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

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