Suppose $A \in \mathbf{R}^{n \times n}$ is a rank $r$ matrix and $|A_{ij}| \leq B$ for all $i, j$. I can obviously write down the inequality $$ \|A\|_F \leq cB,$$ with $c = n$. But can we write $c = f(r, n)$ where $f(r, n) < n$? (For example, can I take $c = \sqrt{rn}$ or similar?).
As clarification $\|\cdot\|_F$ is the Frobenius norm on the space of $n$ by $n$ real matrices.