Is it true that $\int_{B(z,r-1)} \int_{B(a,1)} |f(w)| \, dw da \leq C \int_{B(z,r)} |f(w)| \, dw$?

Question

Let $B(z,r)$ denote the disc of radius $r$ around $z$. Is it true that if $r>1$ then $$ \int_{B(z,r-1)} \int_{B(a,1)} |f(w)| \, dw \, da \leq C \int_{B(z,r)} |f(w)| \, dw $$ for some constant $C$ that is independent of $r$? This seems to look like a Fubini-type argument for me but I'm not able to prove it.