Analytical Framework for Extended Hadamard Product Operations on Meromorphic Multivalent Function Classes with Positivity Constraints

Abstract

The study of meromorphic multivalent functions has evolved significantly due to their structural richness and applicability in complex analysis and geometric function theory. In particular, Hadamard product operations have played a central role in constructing and analyzing new subclasses of analytic and meromorphic functions with controlled geometric properties. This paper develops an analytical framework for extended Hadamard product operations on meromorphic multivalent function classes under positivity constraints on coefficients. The proposed framework integrates classical convolution techniques with modern generalization strategies to characterize stability, closure properties, and functional behavior of these classes. Building upon foundational work on meromorphic starlike and convex functions, the study systematically extends earlier results to p-valent and multivalent settings with positivity restrictions. Emphasis is placed on coefficient bounds, operator stability, and functional preservation under convolution transformations. The analysis further investigates structural invariants preserved under generalized Hadamard products and identifies sufficient conditions for maintaining starlikeness and convexity in meromorphic domains. The findings reveal that positivity constraints significantly enhance the stability of convolution operations, ensuring bounded geometric distortion under extended transformations. The theoretical contributions are supported through comparative synthesis of established results in meromorphic function theory and convolution analysis. The proposed framework provides a unified perspective for future research in geometric function theory, particularly in the development of operator-based functional classes and their applications in complex analytic modeling.