Is it true that $\sum_{n=0}^{\infty}(\int_n^{n+1} |f| dt)^p \leq \int_{\mathbb{R}}|f|^p\,\mathrm{d}t$ when $1\leq p\leq\infty$?

Can we show that $$\sum_{n=0}^{\infty} \left(\int_n^{n+1} |f| dt\right)^p \leq \int_\mathbb{R}|f|^pdt$$ for $1\leq p\leq\infty$?

• Should the integral be $\int_0^\infty$? – angryavian Mar 21 '18 at 21:55

By writing $\int_0^\infty |f|^p \, dt = \sum_{n = 0}^\infty \int_n^{n+1} |f|^p \, dt$, it suffices to check $$\left(\int_n^{n+1} |f| \, dt\right)^p \le \int_n^{n+1} |f|^p\, dt.$$ When $p \ge 1$, this is a consequence of Jensen's inequality, since $x \mapsto x^p$ is convex.