This paper explores the class $\mathcal{C}_{G}$ , consisting of functions g that satisfy a specific subordination relationship with Gregory coefficients in the open unit disk E. By applying certain conditions to related coefficient functionals, we establish sharp estimates for the first five coefficients of these functions. Additionally, we derive bounds for the second and third Hankel determinants of functions in $\mathcal{C}_{G}$ , providing further insight into the class’s properties. Our study also investigates the logarithmic coefficients of $\log \left ( \frac{g(t)}{t}\right ) $ and the inverse coefficients of the inverse functions $(g^{-1})$ within the same class.
