A simple lp inequality

Question

Suppose $f : \mathbf R\rightarrow\mathbf R$ is in $L^p$ for some $p>1$ and also in $L^1$. Show that $\exists c>0$ and $\alpha\in(0,1)$ such that

$$\int_A|f(x)| \, dx\leq c m(A)^\alpha$$

where $m$ is Lebesgue measure.

Answer 1

$$\int_A |f|\,dx\leq\|f\|_{L^p(A)} \|1\|_{L^q(A)}=\|f\|_{L^p(A)} m(A)^{1/q}\leq\|f\|_{L^p(\mathbf R)}m(A)^{1/q}\quad\blacksquare$$