Analytic functions with conic domains associated with certain generalized q-integral operator

TL;DR

This paper introduces a new subclass of $k$-uniformly starlike functions defined via a generalized $q$-integral operator, exploring their geometric properties, coefficient conditions, inequalities, and radius problems related to conic domains.

Contribution

It defines a novel subclass of $k$-uniformly starlike functions using a generalized $q$-integral operator and investigates their geometric and analytic properties, including coefficient bounds and inequalities.

Findings

Established $q$-sufficient coefficient conditions.

Derived $q$-Fekete-Szeg"o$ inequalities.

Analyzed radius problems for the new class.

Abstract

In this paper, we define a new subclass of $k$-uniformly starlike functions of order $\gamma,\ (0\leq\gamma<1)$ by using certain generalized $q$-integral operator. We explore geometric interpretation of the functions in this class by connecting it with conic domains. We also investigate $q$-sufficient coefficient condition, $q$-Fekete-Szeg\"{o} inequalities, $q$-Bieberbach-De Branges type coefficient estimates and radius problem for functions in this class. We conclude this paper by introducing an analogous subclass of $k$-uniformly convex functions of order $\gamma$ by using the generalized $q$-integral operator. We omit the results for this new class because they can be directly translated from the corresponding results of our main class.