Coefficient estimates for some classes of functions associated with \(q\)-function theory

TL;DR

This paper explores coefficient estimates and classical function theory problems for classes of q-analog functions, extending known results and proposing new conjectures in q-function theory.

Contribution

It establishes the Herglotz representation, addresses Bieberbach, Fekete-Szegö, and Hankel determinant problems for q-classes, and introduces conjectures for q-starlike functions.

Findings

Derived Herglotz representation theorem for q-functions

Solved Bieberbach type problem for q-convex functions

Proposed conjectures for q-starlike functions of order α

Abstract

In this paper, for every $q\in(0,1)$, we obtain the Herglotz representation theorem and discuss the Bieberbach type problem for the class of $q$-convex functions of order $\alpha, 0\le\alpha<1$. In addition, we discuss the Fekete-szeg\"o problem and the Hankel determinant problem for the class of $q$-starlike functions, leading to couple of conjectures for the class of $q$-starlike functions of order $\alpha, 0\le\alpha<1$.