Convex hulls of polynomial Julia sets

We prove P. Alexandersson’s conjecture that for every complex polynomial p p of degree d ≥ 2 d \geq 2 the convex hull H p H_p of the Julia set J p J_p of p p satisfies p − 1 ( H p ) ⊂ H p p^{-1}(H_p) \subset H_p . We further prove that the equality p − 1 ( H p ) = H p p^{-1}(H_p) = H_p is achieved if and only if p p is affinely conjugated to the Chebyshev polynomial T d T_d of degree d d , to − T d -T_d , or to a monomial c z d c z^d with | c | = 1 |c|=1 .