proof of Minkowski inequality

For p=1 the result follows immediately from the triangle inequalityMathworldMathworldPlanetmathPlanetmath, so we may assume p>1.

We have


by the triangle inequality. Therefore we have


Set q=pp-1. Then 1p+1q=1, so by the Hölder inequalityMathworldPlanetmath we have


Adding these two inequalities, dividing by the factor common to the right sides of both, and observing that (p-1)q=p by definition, we have


Finally, observe that 1-1q=1p, and the result follows as required. The proof for the integral version is analogous.

Title proof of Minkowski inequality
Canonical name ProofOfMinkowskiInequality
Date of creation 2013-03-22 12:42:14
Last modified on 2013-03-22 12:42:14
Owner Andrea Ambrosio (7332)
Last modified by Andrea Ambrosio (7332)
Numerical id 10
Author Andrea Ambrosio (7332)
Entry type Proof
Classification msc 26D15
Related topic HolderInequality