Let $b,c \in (0,1)$ be such that $b+c<1.$ Define the following function for $p \in (0,1) :$ $ I(p;b,c):=(b+c)[p \log\frac{1}{p}+(1-p)\log\frac{1}{1-p}]-cH(p;b,c)-bH(p;c,b)$ where $H(x;\epsilon,\delta):= p \log\frac{\epsilon (1-p)+ \delta p}{\delta p}+ (1-p) \log\frac{\delta (1-p)+\epsilon p}{\delta (1-p)} .$
Ploting this function for various values of $b,c$ we see that the maximum is obtained at $p=\frac{1}{2}.$ Can we prove rigorously this fact for any $b,c$ described before ?
(Since $I(p;b,c)=I(1-p;b,c)$ it suffices to show for example that H'(p;b,c)>0 for $0 However this seems more difficult than the original question.)