Does the following Poincaré inequality hold

$$\int_{B_{r_2}\setminus B_{r_1}} |f-\bar{f}|^2 dx \leq C(r_2-r_1)^2 \int_{B_{r_2}\setminus B_{r_1}} |\nabla f|^2 dx,$$ where $B_r$ denotes a ball of radius $r>0$ centered at the origin, $f\in C^{\infty}_c(B_{r_2}\setminus B_{r_1})$, $C>0$ is some positive constant and $\bar{f}$ is the average value on the annular region?