Young's Inequality for Products
Contents
Theorem
Let $p, q \in \R_{> 0}$ be strictly positive real numbers such that:
- $\dfrac 1 p + \dfrac 1 q = 1$
Then, for any $a, b \in \R_{\ge 0}$:
- $a b \le \dfrac {a^p} p + \dfrac{b^q} q$
Equality occurs if and only if:
- $b = a^{p-1}$.
Proof 1
The result follows directly if $a = 0$ or $b = 0$.
Without loss of generality, assume that $a > 0$ and $b > 0$.
Then:
| \(\displaystyle ab\) | \(=\) | \(\displaystyle \exp \left({\ln\left({ab}\right)}\right)\) | Exponential of Natural Logarithm | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \exp \left({ \ln a + \ln b }\right)\) | Sum of Logarithms | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \exp \left({ \frac 1 p p \ln a + \frac 1 q q \ln b }\right)\) | Definitions of Multiplicative Identity and Multiplicative Inverse | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \exp \left({ \frac 1 p \ln \left( {a^p} \right) + \frac 1 q \ln \left( {b^q} \right) }\right)\) | Logarithms of Powers | ||||||||||
| \(\displaystyle \) | \(\le\) | \(\displaystyle \frac 1 p \exp \left( {\ln \left( {a^p} \right)} \right) + \frac 1 q \exp \left( {\ln \left( {b^q} \right)} \right)\) | Exponential is Strictly Convex and the hypothesis that $\dfrac 1 p + \dfrac 1 q = 1$ | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \frac{a^p} p + \frac{b^q} q\) | Exponential of Natural Logarithm |
$\blacksquare$
Proof 2
In order for $\dfrac 1 p + \dfrac 1 q = 1$ it is necessary for both $p > 1$ and $q > 1$.
| \(\displaystyle \frac 1 p + \frac 1 q\) | \(=\) | \(\displaystyle 1\) | |||||||||||
| \(\displaystyle \implies \ \ \) | \(\displaystyle p + q\) | \(=\) | \(\displaystyle p q\) | multiplying both sides by $p q$ | |||||||||
| \(\displaystyle \implies \ \ \) | \(\displaystyle p + q - p - q + 1\) | \(=\) | \(\displaystyle p q - p - q + 1\) | adding $1 - p - q$ to both sides | |||||||||
| \(\displaystyle \implies \ \ \) | \(\displaystyle 1\) | \(=\) | \(\displaystyle \left({p - 1}\right) \left({q - 1}\right)\) | elementary algebra | |||||||||
| \(\displaystyle \implies \ \ \) | \(\displaystyle \frac 1 {p - 1}\) | \(=\) | \(\displaystyle q - 1\) |
Accordingly:
- $u = t^{p-1} \iff t = u^{q-1}$
Let $a, b$ be any positive real numbers.
Since $a b$ is the area of the rectangle in the given figure, we have:
- $\displaystyle a b \le \int_0^a t^{p-1} \ \mathrm d t + \int_0^b u^{q-1} \ \mathrm d u = \frac {a^p} p + \frac {b^q} q$
Note that even if the graph intersected the side of the rectangle corresponding to $t = a$, this inequality would hold.
Also note that if either $a = 0$ or $b = 0$ then this inequality holds trivially.
$\blacksquare$
Source of Name
This entry was named for William Henry Young.
Sources
- 1953: Walter Rudin: Principles of Mathematical Analysis: Exercise $6.10 a$
