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Geometry of nowhere vanishing, point separating sub-algebras of $\mathcal{H}ol(Γ\cup\text{Int}(Γ))$ and zeros of Holomorphic functions

We study $ \mathcal{H}ol(Γ\cup\text{Int}(Γ)) $, the normed algebra of all holomorphic functions defined on some simply connected neighbourhood of a simple closed curve $Γ$ in $\mathbb{C} $, equipped with the supremum norm on $ Γ$. We explore the geometry of nowhere vanishing, point separating sub-algebras of $ \mathcal{H}ol(Γ\cup \text{Int}(Γ)) $. We characterize the extreme points and the exposed points of the unit balls of the said sub-algebras for $Γ$ analytic. We also characterize the smoothness of an element in these sub-algebras by using Birkhoff-James orthogonality techniques without any restriction on $Γ$. As a culmination of our study, we assimilate the geometry of the aforesaid sub-algebras with some classical concepts of complex analysis and establish a connection between Birkhoff-James orthogonality and zeros of holomorphic functions.