Geometric Characterizations of Imaginary Error Functions in Subclasses of Spirallike Analytic Functions

Abstract

In this paper, we investigate the geometric behavior of the generalized normalized imaginary error function $\Upsilon i_{k}\left( z\right)$ and the associated convolution operator $\mathcal{I}i_{k}(z)$ within the framework of analytic function theory. Specifically, we establish necessary and sufficient conditions under which these functions belong to the subclasses $\mathcal{SPE}(\vartheta ,\zeta )$ and $\mathcal{CSPE}(\vartheta,\zeta )$ of spirallike and convex spirallike analytic functions, respectively. Additionally, we derive sharp criteria for an integral operator involving $\Upsilon i_{k}\left( z\right)$ to be a member of these subclasses. These results extend and generalize several known findings and may inspire further applications of the imaginary error function in geometric function theory.