Young's Inequality
Theorem
Young's inequality states that for and with
Proof
The condition that gives motivation for using the definition of convex functions. The fact that guarantee that as required, and so we have that for any convex function and real numbers :
Then we choose , since the left hand side looks kind of like what we want, and by choosing as above and taking the exponential function, we can take and to be powers on the right and end up with a product.
Now all we must do is let and to get Young's inequality:
The above proof is somewhat more motivated than the one we include below, however the proof below has the benefit of depending on less machinery.
Proof
Consider the function defined by
where and and satisfy the conditions of the theorem statement.
We will show that this function is non-negative for all values of . Firstly, notice that the derivative given by
is zero at exactly noting that we are only considering positive values of .
Furthermore, we know that
so this point corresponds with , at which point
Furthermore, the function is convex everywhere it is defined since