On moduli of boundary values of holomorphic functions

Some maximum modulus polynomial rings and constant modulus spaces

“…Observation 2.10 Let M, K be real or complex Euclidean manifolds, let Ω M be a domain and denote by M ⊂ C 1 (M, K) the subspace of functions which satisfy the strong maximum modulus principle bilaterally and do not attain weak local maximum on Ω. Any function f : M → K which can be uniformly approximated on a neighborhood 5 of Ω by a family {P j } j∈N , P j ∈ M, satisfies on any subdomain ω ⊆ Ω, max z∈ω |f (z)| = max z∈∂ω |f (z)|. The proof of this observation is quite simple so we give it as an appendix.…”

“…We also mention that Globenvik [5] proved a result which implies that for every ϕ : ∂D such that log ϕ is integrable defines the norm f : ∂D → R ≥0 , for some analytic disc We shall use the following lemma.…”