Communications of the
Korean Mathematical Society CKMS

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