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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.
[1][2][3][4]
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
The logarithmic mean is obtained as the value of ξ by substituting ln for f and similarly for its corresponding derivative:
and solving for ξ:
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 x and y. The homogeneity of the integral operator is transferred to the mean operator, that is .
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B. Ostle & H. L. Terwilliger (1957). "A comparison of two means". Proc. Montana Acad. Sci. 17: 69–70.
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Tung-Po Lin (1974). "The Power Mean and the Logarithmic Mean". The American Mathematical Monthly. 81 (8): 879–883. doi:10.1080/00029890.1974.11993684.
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Frank Burk (1987). "The Geometric, Logarithmic, and Arithmetic Mean Inequality". The American Mathematical Monthly. 94 (6): 527–528. doi:10.2307/2322844. JSTOR2322844.