Let $f:\mathbb{R} \to \mathbb{R}$ be an integrable function. For every $h>0$ we define
$$f_h (t) = \frac{1}{h} \int_{t-\frac{h}{2}}^{t+\frac{h}{2}}f(x)dx$$ If $f\in L^p$, show that $$\|f_h\|_{\infty} \leq h^{-\frac{1}{p}}\|f\|_p$$
I couldn't prove it for $p<1$. For the case $1\leq p$, I used that, if $0<|E|< \infty$, $\displaystyle N_p [f] = \left(\frac{1}{|E|}\int_E |f|^p\right)^{\frac{1}{p}}$ is monotonically increasing over $p$. Any help will be appreciated.