On the inverses of permutation polynomials of the form $h(ψ(x))\varphi(x)+g(ψ(x))$ over finite fields

In this paper, we investigate the compositional inverses of permutation polynomials of the form \[ F(x)=h(ψ(x))\varphi(x)+g(ψ(x)) \in \mathbb{F}_{q^n}[x], \] where \(ψ(x),\varphi(x) \in \mathbb{F}_{q^n}[x]\) are additive polynomials, \(h(x), g(x) \in \mathbb{F}_{q^n}[x]\) satisfy $ h(ψ(\mathbb{F}_{q^n})) \subseteq \mathbb{F}_q^*, $ and there exists a polynomial \(\barψ(x) \in \mathbb{F}_{q^n}[x]\) such that $ \barψ(F(x)) = ψ(x). $