Hölder inequality for n functions

Question

I am trying to prove Hölder's inequality for a series of $N$ functions. To be more precise, assume $1<p_k<\infty$ for $k=1,\dots,N$ and $\sum^N_{k=1} (1/p_k) = 1$

$\int f_1f_2f_3....f_Ndu \leq ||f_1||_{p_1}.||f_2||_{p_1}.||f_3||_{p_3}.......||f_N||_{p_N}$

I know how to prove it for N= 3, and my idea was to apply induction to prove for N.

So far I have done till this point

$\int \Pi_{n=1}^N f_n du = \int f_1. \Pi_{n=2}^N f_n du \leq ||f_1||_{p_1}.||\Pi_{n=2}^N f_n||_{p_1/(p_1-1)}$

However to write the above inequality I am using Hölder's inequality , but to use that I have to be able to prove that $\Pi_{n=2}^N f_n \in L_{p_1/(p_1-1)}$.

Which I am not sure how to prove !

Any help will be appreciated

Answer 1

The inequality is true even if $\prod_{2}^Nf_n \notin L^{p_1'}$; indeed it would just read $\int \prod_1^N f_n \le \infty$, which is trivially true.