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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 inducebijections either on $\mu_{q+1}$ or between $\mu_{q+1}$ and $\mathbb{F}_q \cup\{\infty\}$. In particular, we generalize results from Zieve.

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