This is the current revision of this page, as edited by 180.248.10.190(talk) at 07:02, 22 August 2024(→Inequalities: Fixed typo. Actually though, this whole part is a mess. A random dude did this, cool, but without any mention of the mathematical importance of this inequality, what is the merit of even acknowledging this here? For what it's worth I could just replace everything with a "+n" variant and it would be just as valid. Meaningless, but valid. Should I add that here?). The present address (URL) is a permanent link to this version.
Revision as of 07:02, 22 August 2024 by 180.248.10.190(talk)(→Inequalities: Fixed typo. Actually though, this whole part is a mess. A random dude did this, cool, but without any mention of the mathematical importance of this inequality, what is the merit of even acknowledging this here? For what it's worth I could just replace everything with a "+n" variant and it would be just as valid. Meaningless, but valid. Should I add that here?)
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 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 .
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
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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.