Exploring Subordination and Argument Behavior in a New Class of Analytic Functions

The class (Formula presented.) includes functions with a specific starting term and higher-order terms in their series, and studying these functions involves examining their analytic properties, argument behavior, and subordination characteristics, which lead to various interesting and significant results in complex analysis. The functions within (Formula presented.) are designed to explore properties that arise from their structure and the way their series are constructed, and the main interest in this class lies in analyzing how these functions behave in various mathematical contexts, particularly in relation to argument shifts and the relationships between different functions. In a nutshell, the class (Formula presented.) offers a framework for examining a specific type of analytic function with a structured series expansion, thereby allowing for a deeper investigation into their mathematical properties and applications. The new class (Formula presented.), which is related to the class (Formula presented.), is characterized by functions that are analytic in the open unit disk and have a specific series expansion involving fractional powers. The object of this paper is to discuss some interesting applications of (Formula presented.) concerning argument problems and subordinations, and furthermore, prominent examples for the main results are given.