Fekete–Szegö Variations for Some New Classes of Analytic Functions Explained Over Poisson and Borel Distribution Series

The motivation of this paper is to explore and generalize Sakaguchi‐type functions, which play a significant role in geometric function theory. In this context, we introduce four new classes of analytic univalent functions: , and . These classes are defined to extend the study of these functions and their applications. First, we define the class , associated with generalized telephone numbers, using the concept of subordination. For functions in this class, we derive initial coefficient estimates and address the Fekete–Szegö functional problem, as well as explore results for the inverse function . Next, we introduce the remaining classes , and which are connected to through convolution and Borel and Poisson distribution series. For these classes, we obtain variations of the Fekete–Szegö inequality and discuss their relationships with previously studied function classes.