I have two pairs of Random Variables, $(\mathbb{X},\mathbb{Y})$ and $(\mathbb{M},\mathbb{N})$ which satisfies, $H(\mathbb{X})>H(\mathbb{Y})$ and $H(\mathbb{M})>H(\mathbb{N})$.
For some convex combination, with mixing ratio $\eta$, is there any inequality that holds for $H((1-\eta)\mathbb{X}+\eta\mathbb{M})$ and $H((1-\eta)\mathbb{Y}+\eta\mathbb{N})$.
In particular is it true that, $H((1-\eta)\mathbb{X}+\eta\mathbb{M}) >H((1-\eta)\mathbb{Y}+\eta\mathbb{N})$
False. Take $\mathbb{X} \sim B(0.4)$, $\mathbb{M} \sim B(0.4)$, $\mathbb{Y} \sim B(0.1)$, $\mathbb{N} \sim B(0.9)$, $\eta = 1/2$
(Here $B(p)$ is a Bernoulli distribution with parameter $p$)
I doubt that anything can be said in general.