### Young inequality

Let be positive real numbers. Assume further that The following inequality is called the Young inequality.

Since both are nonzero, we can divide both side of the above inequality by and obtain the following equivalent form:

Let us define a new variable Using the relation we can see that Thus the inequality can be rewritten as

Let us define a function on by

If we can show that for any then the Young inequality holds. This fact can be proved by differential calculus. Solving , we can see that is the only critical point of the function Moreover, if and if . We can also show that for any We find that is the global minimum of the function if you analyze the slop and the convexity of the function by The following picture is the graph of the function in the case when

Thus we conclude that for any 。We coomplete the proof of the Young inequality

### Application in Lebesgue integral theory

Let be a measure space and be the (complex-valued) integrable functions on Define the -norm of by

Using the Young inequality, we can show that

** Proposition**: (Holder inequality)Let and If then

Proof:Dividing the inequality by we have

where We have and Using the Young inequality, we obtain

Integrating both side of the inequality, we get

**Cor**: (Discrete Holder inequality) Assume that are complex numbers. Then