# Introduction

I have seen Hölder’s inequality and Minkowski’s inequality proved in several ways but this seems the most perspicuous (to me at any rate).

# Young’s Inequality

If and such that

then

A and satisfying the premise are known as conjugate indices.

**Proof**

Since is convex we have

Substituting in appropriate values gives

or

Now take exponents.

# Hölders’s Inequality

Let and be conjugate indices with and let and then and

**Proof**

By Young’s inequality

By applying a counting measure to we also obtain

# Minkowski’s Inequality

**Proof**

By Hölder’s inequality

where

and is finite since is a vector space.