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Definition

The logarithmic mean is defined as:

M lm ( x , y ) = lim ( ξ , η ) → ( x , y ) η − ξ ln ⁡ ( η ) − ln ⁡ ( ξ ) = { x if  x = y , y − x ln ⁡ y − ln ⁡ x otherwise, {\displaystyle {\begin{aligned}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{aligned}}}

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. ¹²³⁴ More precisely, for x , y ∈ R {\displaystyle x,y\in \mathbb {R} } with x ≠ y {\displaystyle x\neq y} , we have 2 x y x + y ≤ x y ≤ x − y ln ⁡ x − ln ⁡ y ≤ x + y 2 ≤ ( x 2 + y 2 2 ) 1 / 2 . {\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}.} Sharma⁵ showed that, for any whole number n {\displaystyle n} and x , y ∈ R {\displaystyle x,y\in \mathbb {R} } with x ≠ y {\displaystyle x\neq y} , we have x y   ( ln ⁡ x y ) n − 1 ( n + ln ⁡ x y ) ≤ x ( ln ⁡ x ) n − y ( ln ⁡ y ) n ln ⁡ x − ln ⁡ y ≤ x ( ln ⁡ x ) n − 1 ( n + ln ⁡ x ) + y ( ln ⁡ y ) n − 1 ( n + ln ⁡ y ) 2 . {\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 n = 0 {\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:

∃ ξ ∈ ( x , y ) :   f ′ ( ξ ) = f ( x ) − f ( y ) x − y {\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:

1 ξ = ln ⁡ x − ln ⁡ y x − y {\displaystyle {\frac {1}{\xi }}={\frac {\ln x-\ln y}{x-y}}}

and solving for ξ:

ξ = x − y ln ⁡ x − ln ⁡ y {\displaystyle \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 ) = ∫ 0 1 x 1 − t y t   d t = ∫ 0 1 ( y x ) t x   d t = x ∫ 0 1 ( y x ) t d t = x ln ⁡ y x ( y x ) t | t = 0 1 = x ln ⁡ y x ( y x − 1 ) = y − x ln ⁡ y x = y − x ln ⁡ y − ln ⁡ x {\displaystyle {\begin{aligned}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{aligned}}}

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 x , c y ) = c L ( x , y ) {\displaystyle L(cx,cy)=cL(x,y)} .

Two other useful integral representations are 1 L ( x , y ) = ∫ 0 1 d t t x + ( 1 − t ) y {\displaystyle {1 \over L(x,y)}=\int _{0}^{1}{\operatorname {d} \!t \over tx+(1-t)y}} and 1 L ( x , y ) = ∫ 0 ∞ d t ( t + x ) ( t + y ) . {\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

L MV ( x 0 , … , x n ) = ( − 1 ) n + 1 n ln ⁡ ( [ x 0 , … , x n ] ) − n {\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 ln ⁡ ( [ x 0 , … , x n ] ) {\displaystyle \ln \left(\left[x_{0},\,\dots ,\,x_{n}\right]\right)} denotes a divided difference of the logarithm.

For n = 2 this leads to

L MV ( x , y , z ) = ( x − y ) ( y − z ) ( z − x ) 2 ( ( y − z ) ln ⁡ x + ( z − x ) ln ⁡ y + ( x − y ) ln ⁡ z ) . {\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 S {\textstyle S} with S = { ( α 0 , … , α n ) : ( α 0 + ⋯ + α n = 1 ) ∧ ( α 0 ≥ 0 ) ∧ ⋯ ∧ ( α n ≥ 0 ) } {\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 d α {\textstyle \mathrm {d} \alpha } which assigns the simplex a volume of 1, we obtain

L I ( x 0 , … , x n ) = ∫ S x 0 α 0 ⋅ ⋯ ⋅ x n α n   d α {\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

L I ( x 0 , … , x n ) = n ! exp ⁡ [ ln ⁡ ( x 0 ) , … , ln ⁡ ( x n ) ] {\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:

L I ( x , y , z ) = − 2 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 ) . {\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: L ( x 2 , y 2 ) L ( x , y ) = x + y 2 {\displaystyle {\frac {L\left(x^{2},y^{2}\right)}{L(x,y)}}={\frac {x+y}{2}}}

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

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

Citations Bibliography

• Oilfield Glossary: Term 'logarithmic mean'
• Weisstein, Eric W. "Arithmetic-Logarithmic-Geometric-Mean Inequality". MathWorld.
• Stolarsky, Kenneth B. (1975). "Generalizations of the Logarithmic Mean". Mathematics Magazine. 48 (2): 87–92. doi:10.2307/2689825. ISSN 0025-570X. JSTOR 2689825.

References

1. B. C. Carlson (1966). "Some inequalities for hypergeometric functions". Proc. Amer. Math. Soc. 17: 32–39. doi:10.1090/s0002-9939-1966-0188497-6. https://doi.org/10.1090%2Fs0002-9939-1966-0188497-6 ↩

2. B. Ostle & H. L. Terwilliger (1957). "A comparison of two means". Proc. Montana Acad. Sci. 17: 69–70. ↩

3. Tung-Po Lin (1974). "The Power Mean and the Logarithmic Mean". The American Mathematical Monthly. 81 (8): 879–883. doi:10.1080/00029890.1974.11993684. /wiki/Doi_(identifier) ↩

4. Frank Burk (1987). "The Geometric, Logarithmic, and Arithmetic Mean Inequality". The American Mathematical Monthly. 94 (6): 527–528. doi:10.2307/2322844. JSTOR 2322844. /wiki/Doi_(identifier) ↩

5. T. P. Sharma (2022). "A generalisation of the Arithmetic-Logarithmic-Geometric Mean Inequality". Parabola Magazine. 58 (2): 25–29. doi:10.1090/s0002-9939-1966-0188497-6. https://doi.org/10.1090%2Fs0002-9939-1966-0188497-6 ↩