Derivation of a New Differential Operator on Bi-Bounded Turning Functions

In this research, a novel generalized differential operator will be used to derive a new subclass of bi-univalent functions, specifically those that are subordinated to bounded turning functions with Gregory coefficients. This subclass is expected to provide precise estimates for various coefficient problems, such as coefficient estimates, the second Hankel determinant, and the Fekete-Szegő inequality. The identification of an extremal function will be crucial in establishing the validity and sharpness of the derived bounds. The introduction of this new subclass is expected to bridge the gap in the current literature on bi-univalent functions associated with Gregory coefficients and generalized bounded turning functions. These functions and the operator hold significant relevance across diverse technological and scientific disciplines, including, but not limited to, electromagnetic theory, plasma physics, mathematical biology, dynamical systems and optics.