In this study, we investigate a class of analytic and univalent functions defined in the open unit disc U=z∈C: ∣z∣<1U = \ z C: |z| < 1 \U=z∈C: ∣z∣<1, denoted by HHH, and consider the subclass S⊂HS HS⊂H of normalized univalent functions with f (0) =0f (0) = 0f (0) =0 and f′ (0) =1f' (0) = 1f′ (0) =1. We explore the geometric properties of these functions, including starlikeness, convexity, spirallikeness, close-to-convexity, and bounded turning, which characterize important aspects of their mapping behavior. The concept of subordination, f (z) ≺g (z) f (z) g (z) f (z) ≺g (z), is employed to relate functions via an analytic Schwarz function ω (z) (z) ω (z) satisfying ω (0) =0 (0) = 0ω (0) =0 and ∣ω (z) ∣<1| (z) | < 1∣ω (z) ∣<1, enabling a framework for generalized distribution analysis. Building upon prior results by Porwal 1, we establish novel properties of these distributions and examine conditions under which univalent functions exhibit specific geometric behavior. The findings provide deeper insights into the structural and functional characteristics of analytic univalent functions in complex analysis and extend the theoretical framework for subordination and geometric function theory.
Samuel O. Adeyemi (Sun,) studied this question.