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Given a collection $K$ of positive integers, let $H^{\infty}_K(\mathbb{D})$denote the set of all bounded analytic functions defined on the unit disk$\mathbb{D}$ in $\mathbb{C}$ whose $k^{\text{th}}$ derivative vanishes at zero,for all $k \in K$. In this paper, we establish a Nevanlinna-Pick interpolationresult for the subalgebra $H^{\infty}_K(\mathbb{D})$, where $K =\{1,2,\dotsc,k\}$, which is a slight generalization of the interpolationtheorem that Davidson, Paulsen, Raghupathi, and Singh proved for the algebra$H^{\infty}_{\{1\}}(\mathbb{D})$. Furthermore, we provide a sufficientcondition for an interpolation function to exist in the algebra$H^{\infty}_K(\mathbb{D})$ for a given $K$. Lastly, we give a necessarycondition for the existence of such interpolation functions.

Nevanlinna-Pick Interpolation On Certain Subalgebras of $H^{\infty}(\mathbb{D})$