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The Arithmetic-Logarithmic-Geometric Mean Inequality
The Arithmetic-Logarithmic-Geometric Mean Inequality
The arithmetic-logarithmic-geometric mean inequality states that if then < .
0<a<b,
ab
<b-a
lnb-lna
a+b
2
Left graphic:
The area under on the interval is .
y=
1
x
(a,b)
lnb-lna
The area under the tangent at is (b-a).
x=
a+b
2
2
(a+b)
Then .
lnb-lna>(b-a)
2
(a+b)
Right graphic:
The area under on the interval is , as in the left graphic.
y=
1
x
(a,b)
lnb-lna
The area of the left trapezoid is +(.
1
2
1
a
1
ab
ab
-a)=b-a
2
ab
The area of the right trapezoid is +(b-.
1
2
1
b
1
ab
ab
)=b-a
2
ab
Then .
lnb-lna<
b-a
ab