Communications of the
Korean Mathematical Society

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024



Commun. Korean Math. Soc. 2023; 38(4): 1111-1126

Online first article October 25, 2023      Printed October 31, 2023

Copyright © The Korean Mathematical Society.

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

OM P. AHUJA, Asena \c{C}etinkaya, NAVEEN KUMAR JAIN

Kent State University; \.{I}stanbul K\"{u}lt\"{u}r University; Aryabhatta College


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.

Keywords: Quantum calculus, $q$-derivative operator, $q$-difference operator, $q$-gamma function, $q$-integral operator, conic domains, $k$-uniformly starlike functions of order gamma, coefficient estimates

MSC numbers: 30C45, 30C50, 30C80