= Solution
Taylor's theorem about $1$ gives, for some $\xi\in[t,1]$,
$$
G_X(t)=1-\mu(1-t)+\frac12G_X''(\xi)(1-t)^2.
$$
Since $G_X''$ is nondecreasing and
$$
G_X''(1)=\mathbb E[X(X-1)]=\sigma^2+\mu^2-\mu=:A,
$$
the stated inequality follows. At the fixed point $d<1$,
$$
0\leq(1-d)\left[1-\mu+\frac A2(1-d)\right].
$$
Therefore the <quadratic Galton-Watson extinction bound> is
$$
d\leq d^*:=1-\frac{2(\mu-1)}{\sigma^2+\mu^2-\mu}<1.
$$
The denominator is at least $2(\mu-1)$ because $X$ is integer-valued and $\mathbb E[(X-1)(X-2)]\geq0$, so $d^*\geq0$.
Back to article page
