The geometric, logarithmic, and arithmetic mean inequality
Chapter 7, Problem 60E(choose chapter or problem)
The geometric, logarithmic, and arithmetic mean inequality a. Show that the graph of ex is concave up over every interval of x-values. b. Show, by reference to the accompanying figure, that if 0 < a < b then c. Use the inequality in part (b) to conclude that This inequality says that the geometric mean of two positive numbers is less than their logarithmic mean, which in turn is less than their arithmetic mean. (For more about this inequality, see “The Geometric, Logarithmic, and Arithmetic Mean Inequality” by Frank Burk, American Mathematical Monthly, Vol. 94, No. 6, June–July 1987, pp. 527–528.)
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