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. The logarithmic mean is defined as: for the positive numbers x, y. 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. Toyesh Prakash Sharma generalizes the arithmetic logarithmic geometric mean inequality for any n belongs to the whole number as Now, for n = 0: This is the arithmetic logarithmic geometric mean inequality. similarly, one can also obtain results by putting different values of n as below For n = 1: for the proof go through the bibliography. From the mean value theorem, there exists a value ξ in the interval between x and y where the derivative f ′ equals the slope of the secant line: The logarithmic mean is obtained as the value of ξ by substituting ln for f and similarly for its corresponding derivative: and solving for χ: 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 x and y. The homogeneity of the integral operator is transferred to the mean operator, that is . Two other useful integral representations areand One can generalize the mean to n + 1 variables by considering the mean value theorem for divided differences for the n-th derivative of the logarithm. We obtain where denotes a divided difference of the logarithm. For n = 2 this leads to 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 v