Logarithmic mean

In mathematics , the logarithmic mean , that is, the logarithmic mean , is a certain mean value that uses the logarithmic function.

The logarithmic mean of two different positive real numbers is given by ${\displaystyle M_{\text{lm}}}$ ${\displaystyle x,y}$

${\displaystyle M_{\text{lm}}(x,y)={\frac {y-x}{\ln {\frac {y}{x}}}}={\frac {y-x}{\ln y-\ln x}}}$

In order to also cover the case, one defines more generally ${\displaystyle x=y}$

${\displaystyle M_{\text{lm}}(x,y)=\lim _{(\xi ,\eta )\to (x,y)}{\frac {\eta -\xi }{\ln \eta -\ln \xi }}}$

Then is . ${\displaystyle M_{\text{lm}}(x,x)=x}$

The logarithmic mean is a strictly monotonically increasing function . Furthermore, the logarithmic mean lies between the arithmetic and geometric mean:

${\displaystyle {\sqrt {x\cdot y}}\leq {\frac {y-x}{\ln y-\ln x}}\leq {\frac {x+y}{2}}}$

This equation applies to ; Equality if and only if ${\displaystyle x,y>0}$ ${\displaystyle x=y.}$

The logarithmic mean is used in various sciences and technical problems. It usually occurs when averaging over driving slopes. This is the case, for example, with the integral consideration of heat or mass transport processes, for example with the procedural design of heat exchangers or separation columns .

Analysis

Mean theorem

According to the mean value theorem of differential calculus, there is a with for a differentiable function ${\displaystyle f\colon [x,y]\rightarrow \mathbb {R} }$ ${\displaystyle \xi \in [x,y]}$

${\displaystyle f'(\xi )={\frac {f(x)-f(y)}{x-y}}.}$

For one gets from it ${\displaystyle f(x)=\ln \,x}$

${\displaystyle {\frac {1}{\xi }}={\frac {\ln x-\ln y}{x-y}}\,\,}$, so . ${\displaystyle \xi ={\frac {x-y}{\ln x-\ln y}}}$

That is so in this case the logarithmic mean of and . ${\displaystyle \xi }$ ${\displaystyle x}$ ${\displaystyle y}$

integration

You also get for integration

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

Generalizations

Multiple variables

The generalizations of the log mean to more than two variables are used less often and are inconsistent.

If one generalizes the idea of ​​the mean theorem is roughly

${\displaystyle L_{\mathrm {MV} }(x_{0},\dots ,x_{n})={\sqrt[{-n}]{(-1)^{(n+1)}\cdot n\cdot \ln[x_{0},\dots ,x_{n}]}}}$

where denotes the divided differences of the logarithm. ${\displaystyle \ln[x_{0},\dots ,x_{n}]}$

For , that is for three variables, this leads to ${\displaystyle n=2}$

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

One generalizes the integral to

${\displaystyle L_{\mathrm {I} }(x_{0},\dots ,x_{n})=\int _{S}x_{0}^{\alpha _{0}}\cdot \dots \cdot x_{n}^{\alpha _{n}}\ \mathrm {d} \alpha }$

with you would get ${\displaystyle S=\{(\alpha _{0},\dots ,\alpha _{n})|\alpha _{0}+\dots +\alpha _{n}=1\ \land \ \alpha _{0}\geq 0\ \land \ \dots \ \land \ \alpha _{n}\geq 0\}}$

${\displaystyle L_{\mathrm {I} }(x_{0},\dots ,x_{n})=n!\cdot \exp[\ln x_{0},\dots ,\ln x_{n}]}$

and as a special case for three variables

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

Other means

The Stolarsky mean, for example, generalizes the logarithmic mean.

swell

• Horst Alzer: Inequalities for mean values . Archive of Mathematics, Vol 47, No. 5 / Nov. 1986, doi: 10.1007 / BF01189983 .
• AO Pittenger: The logarithmic mean in n variables . In: American Mathematical Monthly, 92 (1985), pp. 99-104.
• Gao Jia, Jinde Cao: A New Upper Bound of the Logarithmic Mean . Journal of Inequalities in Pure and Applied Mathematics 4, 4, 2003, 80.

Individual evidence

1. ↑ Eric W. Weisstein: Arithmetic-Logarithmic-Geometric-Mean-Inequality and Napier's Inequality in MathWorld