The compositional inverses of three classes of permutation polynomials over finite fields

R. Gupta, P. Gahlyan and R.K. Sharma presented three classes of permutation trinomials over ${\mathbb{F}}_{q^3}$ in Finite Fields and Their Applications. In this paper, we employ the local method to prove that those polynomials are indeed permutation polynomials and provide their compositional inverses.