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Young's Inequality

​
a
5
b
4
p
2.4
Assume a, b, p, and q are positive, and that
1
p
+
1
q
1.
red area =
a
∫
0
p-1
x
x
p
a
p
orange area =
b
∫
0
q-1
y
y
q
b
q
area of blue rectangle = ab
Young's inequality: ab≤
p
a
p
+
q
b
q
Given positive numbers
p
and
q
whose reciprocals add up to one, the product
ab
of two positive real numbers
a
and
b
is less than or equal to a weighted sum of
th
p
and
th
q
powers of
a
and
b
.
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