Some New Applications of the Quantum Calculus for New Families of Sigmoid Activation Bi-univalent Functions Connected to Horadam Polynomials

The study of q-calculus is becoming increasingly prominent in the field of geometric function theory, reflecting a growing interest in its applications. In this research work, we first develop a new type of modified Sigmoid-Salagean q-differential operator in the open unit disk D, utilizing the concepts of quantum calculus and the Sigmoid activation function. Using this newly defined q analogous differential operator and Horadam polynomials, we introduce new subclasses of bi-univalent functions in D. We determine upper bounds on initial coefficients, as well as the Fekete-Szeg ̈o problems, for functions belonging to these special families. Additionally, we discuss several interesting consequences related to the findings presented in this study.