WOLFRAM|DEMONSTRATIONS PROJECT

The Arithmetic-Logarithmic-Geometric Mean Inequality

​
a
1
b
6
The arithmetic-logarithmic-geometric mean inequality states that if
0<a<b,
then
ab
<
b-a
lnb-lna
<
a+b
2
.
Left graphic:
The area under
y=
1
x
on the interval
(a,b)
is
lnb-lna
.
The area under the tangent at
x=
a+b
2
is
2
(a+b)
(b-a)
.
Then
lnb-lna>
2
(a+b)
(b-a)
.
Right graphic:
The area under
y=
1
x
on the interval
(a,b)
is
lnb-lna
, as in the left graphic.
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
1
2
1
b
+
1
ab
(b-
ab
)=
b-a
2
ab
.
Then
lnb-lna<
b-a
ab
.