Let $a,b,c>0$ be distinct postive reals.
Define four different probability distributions:
$$\mathcal{P}_{ab}:P_{a,ab}=\frac{a}{a+b}=1-P_{b,ab}$$ $$\mathcal{P}_{bc}:P_{b,bc}=\frac{b}{b+c}=1-P_{c,bc}$$ $$\mathcal{P}_{ca}:P_{c,ca}=\frac{c}{c+a}=1-P_{a,ca}$$ $$\mathcal{P}_{abc}:P_{a,abc}=\frac{a}{a+b+c},\mbox{ }P_{b,abc}=\frac{b}{a+b+c},\mbox{ }P_{c,abc}=\frac{c}{a+b+c}$$
Does the entropy of a random variable from distribution $\mathcal{P}_{abc}$ dominate the other three for all $a,b,c\in\Bbb R^+$?
In general when does $\mathcal{P}_{abc}$ dominate?