I have that

$\displaystyle \prod_{i,j} (1-y_{ij}) \geq l $, where $\displaystyle 1> y_{ij}\geq 0$

I need to bound the following using $\displaystyle l $

$\displaystyle \prod_{i,j} (1-a_{ij}y_{ij}) \geq ? $, where $\displaystyle 1> y_{ij}\geq 0$ , and $\displaystyle 0.5 \geq a_{ij}\geq 0$

Such that ? has to be a function of $\displaystyle l $, $\displaystyle f(l) $

Can I use something like Holder's inequality?

I started by taking the logarithm of both sides

$\displaystyle \sum_{i,j} \log (1-y_{ij}) \geq \log(l) $

$\displaystyle \sum_{i,j} \log (1-a_{ij}y_{ij}) \geq ? $

And I believe that the solution is

$\displaystyle \sum_{i,j} \log (1-a_{ij}y_{ij}) \geq \log (\frac{1+l}{2}) $

but I don't know how to prove it. Any pointers?