Hölder and Minkowski Inequalities

Key idea

Key inequalities for Lp spaces.

In depth

Hölder: ∫|fg| ≤ ‖f‖ₚ‖g‖_q with 1/p + 1/q = 1 (Cauchy–Schwarz when p = q = 2). Minkowski: ‖f + g‖ₚ ≤ ‖f‖ₚ + ‖g‖ₚ, the triangle inequality for Lᵖ norms.

Worked example

With p = 2, Hölder gives ∫|fg| ≤ √(∫f² ∫g²).

Real-world uses

• Bounding products and sums of signals in communications and statistics

Common mistake

Using conjugate exponents that do not satisfy 1/p + 1/q = 1.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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