Let $f$ be a non-negative function, $R \geq 0$, and $B_R \subset \mathbb{R}^d$ a ball of radius $R$. Prove that for every $r \in (0,R)$ we have $$\int_{B_R}f\leq C\int_{B_R}\left(\frac{1}{\vert B_r(x)\vert} \int_{B_r(x)} f \right) dx$$ where $C(d)$ is a contant and $B_r(x) = \{ y \in \mathbb{R}^d : | x - y | \leq r \}.$
My thoughts:
As an earlier part to this problem, I was able to show that for any $E \subset \mathbb{R}^d$ with finite measure, we can find an $R > 0$ such that the above is true but instead we're integrating over $E$ instead of $B_R$ and $C = 2$.
I believe this result should be helpful for proving the above, but I'm stuck. It seems intuitive to me that the problem implies we can find an $R$ up to the radius of the ball we're integrating over? But I'm unsure.
Thank you for your time and feedback.