Let $f(\theta)=\sum_i\theta^i P(i)$ where $P(i)$ is a probability distribution of $i=1,\dots, n,\dots$ with $i\in Z$ and $0\leq \theta\leq 1$. Denote $G$ the graph of $f(\theta)$ where $\theta$ ranges in $(0,1)$.
Then $f(0)=P(0)$ and $f(1)=1$. Let $\mu=\sum_i iP(i)$ be the expectation value of $i$.
$\textbf{Q:}$ The book draws two graphs which distinguishes 2 cases. If $\mu\geq 1$. Then $G$ intersects $h(\theta)=\theta$'s graph in $(0,1)$ part at only 1 point $0<\theta'<1$ If $\mu<1$, then $G$ does not intersect $h(\theta)=\theta$'s graph in $(0,1)$. In other words, how do I show that $f(\theta)\geq\theta$ for $\mu<1$ case in $\theta\in (0,1)$ and the other case for only 1 intersection?