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

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