Extension of Holder's Inequality

Question

I found the proof of Generalization of Hölder's inequality from Wikipedia

I get the other part, but I don't get why in case 1: we have the inequality $\Vert f_1f_2\cdots f_n \Vert_r \leq \Vert f_1f_2 \cdots f_{n-1} \Vert_r \Vert f_n \Vert_\infty $

Could someone explain this?

You have copied the inequality incorrectly. The product on the right hand side should end with $f_{n-1}$. — md2perpe, Jun 19 '17 at 21:01

score 0 · Answer 1 · answered Jun 19 '17 at 21:03

0

Take $f = f_1 f_2 \ldots f_{n-1}$ and $g = f_n$. Then, by Hölder's inequality we have $$\|fg\|_r \leq \|f\|_r \|g\|_\infty,$$ i.e. $$\|f_1 f_2 \ldots f_{n-1} f_n\|_r \leq \|f_1 f_2 \ldots f_{n-1}\|_r \|f_n\|.$$

answered Jun 19 '17 at 21:03

md2perpe

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Extension of Holder's Inequality

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