Suppose $x >0, y \in (0,1), y(1+x) \in (0, 1)$, how can I prove that
$(x+y) [\ln (x+y) - \ln y] +(1-x-y) [\ln (1-x-y) - \ln (1-y)] \geq y [x - (1+x) \ln (1+x)]?$
I have no clue about how to start. It is a part that will help solving another problem. $y$ and $y(1+x)$ can be thought as the parameters of two Bernoulli distributions. Thanks in advance!