Question: Given $X\sim\text{Bernoulli}(\alpha)$, $Y\sim\mathcal{N}(0,1)$, and two non-random constants $C_1$ and $C_2$ such that $C_1>C_2$. What can we say about the inequality between the two differential entropies $H(C_1X+Y)$ and $H(C_2X+Y)$? I.e., is it true that $$ H(C_1X+Y)>H(C_2X+Y). $$
Fact: We know from here that differential entropy is shift invariant but is affected by scaling.
Attempt: I am aware that for a single random variable $X$ alone, the entropy $H(X)$ is shift invariant. Hence we can rewrite $$ H(C_1X+Y)=H(\epsilon X+C_1X+Y), $$ for some $\epsilon$. However, this is not a simple shift because we are shifting by a fraction of a random variable.
It should be intuitive that $I(C_1X + Y;X) \ge I(C_2 X + Y;X)$ (but see below). But notice that $I(C_1 X + Y; X) = h(C_1 X + Y) - h(C_1X + Y|X),$ and the second term is just $h(Y)$. So, we can conclude that $$I(C_1X + Y;X) \ge I(C_2X + Y;X) \iff h(C_1 X + Y) \ge h(C_2 X + Y),$$ and so we'd be done if we show the first inequality.
One way to see this is the I-mmse relation that says that for standard Gaussian noise $Y$, with mutual information defined in nats, $$ \frac{\mathrm{d}}{\mathrm{d}\gamma} I(\sqrt{\gamma} X + Y;X) = \frac12\mathrm{mmse}(\gamma),$$ where $\mathrm{mmse}(\gamma)$ is the minimum mean square error $\mathbb{E}[(X - \mathbb{E}[X| \sqrt{\gamma} X + Y])^2].$ See this paper for a proof. The striking thing here is that the result is completely generic, and holds for any law over $X$ (as long as it has a second moment of course). The relation is indelibly connected to the infinite divisibility of the Gaussian (see the paper for more), which you could use directly to show the inequality as well. In any case, the $\mathrm{mmse}$ is nonnegative, so this immediately shows the monotonicity of $I(CX + Y;X)$ in $C$.
As an aside, I think it is good practice (and the standard convention) to always write $h(\cdot)$ when one intends differential entropy, and not $H(\cdot)$ which is reserved for discrete entropy. The two functionals are quite different, and while it is often clear from context which is intended, this context can often be hidden enough that one ends up with spurious manipulations. The small notation change helps prevent this from the get go.