Automorphisms of subalgebras of bounded analytic functions

Let $H^{\mathrm{\infty }}$ denote the algebra of all bounded analytic functions on the unit disk. It is well-known that every (algebra) automorphism of $H^{\mathrm{\infty }}$ is a composition operator induced by disc automorphism. Maurya et al., (J. Math. Anal. Appl. 530 : Paper No: 127698, 2024) proved that every automorphism of the subalgebras $\{f\in H^{\mathrm{\infty }}:f(0)=0\}$ or $\{f\in H^{\mathrm{\infty }}:f^{\mathrm{\prime }}(0)=0\}$ is a composition operator induced by a rotation. In this article, we give very simple proof of their results. As an interesting generalization, for any $\psi \in H^{\mathrm{\infty }}$, we show that every automorphism of $\psi H^{\mathrm{\infty }}$ must be a composition operator and characterize all such composition operators. Using this characterization, we find all automorphism of $\psi H^{\mathrm{\infty }}$ for few choices of $\psi$ with various nature depending on its zeros.