Logarithmic mean


In mathematics, the logarithmic mean is a function of two non-negative numbers which is equal to their difference divided by the logarithm of their quotient.
This calculation is applicable in engineering problems involving heat and mass transfer.

Definition

The logarithmic mean is defined as:
for the positive numbers.

Inequalities

The logarithmic mean of two numbers is smaller than the arithmetic mean and the generalized mean with exponent one third but larger than the geometric mean, unless the numbers are the same, in which case all three means are equal to the numbers.

Derivation

Mean value theorem of differential calculus

From the mean value theorem, there exists a value in the interval between x and y where the derivative equals the slope of the secant line:
The logarithmic mean is obtained as the value of by substituting for and similarly for its corresponding derivative:
and solving for :

Integration

The logarithmic mean can also be interpreted as the area under an exponential curve.
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 and. The homogeneity of the integral operator is transferred to the mean operator, that is.
Two other useful integral representations areand

Generalization

Mean value theorem of differential calculus

One can generalize the mean to variables by considering the mean value theorem for divided differences for the th derivative of the logarithm.
We obtain
where denotes a divided difference of the logarithm.
For this leads to

Integral

The integral interpretation can also be generalized to more variables, but it leads to a different result. Given the simplex with and an appropriate measure which assigns the simplex a volume of 1, we obtain
This can be simplified using divided differences of the exponential function to
Example

Connection to other means