Let $Z_n$ be the Galton Watson Branching Process. Let $Z_n=\sum_{k=1}^{Z_{n-1}}X_{k,n}$ where $X_{k,n}\sim X$ are iid progeny distribution. If $p_0=P(X=0)>0$ then show that $\forall s\in(0,1)$ we have $\phi_n'(s)\to0$ where $\phi_n(s)$ is the generating function of $Z_n$, evaluated at $s$.
I tried to use $\phi_n(s)=\phi_X(\phi_{n-1}(s))$ and then differentiate both sides to get $\phi_n'(s)=\phi_X'(\phi_{n-1}(s))\phi_{n-1}(s)$ and iterate but going nowhere.
Actually, I am trying to prove that $\phi_n(s)\to\eta$ for every $s\in[0,1)$ where $\eta$ is the extinction probability. I know that $\phi_n(0)\to\eta$ as $n\to\infty$, so I can write $\\phi_n(s)=\phi_n(0)+s\phi_n'(s_n)$ where $s_n\in(0,s)$. I will be done if I show $\phi_n'(s_n)\to0$. But note that $0<s_n<s$ so $\phi_n'(s_n)\leq \phi_n(s)$ so it suffices to show $\phi_n'(s)\to0$ for every $s\in(0,1)$.
Expanding on @Did's comment, we have $$\varphi_n(s)=\sum_{j=0}^\infty \mathbb P(Z_n=j)s^j $$ and so $$ s\varphi'_n(s)=\sum_{j=0}^\infty j\ \mathbb P(Z_n=j)s^j = \mathbb E\left[Z_n s^{Z_n}\right].$$ Now, $\{Z_n:n=0,1,2,\ldots\}$ is a Markov chain on $\{0,1,2,\ldots\}$ with $\mathbb P(Z_0=1)=1$ and transition probabilities according to the branching process. Since $P_{00}=1$, $\{0\}$ is an absorbing state. Let $\tau = \inf\{n>0:Z_n=0\}$. Conditional on $\{\tau<\infty\}$, we have that $$\mathbb P\left( \liminf_{n\to\infty} \{Z_n=0\}\right)=1, $$ so $Z_n\stackrel{a.s.}\longrightarrow0$, hence $s^{Z_n}\stackrel{a.s.}\longrightarrow 1$, so that $Z_ns^{Z_n}\stackrel{a.s.}\longrightarrow0$. Conditional on $\{\tau=\infty\}$, we have $$\mathbb P\left(\bigcap_{m=1}^\infty \liminf_{n\to\infty} \{Z_n>m\}\right)=1, $$ so $Z_n\stackrel{a.s.}\longrightarrow\infty$, hence $s^{Z_n}\stackrel{a.s.}\longrightarrow0$ and $Z_ns^{Z_n}\stackrel{a.s.}\longrightarrow0$. Then from the inequality $$us^u \leqslant -(e\log s)^{-1} \tag1,\ u\geqslant 0 $$ we conclude from dominated convergence that $$ \lim_{n\to\infty}\varphi'_n(s) = \lim_{n\to\infty}s^{-1}\mathbb E\left[ Z_ns^{Z_n}\right]=s^{-1}\mathbb E\left[\lim_{n\to\infty} Z_ns^{Z_n}\right]=0.$$
P.S. If someone can help me justify $(1)$ I would be very grateful.