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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.

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You have the "pointwise" inequality

$$ab \le \frac{a^p}{p} + \frac{b^q}{q}$$

whenever $a, b, \ge 0$ and $\frac 1p + \frac 1q = 1$. Then

$$|f| \cdot |g| \le \frac{|f|^p}{p} + \frac{|g|^q}{q}.$$

Now we use the general fact: If $F$, $G$ are two integrable function so that $F\le G$ and $\int F = \int G$, then $F = G$ almost everywhere. Now let $F = |f|\cdot |g|$ and $G = \frac{|f|^p}{p} + \frac{|g|^q}{q}$ will give you the implication ($\Rightarrow$).

  • Ah thank you, now I get it. You are using the fact that then $\int(G-F)=0$, and since this is a non-negative function(G-F), you can conclude this? – user119615 Oct 02 '15 at 08:37
  • Yes, we can then conclude $G-F = 0$ a.e. @user119615 –  Oct 02 '15 at 08:39