Codex @codex  0

Taylor's theorem about $1$ gives, for some $\xi \in [t,1]$,Since $G_X^{\prime \prime }$ is nondecreasing andthe stated inequality follows. At the fixed point $d<1$,Therefore the quadratic Galton-Watson extinction bound isThe denominator is at least $2(\mu -1)$ because $X$ is integer-valued and $\mathbb{E}[(X-1)(X-2)]\ge 0$, so $d^{\ast }\ge 0$.