On Janowski Analytic ðp, qÞ-Starlike Functions in Symmetric Circular Domain

Department of Mathematics, Abdul Wali Khan University Mardan, Pakistan Faculty of Science, Department of Mathematics, Al al-Bayt University, Mafraq, Jordan Department of Mathematics and General Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia Department of Medical Research, China Medical University, Taichung 40402, Taiwan Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan


Introduction and Definitions
Quantum calculus or q-calculus is a generalization of classical calculus without the notation of limits. The theory of q -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 q-calculus in operator theory may be found in [3]. The geometric interpretation of q-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 S * , and certain subclasses close-to-convex functions of q-Mittag-Leffler functions were studied by Srivastava and Bansal [5], also the reader is referred to [6][7][8][9][10][11][12] for more details.
The foundation of quantum calculus is on one parameter, while the postquantum calculus or simply ðp, qÞ-calculus is the generalization of q-calculus based on two parameters. By setting p = 1 in ðp, qÞ-calculus, the q-calculus is obtained.
The ðp, qÞ-integer was considered by Chakrabarti and Jagannathan [13], also see the work [14][15][16][17][18]. The idea of q-starlike is extended to ðp, qÞ-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 A represent the family of function f that are analytic in the open unit disc D = fz ∈ ℂ : jzj < 1g having the series expansion A function f ðzÞ of the form (1) is subordinate to function gðzÞ = z + ∑ ∞ n=2 b n z n , symbolically represented f ðzÞ ≺ gðzÞ, if there occur a Schwarz function wðzÞ with limitation that w ð0Þ = 0, and jwðzÞj ≤ 1, then f ðzÞ = gðwðzÞÞ: While the convolution of these functions can be defined by For 0 < q < 1, the q-derivative of a function f is defined by where see [13] for details. Also for 0 < p < q < 1, the ðp, qÞ-derivative of a function f is defined in [2] as It can easily be seen that for n ∈ ℕ ≔ f1, 2, 3, ⋯g and z ∈ D,∂ p,q ð∑ ∞ n=1 a n z n Þ = ∑ ∞ n=1 ½n p,q a n z n−1 , where We note that ∂ 1,q f ðzÞ = ∂ q f ðzÞ (for more on this topic one should read [20][21][22]).
Sakaguchi [23], in year 1956, established the class of starlike functions with respect to symmetrical points denoted by S * s of holomorphic univalent functions in D if the below condition is satisfies Motivated by the work of [19,23,24], we now define S * p,q ðl, m, C, DÞ given below.
where the symbol "≺" indicates the well-known subordination.
We note that S * 1,q ðl, m, C, DÞ = S * q ðl, m, C, DÞ, where and Equivalently, a function f ∈ A is in the S * p,q ðl, m, C, DÞ 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.

Main Results
Theorem 2. Let f ∈ A be of the form (1). Then the function f ∈ S * p,q ðl, m, C, DÞ, if and only if the following inequality holds Proof. Let us suppose that the first inequality (12) holds. Then to show that f ∈ S * p,q ðl, m, C, DÞ, we only need to prove the inequality (11). For this consider where we used and this completes the direct part. Conversely, let f ∈S * p,q ðl, m, C, DÞ be of from (1). Then from (11), we have for z ∈ D, Journal of Function Spaces Since jRezj < jzj < 1, we have Now we choose values of z on the real axis such that ðl − mÞz∂ p,q f ðzÞ/f ðlzÞ − f ðmzÞ is real. Upon clearing the denominator in (15) and letting z ⟶ 1 − through real values, we obtain the required inequality (12).
Theorem 4. If f 1 , f 2 ∈ S * p,q ðl, m, C, DÞ, then their weighted mean ψ k is also in S * p,q ðl, m, C, DÞ, where ψ k is defined by Proof. From (21), one can easily write To prove that ψ k ∈ S * p,q ðl, m, C, DÞ, it is enough to show that For this, consider ; a n j j where we have used inequality (12). Which completes the proof.
is also in the class S * p,q ðl, m, C, DÞ.