Let $Z_n$ be a nonnegative discrete random variable (Galton-Watson Process in this case). I need to show that:
$$\frac{f(\phi_{n}(\frac{u}{m}))-f(\phi_{n-1}(\frac{u}{m}))}{u} \le \frac{\phi_{n}(\frac{u}{m})-\phi_{n-1}(\frac{u}{m})}{\frac{u}{m}} $$
where f is the probability generating function of $Z_n$, $1<E(Z_1) = m<\infty$, $u > 0$, $0\le\phi_n(u) = E(e^{-uZ_nm^{-n}})\le 1$ and $\phi_{n+1}(u)=f(\phi_n(\frac{u}{m}))$.
Write $a=\phi_{n-1}(\frac{u}{m})$ and $b=\phi_{n}(\frac{u}{m})$. So I know that there exists a $x_0\in (a,b)\subsetneq [0,1]$ such that $$\frac{f(b)-f(a)}{b-a} = f'(x_0) \\\Leftrightarrow f(b)-f(a)=(b-a)f'(x_0) \le m(b-a)f'(x_0)$$ I know that $f(x)\le 1$ for all $x\in[0,1]$ but $f'(x)=\sum_{j=0}^\infty p_jjx^{j-1}$ so i think that $f'(x)$ can be $\ge1$ because $j\in \mathbb{N}$ if i'm not mistaken. So where am I wrong here ?