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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!

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    I would suggest you put your original problem here and regard this question as part of your thought since you put a tag `probability` to your question.2012-09-20
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    Thanks! The question is motivated under probability background. However, the question itself is purely real analysis.2012-09-20

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