{xy\}\backslash right)$$

\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 $n\; =\; 0$.

From the mean value theorem, there exists a value {{mvar|ξ}} in the interval between {{mvar|x}} and {{mvar|y}} where the derivative {{mvar|f ′}} equals the slope of the secant line:

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

The logarithmic mean is obtained as the value of {{mvar|ξ}} by substituting {{math|ln}} for {{mvar|f}} and similarly for its corresponding derivative:

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

and solving for {{mvar|ξ}}:

:$\backslash xi\; =\; \backslash frac\{x-y\}\{\backslash ln\; x\; -\; \backslash ln\; y\}$

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

$$\backslash begin\{align\}$$

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 {{mvar|x}} and {{mvar|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$$\{1\; \backslash over\; L(x,y)\}\; =\; \backslash int\_0^1\; \{\backslash operatorname\{d\}\backslash !t\; \backslash over\; t\; x\; +\; (1-t)y\}$$and$$\{1\; \backslash over\; L(x,y)\}\; =\; \backslash int\_0^\backslash infty\; \{\backslash operatorname\{d\}\backslash !t\; \backslash over\; (t+x)\backslash ,(t+y)\}.$$

One can generalize the mean to {{math|n + 1}} variables by considering the mean value theorem for divided differences for the {{mvar|n}}-th derivative of the logarithm.

We obtain

:$L\_\backslash text\{MV\}(x\_0,\backslash ,\; \backslash dots,\backslash ,\; x\_n)\; =\; \backslash sqrt[-n]\{(-1)^\{n+1\}\; n\; \backslash ln\backslash left(\backslash left[x\_0,\backslash ,\; \backslash dots,\backslash ,\; x\_n\backslash right]\backslash right)\}$

where $\backslash ln\backslash left(\backslash left[x\_0,\backslash ,\; \backslash dots,\backslash ,\; x\_n\backslash right]\backslash right)$ denotes a divided difference of the logarithm.

For {{math|1=n = 2}} this leads to

:$L\_\backslash text\{MV\}(x,\; y,\; z)\; =\; \backslash sqrt\{\backslash frac\{(x-y)(y-z)(z-x)\}\{2\; \backslash bigl((y-z)\; \backslash ln\; x\; +\; (z-x)\; \backslash ln\; y\; +\; (x-y)\; \backslash ln\; z\; \backslash bigr)\}\}.$

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

:$L\_\backslash text\{I\}\backslash left(x\_0,\backslash ,\; \backslash dots,\backslash ,\; x\_n\backslash right)\; =\; \backslash int\_S\; x\_0^\{\backslash alpha\_0\}\; \backslash cdot\; \backslash ,\backslash cdots\backslash ,\; \backslash cdot\; x\_n^\{\backslash alpha\_n\}\backslash \; \backslash mathrm\{d\}\backslash alpha$

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

:$L\_\backslash text\{I\}\backslash left(x\_0,\backslash ,\; \backslash dots,\backslash ,\; x\_n\backslash right)\; =\; n!\; \backslash exp\backslash left[\backslash ln\backslash left(x\_0\backslash right),\backslash ,\; \backslash dots,\backslash ,\; \backslash ln\backslash left(x\_n\backslash right)\backslash right]$.

Example {{math|1=n = 2}}:

:$L\_\backslash text\{I\}(x,\; y,\; z)\; =\; -2\; \backslash frac\{x(\backslash ln\; y\; -\; \backslash ln\; z)\; +\; y(\backslash ln\; z\; -\; \backslash ln\; x)\; +\; z(\backslash ln\; x\; -\; \backslash ln\; y)\}$

{(\ln x - \ln y)(\ln y - \ln z)(\ln z - \ln x)}.

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

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

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

• 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

;Citations

{{Reflist}}

;Bibliography

• [https://web.archive.org/web/20060215011645/http://jipam-old.vu.edu.au/v4n4/088_03.html Oilfield Glossary: Term 'logarithmic mean']
• {{mathworld|Arithmetic-Logarithmic-GeometricMeanInequality|Arithmetic-Logarithmic-Geometric-Mean Inequality}}
• {{Cite journal |last=Stolarsky |first=Kenneth B. |date=1975 |title=Generalizations of the Logarithmic Mean |url=https://www.jstor.org/stable/2689825 |journal=Mathematics Magazine |volume=48 |issue=2 |pages=87–92 |doi=10.2307/2689825 |jstor=2689825 |issn=0025-570X|url-access=subscription }}

Mean

Category:Means