Young's Inequality — Definition, Formula & Examples
Young's inequality states that for positive real numbers and , and conjugate exponents and , the product is bounded above by a weighted sum of powers: . It provides a fundamental upper bound on products that appears throughout analysis.
Let and let satisfy . Then , with equality if and only if . This follows from the concavity of the logarithm (or equivalently, from the convexity of the exponential function).
Key Formula
Where:
- = Non-negative real numbers
- = Exponent with $p > 1$
- = Conjugate exponent satisfying $\frac{1}{p} + \frac{1}{q} = 1$
How It Works
Young's inequality converts a product into a sum, which is often easier to estimate. You identify the conjugate exponents and satisfying , then apply the bound directly. A common special case is , which gives , the AM-GM inequality for two terms. In proofs involving spaces, you typically set and to be strategically chosen functions or norms, then integrate both sides to derive Hölder's inequality.
Worked Example
Problem: Verify Young's inequality for , , , .
Check conjugate exponents: Confirm that .
Compute the left side: Calculate the product .
Compute the right side: Evaluate .
Answer: Both sides equal 27, confirming with equality. Equality holds because .
Why It Matters
Young's inequality is the key step in proving Hölder's inequality, which in turn underpins the Minkowski inequality and much of space theory. If you study real analysis, functional analysis, or PDE theory, you will encounter it repeatedly as a tool for decoupling products into manageable sums.
Common Mistakes
Mistake: Forgetting that and must be conjugate exponents () and using arbitrary exponents instead.
Correction: Always verify the conjugate condition first. Given , compute before applying the inequality.