Closed ideals in a new class of algebras of holomorphic functions on the disc

We define a new class of Banach algebras of holomorphic functions on the unit disc \(\mathbb{D}\) which contains the algebras studied in [GMR2] and [GW]. To a function G of the class \(\mathcal{A}^1 (\mathbb{D})\) nowhere vanishing in \(\mathbb{D}\) we associate a Banach algebra \(\mathcal{A}^n_G (\mathbb{D})\) contained in the disc algebra \(\mathcal{A}(\mathbb{D})\). We prove that all closed ideals of \(\mathcal{A}^n_G (\mathbb{D})\) of at most countable hull are of the standard form.