Abstract
The main object of the present paper is to apply the concepts of -derivative by establishing a new subclass of analytic functions connected with symmetric circular domain. Further, we investigate necessary and sufficient conditions for functions belonging to this class. Convex combination, weighted mean, arithmetic mean, growth theorem, and convolution property are also determined.
1. Introduction and Definitions
Quantum calculus or -calculus is a generalization of classical calculus without the notation of limits. The theory of -calculus is established by Jackson, for details see [1, 2]. Due to its numerous applications in various branches of applied sciences and mathematics, for example, physics, operator theory, numerical analysis, and differential equations, attracted researchers to this field. A detailed study on applications of -calculus in operator theory may be found in [3]. The geometric interpretation of -calculus has been recognized through studies on quantum groups. Starlikeness and convexity are two major properties of analytic functions. Ismail et al. [4] investigated the generalized starlike function and certain subclasses close-to-convex functions of -Mittag-Leffler functions were studied by Srivastava and Bansal [5], also the reader is referred to [6–12] for more details.
The foundation of quantum calculus is on one parameter, while the postquantum calculus or simply -calculus is the generalization of -calculus based on two parameters. By setting in -calculus, the -calculus is obtained. The -integer was considered by Chakrabarti and Jagannathan [13], also see the work [14–18]. The idea of -starlike is extended to -stalikeness by Raza et al. [19]. Before we define our new class in this field, we give some basics for a better understanding of the work to follow.
Let represent the family of function that are analytic in the open unit disc having the series expansion
A function of the form (1) is subordinate to function symbolically represented if there occur a Schwarz function with limitation that and then While the convolution of these functions can be defined by
For the -derivative of a function is defined by where see [13] for details.
Also for the -derivative of a function is defined in [2] as
It can easily be seen that for and
where
We note that (for more on this topic one should read [20–22]).
Sakaguchi [23], in year 1956, established the class of starlike functions with respect to symmetrical points denoted by of holomorphic univalent functions in if the below condition is satisfies
Motivated by the work of [19, 23, 24], we now define given below.
Definition 1. Let and then the function is in the class if it satisfies where the symbol “” indicates the well-known subordination.
We note that where
and
Equivalently, a function is in the if and only if
In our main results, in the next section, we evaluate the criteria for functions belonging to this newly defined class. After that, the convex combination property for this class will be discussed. Then utilizing these results, the weighted mean and arithmetic mean properties will be investigated. Further, convolution type results will be discussed in the form of two theorems. At the end of this article, a conclusion and future work will be presented.
2. Main Results
Theorem 2. Let be of the form (1). Then the function if and only if the following inequality holds
Proof. Let us suppose that the first inequality (12) holds. Then to show that we only need to prove the inequality (11). For this consider where we used and this completes the direct part. Conversely, let be of from (1). Then from (11), we have for Since we have Now we choose values of on the real axis such that is real. Upon clearing the denominator in (15) and letting through real values, we obtain the required inequality (12).
Theorem 3. Let and having power series representations Then where
Proof. By Theorem 2, one can write Therefore however, then Hence, the proof is completed.
Theorem 4. If , then their weighted mean is also in , where is defined by
Proof. From (21), one can easily write To prove that it is enough to show that For this, consider where we have used inequality (12). Which completes the proof.
Theorem 5. Let with . Then, their arithmetic mean of is also in the class .
Proof. From (25), we can write Since for every using (12), we have which complete the proof.
Theorem 6. Let Then for where where
Proof. To prove (28), consider as so hence Similarly, Hence complete the proof of (28). Similarly, we can prove (30).
Theorem 7. Let such that with condition then
Proof. Since form (35), we have Then convolution is defined as Since with limitation that Therefore Hence
Theorem 8. Let . Then
Proof. Let Then by definition of subordination, there exists a Schwarz function such that and equivalently, using the relations now (42), becomes which completes the proof.
3. Conclusions
Utilizing the concepts of postquantum calculus, we defined a new subclass of analytic functions associated with symmetric circular domain. For this class, we investigated some useful results such as necessary and sufficient problem, convex combination, weight mean, arithmetic mean, distortion bounds, and convolution property. There are some problems open for researchers such as radii problems, extreme point theorem, analytic criteria, and integral mean of inequality. Moreover, this concept is new and can be extended to meromorphic functions and harmonic functions.
Data Availability
Data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no competing interests.
Authors’ Contributions
All authors jointly worked on the results and they read and approved the final manuscript.
Acknowledgments
The authors would like to express their sincere thanks to the reviewers for their helpful comments and suggestions. This work is supported by the Department of Mathematics and General Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia.