Logarithmic mean
In mathematics, the logarithmic means, that the logarithmic mean, a certain average value of the logarithm used.
The logarithmic means of two different positive real numbers is given by
In order to record the event to define a general
Then is.
The logarithmic means is a strictly increasing function. Furthermore, the logarithmic mean between the arithmetic and geometric means is:
This equation is valid for; Equality if and only if
The logarithmic mean value found in various sciences and technical problems using. It usually occurs when averaged over the driving gradient. This is for example the integral consideration of heat or mass transport processes is the case, for example in the process design of heat exchangers or separation columns.
- 2.1 Several Variables
- 2.2 Other Means
Analysis
Mean value theorem
By the mean value theorem of differential calculus there is a differentiable function with a
For one obtains
Which in this case, therefore, the logarithmic mean of and.
Integration
In addition, we obtain for the integration
Generalizations
Several variables
The generalizations of the logarithmic agent on more than two variables is less often used and is non-uniform.
Generalizing the idea of the mean value theorem is about
The divided differences of the logarithm, respectively.
For, that is, for three variables, the result is
Generalizing the integral to
Would obtain with
And as a special case for three variables
Another idea is
Other averages
The Stolarsky agent about the generalized logarithmic means.
Swell
- Horst Alzer: inequalities for mean values. Archives of Mathematics, Vol 47, No. 5 / Nov. 1986. Springer link- PDF
- 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, 80th