According to an article entitled "On the Univalency of Certain Analytic Functions" by Wang et al. (2006), we have to show that $|f'(z)-1|<1$ in order to find the radius of univalency for the class $Q(\alpha,\beta,\gamma)$. Note that $Q(\alpha,\beta,\gamma)$ denotes the class of functions of the form $f(z)=z+a_{2}z^{2}+\cdots$ which are analytic in open unit disk, $D=\{z:|z|<1\}$, and satisfy the condition
$\mathfrak{Re} \left\{\frac{\alpha f(z)}{z}+\beta f'(z)\right\}>\gamma \qquad (\alpha, \beta >0;\ 0 \leq \gamma<\alpha+ \beta\leq 1;\ z\in D)$
Why it is sufficient to show that $|f'(z)-1|<1$?