Vandermonde Determinant for the Generalized Bounded Turning Functions Associated with Gregory Coefficients

This paper explores the class ${G_{\rm{G}}}\left( {\alpha ,\delta } \right)$ of analytic functions, which is associated with generalized bounded turning and the generating functions of Gregory coefficients. By using bounds on certain coefficient functionals for functions with a positive real part, we obtain initial Taylor coefficient bounds and logarithmic coefficient bounds of functions and inverse functions within this class. Consequently, we establish upper bounds of the second-order for the Vandermonde determinant, where the entries are Taylor coefficients and logarithmic coefficients of functions and inverse functions. Additionally, we highlight several interesting implications of these results, contributing new insights to this generalized class.