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 by
for, such that.

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.
More precisely, for with and, we have
where the expressions in the chain of inequalities are, in order: the harmonic mean, the geometric mean, the logarithmic mean, the arithmetic mean, and the generalized arithmetic mean with exponent.

Derivation

Mean value theorem of differential calculus

From the mean value theorem, there exists a value in the interval between and 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 is also given by the integral
This interpretation allows the derivation of some properties of the logarithmic mean. Since the exponential function is monotonic, the integral over an interval of length 1 is bounded by and.
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 expressed as the divided differences of the exponential function by
In the case of, it is

Connection to other means

Some other means can be expressed in terms of the logarithmic mean.

Name	Mean	Expression
Arithmetic mean
Geometric mean
Harmonic mean