This question is in regard to an answer here: On the equality case of the Hölder and Minkowski inequalites The writer has written in his profile that he is on leave, so there is not much point in me asking a comment for him.

My question is: why is it that we have in the last line:

$\int f g =1/p \int f^p +1/q \int g^q \leftarrow \rightarrow f g=1/p \cdot f^p+1/q \cdot g^q$ ae

I get the left implication, but why do we have the right?

PS: I do not mean to say that the poster is wrong, I am sure it is correct, but I just want to understand why.

The reason I am asking is that I am having problems proving the equality case in Holder, it is iff $\alpha |f|^p=\beta |g|^q$ a.e. I can do the if, but not the only if, and it is the only if part that I am also having trouble understanding in this proof.