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The Arithmetic-Logarithmic-Geometric Mean Inequality

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The arithmetic-logarithmic-geometric mean inequality states that if then .

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Left graphic:

The area under on the interval is .

The area under the tangent at is .

Then .

Right graphic:

The area under on the interval is , as in the left graphic.

The area of the left trapezoid is .

The area of the right trapezoid is .

Then .

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Contributed by: Soledad Mª Sáez Martínez and Félix Martínez de la Rosa (March 2011)
Open content licensed under CC BY-NC-SA

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Reference: R. B. Nelsen, "Proof without Words: The Arithmetic-Logarithmic-Geometric Mean Inequality," Mathematics Magazine 68(4), 1995 p. 305.

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