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The arithmetic-logarithmic-geometric mean inequality states that if

then

Left graphic:

The area under

on the interval

The area under the tangent at

Then

Right graphic:

The area under

on the interval

, as in the left graphic.

The area of the left trapezoid is

The area of the right trapezoid is

Then

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

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The Arithmetic-Logarithmic-Geometric Mean Inequality