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.