Thread: bound on sum of logarithms

1. bound on sum of logarithms

I have that

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

I need to bound the following using $l$

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

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

Can I use something like Holder's inequality?

I started by taking the logarithm of both sides

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

And I believe that the solution is

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

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