Newton-Like Components in the Chebyshev–Halley Family of Degree n Polynomials

Abstract We study the Chebyshev–Halley methods applied to family of polynomials $$f_{n,c}(z)=z^n+c$$ f n , c ( z ) = + , for $$n\ge 2$$ ≥ 2 and $$c\in \mathbb {C}^{*}$$ ∈ C ∗ . prove existence parameters such that immediate basins attraction corresponding roots unity are infinitely connected. also that, $$n \ge dynamical plane contains a connected component Julia set, which is quasiconformal deformation set map obtained by applying Newton’s method $$f_{n,-1}$$ - 1