Generalization of Hölder's inequality

Question

Assume $1<p_k< \infty$ for $k=1,\ldots,N$ , and $\displaystyle\sum^N_{k=1}\frac{1}{p_k} =1$.

I want to prove that $$\left|\int_X f_1 f_2\cdots f_N\; d\mu \right| \le \lVert f_1\rVert_{p_1} \lVert f_2\rVert_{p_2} \cdots \lVert f_N\rVert_{p_N}.$$

How can I directly adjust Hölder's inequality for it?

Answer 1

Hint

1. Start with Hölder on the function $f_1$ and $g_1=f_2f_3\dots f_n$ using $p_1$ and $p_1'$. That is $$\left|\int f_1 g_1 dx\right|\leq\left(\int |f_1|^{p_1}dx\right)^{1/p_1}\left(\int |g_1|^{p_1'}dx\right)^{1/p_1'}$$ where $p_1'=p_1/(p_1-1)$.

2. Apply Hölder on $|f_2|^{p_1'}$ and $g_2 = |f_3f_4\dots f_n|^{p_1}$ using $p_2/p_1'$ and $(p_2/p_1')'$...