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
.

References

[1] R. B. Nelson, Proofs without Words II, More Exercises in Visual Thinking, Washington, DC: Mathematical Association of America, 2000.
[2] W. H. Young, "On Classes of Summable Functions and Their Fourier Series," Proceedings of the Royal Society A, 87, 1912 pp. 225–229.

Permanent Citation

Ken Levasseur
​
​"Young's Inequality"​
​http://demonstrations.wolfram.com/YoungsInequality/​
​Wolfram Demonstrations Project​
​Published: December 9, 2013