Young's Inequality

• categories: Real Analysis, Theorem

Young’s Inequality

Young’s inequality provides a bound for the product of two non-negative numbers in terms of their powers, using conjugate exponents. It is a useful result in analysis, particularly in the context of Hölder’s Inequality and convex functions.

Statement (Real Numbers)

For any $a,b\ge 0$ and $p,q>1$ such that $\frac{1}{p}+\frac{1}{q}=1$, we have:

$$ab\le \frac{a^p}{p}+\frac{b^q}{q}.$$

Equality holds if and only if $a^p=b^q$.

Proof Sketch

1. Convex Function: Consider the function $f(x)=x^p/p$ for $x\ge 0$, which is convex since its second derivative is non-negative.
2. Tangent Line Argument: By the convexity of $f$, for any $b\ge 0$ we have: $$f(b)\ge f(a)+f^{\prime }(a)(b-a).$$ Setting up this inequality with the appropriate values of $a$ and $b$ yields the desired result.