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
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