  Let $\mu_{q+1}$ denote the set of $(q+1)$-th roots of unity in
$\mathbb{F}_{q^2 }$. We construct permutation polynomials over
$\mathbb{F}_{q^2}$ by using rational functions of any degree that induce
bijections either on $\mu_{q+1}$ or between $\mu_{q+1}$ and $\mathbb{F}_q \cup
\{\infty\}$. In particular, we generalize results from Zieve.
