Logarithmic mean

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Three-dimensional plot showing the values of the 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[edit]

The logarithmic mean is defined symbolically as:

for the positive numbers .

Inequalities[edit]

The logarithmic mean of two numbers is smaller than the arithmetic mean but larger than the geometric mean (unless the numbers are the same, in which case all three means are equal to the numbers):

[1][2]

Derivation[edit]

Mean value theorem of differential calculus[edit]

From the mean value theorem, there exists a value in the interval between x and y where the derivative equals 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[edit]

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 are

and

Generalization[edit]

Mean value theorem of differential calculus[edit]

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[edit]

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[edit]

  • Arithmetic mean:
  • Geometric mean:
  • Harmonic mean:

See also[edit]

References[edit]

  1. ^ B. C. Carlson (1966). "Some inequalities for hypergeometric functions". Proc. Amer. Math. Soc. 17: 32–39. doi:10.1090/s0002-9939-1966-0188497-6.
  2. ^ B. Ostle & H. L. Terwilliger (1957). "A comparison of two means". Proc. Montana Acad. Sci. 17: 69–70.