9. $f(x)$ is a function defined on $\mathbf{R}$, if $f(0)=1008$, and for any $x \in \mathbf{R}$, it satisfies $f(x+4)-f(x) \leqslant 2(x+1), f(x+12)-f(x) \geqslant 6(x+5)$, then $\frac{f(2016)}{2016}=$ $\qquad$
$$
\begin{array}{l}
f(x+12)-f(x+8) \leqslant 2(x+9), f(x+8)-f(x+4) \leqslant 2(x+5), \\
f(x+4)-f(x) \leqslant 2(x+1) \Rightarrow f(x+12)-f(x) \leqslant 6 x+30,
\end{array}
$$

Then $f(x+12)-f(x)=6 x+30$. And $2016=12 \times 168$,
$$
\begin{array}{l}
\text { Thus } f(2016)-f(0)=6 \times 12 \sum_{i=0}^{167} i+30 \times 168=(36 \times 167+30) \times 168 \\
\Rightarrow f(2016)=(36 \times 167+30) \times 168+6 \times 168=36 \times 168 \times 168 \\
\Rightarrow \frac{f(2016)}{2016}=3 \times 168=504 .
\end{array}
$$