FRACTIONAL ANALYSIS OF COEFFICIENT ESTIMATES FOR STARLIKE FUNCTIONS IN SHELL-LIKE DOMAINS

Recent studies in analytic functions have increasingly focused on shell-like domains and their connections to fractional operators. This work introduces a novel class of analytic functions, [Formula: see text], and explores the Fekete–Szegö inequality and new coefficient values for this class. To align with fractional analysis, we explore the applications of the Erdélyi–Kober fractional integral operator within this framework. Additionally, we examine the function [Formula: see text] and its inverse [Formula: see text], extending the analysis to fractional-order settings. A new class, [Formula: see text], is defined and studied in relation to convolution with normalized analytic functions. Notably, Fekete–Szegö inequalities are derived for specific subclasses involving the Poisson distribution series, demonstrating how fractional calculus enriches geometric function theory and extends its applications across broader mathematical and physical domains. Recent studies in analytic functions have increasingly focused on shell-like domains. An innovative class of analytic functions is presented in this study, [Formula: see text], and the Fekete–Szegö inequality is investigated for this class and new coefficient estimates. We extend our analysis to include the function [Formula: see text] and its inverse [Formula: see text]. Additionally, we define and study the class [Formula: see text] and explore its applications to functions obtained by convolution with normalized analytic functions. Notably, we expand our method to more general function theory applications by deriving Fekete–Szegö inequalities for particular subclasses of functions defined via the Poisson distribution series.