Let $X$ and $Y$ be two $\mathbb{N}$-valued random variables. We say $Y$ stochastically dominates $X$ in the first order, written $X\le Y,$ if and only if $\mathbb{P}(Y>k) \ge \mathbb{P}(X>k)$ for all $k \in \mathbb{N}$.
It is easy to show that if $X \le Y,$ and $G_X(z), G_Y(z)$ are the corresponding probability generating functions, then $G_X(z) \ge G_Y(z)$ for all $z\in[0,1]$. Let $\rho_X$ and $\rho_Y$ denote the extinction probabilities for a branching process with offspring distribution $X$ and $Y$. Then it follows immediately that $$\rho_X = G_X(\rho_X) \le G_Y(\rho_Y) = \rho_y.$$
My question concerns the derivative of $G_x$ and $G_Y$ at the extinction probability. I want to know: when is it true that $$ G_Y^\prime(\rho_Y) \le G_X^\prime(\rho_X) \tag{1} $$ Note that $G_Y^\prime(1) \ge G_X^\prime(1)$ because $Y\ge X$ implies that $\mathbb{E}[Y]\ge \mathbb{E}[X]$. But for small $z,$ and "nice" FOSD shifts, we have $G_Y^\prime(z) \le G_X^\prime(z)$. Perhaps the key here is working out exactly what these "nice" FOSD shifts are (I have percolation in mind if that helps).
The reason I care about this is that I am trying to look at the behavior of finite component sizes in a random graph with an arbitrary degree distribution, which can be approximated by a branching process. Suppose the degree distribution is such that the graph has a giant component with high probability. If $X$ is the distribution of the "forward" degree distribution in a random graph with an arbitrary degree distribution, then one can show that (assuming the graph can be locally approximated by a branching process) the expected finite component size is given by (e.g. see page 28 here) $$ \frac{1}{1-G_X^\prime(\rho_X)}. $$ So (1) translates to saying that the expected finite component size reverses the stochastic order. Intuitively, because $Y\ge X,$ more vertices in large finite components get joined up to the giant component, and so I expect (1) to be true. An obvious example would be a Poisson distribution, where we can parameterize FOSD shifts by the mean $\lambda$. If $X\sim \mathrm{Po}(\lambda)$ then in this case we know that $$ G_X^\prime(\rho) = \lambda\rho, $$ and one can show that this is strictly decreasing in $\lambda$ (treating $\rho$ as an implicit function of $\lambda$). A similar statement can be made for a random regular graph and a geometric degree distribution, which leads me to think there's a much more general statement to be made. But these distributions are all nicely parameterized, so it's not clear that (1) holds for ANY FOSD shift in the distribution. Is there a better condition I could appeal to here? Is there any reference/book I could check for this?
