General upper bound of extinction probability We consider here a Galton–Watson process with an offspring distribution $X$, where $\mathbb{E}X = \mu$ and $\operatorname{Var} X = \sigma^{2} < \infty$ and $q = \mathbb{P}(\text{extinction})$, i.e., the extinction probability. 
I want to prove that the general upper bound of the extinction probability is given by:
$$q \leq 1 - \frac{\mu - 1}{\sigma^{2} + \mu^{2} - \mu}.$$ 
So if this helps, I can rewrite the above form as follows:
$$q \leq 1 - \frac{G_X'(1) - G_X(1)}{G_X''(1)},$$
where $G_X(s)$ is the probability generating function of the random variable $X$.
But here I am stuck.
I know the question is not well-asked, but hope that anyone can give me some hints.
 A: We have 
$$G(s):=G_X(s)=Es^X,$$
with the convention $0^0:=1$, where $X$ is a random variable with values in $\{0,1,\dots\}$ and $EX^2<\infty$. So, $G$ is a nonnegative nondecreasing convex function from $[0,1]$ to $[0,1]$ with nondecreasing $G''$. Also, $G(0)=P(X=0)$ and $G'(0)=P(X=1)$. So, excepting the case when $P(X=0)=0$ and $P(X=1)=1$ (and hence $G''=0$, which makes your inequality devoid of meaning), the extinction probability is the smallest root of the equation 
$$G(q)=q. \tag{1}$$
Also, $G(1)=1$. 
So, by the convexity of $G$ and (1), we have $G(s)\le s$ for $s\in[q,1]$ and $G(s)\ge s$ for $s\in[0,q]$. So,
$$G'(q)\le1.$$
So,
$$G'(1)-G(1)=G'(1)-1\le G'(1)-G'(q)=\int_q^1 G''(s)\,ds
\le \int_q^1 G''(1)\,ds=(1-q)G''(1),$$
which implies that indeed
$$q\le1-\frac{G'(1)-G(1)}{G''(1)},$$
as desired.

To illustrate this, here are the graphs $\{(s,s)\colon0\le s\le1\}$ (blue) and $\{(s,G(s))\colon0\le s\le1\}$ (gold) for the case when $X$ takes values $0,1,2$ with probabilities $\frac2{10},\frac1{10},\frac7{10}$, respectively, so that here $q=\frac27$. 
 
