## Bound on the sum of products

Let's say that I know that:

$\sum_{i,j} a_{ij} \leq r$, where $0 \leq a_{ij} \leq 0.5$ and $r \geq 0$ and $1 \leq i,j \leq n$

and I also know that

$\sum_{i,j} y_{ij} \leq d$, where $y_{ij} \geq 0$ and $d \geq 0$ and $y$ is zero-diagonal and $1 \leq i,j \leq n$

$\sum_{i,j} a_{ij} y_{ij} \leq ?$
$\sum_{i,j} a_{ij} y_{ij} \leq \frac{d}{2}$