$$

\max_{a\leq \frac{b}{1-db}}\frac{1}{1+da}(-a\log(a)-(1-a)\log(1-a))

$$

where $b\rightarrow 0$, $d=\frac{n}{b}$ and $n\in\mathbb{R}$ is a constant.

___________________________________________________________________________

Here is what I've done,

first I tried to simplify the problem, so I studied the case where $b$ is still going to $0$ but $d$ is a constant. And the maximum is (obviously) archived just taking $a$ to be the higher bound.

With the complete problem (where $d=\frac{n}{b}$ and $b$ is going to $0$), I started for doing some plots of specific cases, and it seems that the evaluation of the higher bound is again the maximum... (At least in the cases I saw) but of course this is just to have an idea, and it is not a proof...

For the proof I tried to optimize the expression evaluated in some $xb$ (because the value of $a$ I think will depend on $b$) but I didn't get any interesting.

Any help will be appreciated!