Functional inequalities for starlike and convex functions associated with $g$-derivative operator

Abstract

In this paper we first introduce a new class of derivative operators, termed $g$-derivative operators, which unifies the $q$-derivative, $(p,q)$-derivative and $(\hat{I}\pm ,{\hat{I}}^2,{\hat{I}}^3)$-derivative operators, even classical derivative in the literature. Based on this generalized framework, we define two novel function classes, specifically $g$-starlike functions and $g$-convex functions. Further, we employ the subordination principle of analytic functions to conduct the coefficient estimations for such function classes, and subsequently derive the bounds for the corresponding Fekete-Szegö inequality, Toeplitz determinants and Hankel determinants.