Permutation polynomials over $\mathbb{F}_{q^2}$ from rational functions

Abstract: 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.