FRACTIONAL CALCULUS OPERATORS IN GEOMETRIC FUNCTION THEORY AND THEIR SEMIGROUP STRUCTURES

Abstract

Fractional calculus operators Riemann–Liouville, Caputo, Erdélyi–Kober, Hadamard, and their q-analogues have emerged as powerful tools in geometric function theory, enabling the study of analytic functions and their subclasses through non-integer order differintegral transforms that capture memory effects, scale-invariant properties, and generalized subordination. This review systematically examines the application of fractional operators to define and characterize new subclasses of univalent, bi-univalent, and multivalent functions in the open unit disk, focusing on their geometric properties (starlikeness, convexity, close-to-convexity, spirallikeness) and subordination relationships. Key developments include fractional integral/differential operators used to construct coefficient bounds, Fekete–Szegő inequalities, radius constants, and inclusion theorems for families such as fractional q-starlike, fractional convex, and fractional close-to-convex functions. The semigroup structures inherent in these operators’ closure under composition, identity element, inverses, and algebraic properties are analyzed, revealing that many fractional differintegral operators form continuous semigroups under suitable function spaces and parameter regimes, facilitating the study of evolution equations and operator semigroups in complex analysis. Connections to special functions (q-Fibonacci, q-hypergeometric, Mittag-Leffler) and their convolution operators further enrich the theory, yielding sharp estimates and geometric characterizations. Applications extend to control theory (fractional-order PID/sliding-mode controllers) and signal processing, where semigroup properties ensure stability and robustness. Challenges non-locality, computational complexity, and lack of unified algebraic frameworks are addressed through recent advances in q-fractional calculus and operator theory. The integration of fractional calculus into geometric function theory not only generalizes classical results but also opens avenues for novel subclasses and semigroup-based techniques in analytic function spaces.