# A Cauchy-Schwarz-type inequality for $\int\prod_n|f_n|$

If $$X_1,X_2$$ have finite second moments then Cauchy-Schwarz gives $$\langle |X_1||X_2|\rangle^2 \leq \langle |X_1|^2\rangle \langle |X_2|^2\rangle$$

If $$(X_n)_{n=1}^N$$ have their $$N$$th moments is it so that $$\langle\prod_n|X_n|\rangle^N \leq \prod_n\langle |X_n|^N\rangle$$?

• I think this is probably what you are looking for: en.wikipedia.org/wiki/… – b00n heT Jan 2 at 8:33
• Thanks, it does appear much closer to Holder's inequality. I will think about it in the morning. – enthdegree Jan 2 at 8:36
• Cauchy-Schwarz is indeed a special case of Hölder's inequality. Good luck :) – b00n heT Jan 2 at 8:37

This can be proven using Holder's inequality. Let's do it by induction, obviously this is true for $$N=1,2$$ so I focus on the induction step. Suppose we are given $$(X_n)_{n=1}^{N+1}$$ and have finite $$N+1$$'th moments, then \begin{align*} \left\langle \prod_{n=1}^{N+1} |X_n|\right\rangle&\leq \left\langle \prod_{n=1}^N |X_n|^{\frac{N+1}{N}} \right\rangle^{\frac{N}{N+1}}\cdot\langle |X_{N+1}|^{N+1} \rangle^{\frac{1}{N+1}}\\ &\leq \prod_{n=1}^{N+1} \langle |X_n|^{N+1} \rangle^{\frac{1}{N+1}} \cdot \langle|X_{N+1}|^{N+1} \rangle^{\frac{1}{N+1}}\\ \end{align*} where the first inequality is Holder's inequality with $$p=\frac{N+1}{N}$$ and $$q=N+1$$, the second one is induction hypothesis.
Taking all this to the power $$N+1$$ you get your desired result.