Some maximum modulus polynomial rings and constant modulus spaces

Abstract

Let F ∈ C 1 (Ω, C) be a not necessarily open function on a Euclidean manifold Ω such that F obeys the strong maximum modulus principle in a bilateral sense defined in the paper and does not attain weak local minimum on a submanifold M ⊂ Ω. We prove that the polynomial ring C[F ] satisfies the strong maximum modulus principle on M. We also give a sufficient condition for subspaces of polyanalytic functions to have constant modulus spaces containing only constants.Mathematics Subject Classification: 30C80, 30A99, 46E25