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The Arithmetic-Logarithmic-Geometric Mean Inequality
The arithmetic-logarithmic-geometric mean inequality states that if
then
.
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
.
Contributed by:
Soledad Mª Sáez Martínez
and
Félix Martínez de la Rosa
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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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"
The Arithmetic-Logarithmic-Geometric Mean Inequality
" from
the Wolfram Demonstrations Project
http://demonstrations.wolfram.com/TheArithmeticLogarithmicGeometricMeanInequality/
Contributed by:
Soledad Mª Sáez Martínez
and
Félix Martínez de la Rosa
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