New class of symmetric starlike functions subordinate to the generating function of Gregory coefficients

Function theory research has long struggled with the challenge of deriving sharp estimates for the coefficients of analytic and univalent functions. Researchers have advanced the field by developing and applying a variety of approaches to get these bounds. In the current paper, we apply the technique of subordination, we define the family of symmetric starlike functions which is related to generating function of Gregory coefficients. We provide sharp bounds for the problem concerning the coefficients of the family of symmetric starlike functions connected to the generating function of Gregory coefficients by utilizing the notion of functions with positive real component. These problems include first five sharp coefficient bounds and Fekete-Szego problem along with the Hankel determinant of order three. Additionally, we explore the optimal bounds (sharp bounds) for two important functions, the logarithmic function and the inverse function within the same class of symmetric starlike functions which is related to generating function of Gregory coefficients.