The Faber polynomial expansion method and its application to the general coefficient problem for some subclasses of bi-univalent functions associated with a certain $q$-integral operator

In this article, by using the concept of the quantum (or q-) calculus and a general conic domain ${\Omega }_{k,q}$ , we study a new subclass of normalized analytic functions with respect to symmetrical points in an open unit disk. We solve the Fekete-Szegö type problems for this newly-defined subclass of analytic functions. We also discuss some applications of the main results by using a q-Bernardi integral operator.

In this paper, the concepts of quantum (or q-) calculus and conic regions are combined to define a new domain Ω_{k, q, γ} which represents the generalized conic regions. Then by using a certain generalized conic domain Ω_{k, q, γ} we define and investigate a new subclass of normalized analytic functions in open unit disk E. We also investigate a number of useful properties and characteristics of this subclass such as, structural formula, necessary and sufficient condition, coefficient estimates, Feketo-Szego problem, distortion inequalities, closure theorem, and subordination result. We also highlight some known consequences of our main results as corollaries.

In this paper, we introduce a new class of harmonic univalent functions with respect to k-symmetric points by using a newly-defined q-analog of the derivative operator for complex harmonic functions. For this harmonic univalent function class, we derive a sufficient condition, a representation theorem, and a distortion theorem. We also apply a generalized q-Bernardi–Libera–Livingston integral operator to examine the closure properties and coefficient bounds. Furthermore, we highlight some known consequences of our main results. In the concluding part of the article, we have finally reiterated the well-demonstrated fact that the results presented in this article can easily be rewritten as the so-called $(p,q)$-variations by making some straightforward simplifications, and it will be an inconsequential exercise, simply because the additional parameter p is obviously unnecessary.

This paper considers the basic concepts of q-calculus and the principle of subordination. We define a new subclass of q-starlike functions related to the Salagean q-differential operator. For this class, we investigate initial coefficient estimates, Hankel determinants, Toeplitz matrices, and Fekete-Szegö problem. Moreover, we consider the q-Bernardi integral operator to discuss some applications in the form of some results.

In this paper, the concepts of symmetric q-calculus and conic regions are used to define a new domain $\widetilde{{\mathsf{\Omega }}_{k,q,\alpha }},$ which generalizes the symmetric conic domains. By using the domain $\widetilde{{\mathsf{\Omega }}_{k,q,\alpha }}$, we define a new subclass of analytic and q-starlike functions in the open unit disk U and establish some new results for functions of this class. We also investigate a number of useful properties and characteristics of this subclass, such as coefficients estimates, structural formulas, distortion inequalities, necessary and sufficient conditions, closure and subordination results. The proposed approach is also compared with some existing methods to show the reliability and effectiveness of the proposed methods.