A subclass of meromorphic Janowski-type multivalent q-starlike functions involving a q-differential operator

Keeping in view the latest trends toward quantum calculus, due to its various applications in physics and applied mathematics, we introduce a new subclass of meromorphic multivalent functions in Janowski domain with the help of the q-differential operator. Furthermore, we investigate some useful geometric and algebraic properties of these functions. We discuss sufficiency criteria, distortion bounds, coefficient estimates, radius of starlikeness, radius of convexity, inclusion property, and convex combinations via some examples and, for some particular cases of the parameters defined, show the credibility of these results.


Introduction and motivation
In the classical calculus, if the limit is replaced by familiarizing the parameter q with limitation 0 < q < 1, then the study of such notions is called quantum calculus (q-calculus). This area of study has attracted the researchers due to its applications in various branches of mathematics and physics; for details, see [10,11]. Jackson [19,20] was the first to give some applications of q-calculus and introduced the q-analogues of the derivative and integral.
Using the notion of q-beta functions, Aral and Gupta [10][11][12] established a new q-Baskakov-Durrmeyer-type operator. Furthermore, Aral and Anastassiu [7][8][9] discussed a generalization of complex operators, known as the q-Picard and q-Gauss-Weierstrass singular integral operators. Lately, a q-analogue version of Ruscheweyh-type differential operator was defined by Kanas and Răducanu [21] using the convolution notions and examined some its properties. For more applications of this operator, see [5]. Moreover, Ahuja et al. [2] investigated a q-analogue of Bieberbach-de Branges and Fekete-Szegö theorems for certain families of q-convex and q-close-to-convex functions. Also, Khan et al. [22] studied some families of multivalent q-starlike functions involving higher-order qderivatives. For more recent work related to q-calculus, we refer the reader to [25,38,39].
Let M p denote the class of p-valent meromorphic functions f that are regular (analytic) in the punctured disc D = {ζ ∈ C : 0 < |ζ | < 1} and satisfy the normalization Also, let MS * p (α) and MC p (α) denote the popular classes of meromorphic p-valent starlike and meromorphic p-valent convex functions of order α (0 ≤ α < p), respectively.

Definition 1
For two analytic functions f j (j = 1, 2) in D, the function f 1 is said to be subordinate to the function f 2 , written as if there is a Schwartz function w, analytic in D, such that Further, if the function f 2 is univalent in D, then we have the following equivalence relation: For q ∈ (0, 1), the q-difference operator or q-derivative of a function f is defined by We can observe that for k ∈ N (where N is the set of natural numbers) and ζ ∈ D, The q-number shift factorial for any nonnegative integer k is defined as Furthermore, for x ∈ R, the q-generalized Pochhammer symbol is defined as We now recall the differential operator D μ,q : M p → M p defined by Ahmad et al. [1] by where μ ≥ 0. Now using (1.1), we get We define this operator in such a way that and In the identical way, for m ∈ N , we get From (1.4) and (1.5) after some simplification, we get the identity Now as of q → 1-, the q-differential operator defined in (1.4) reduces to the well-known differential operator defined in [28]. For details on q-analogues of differential operators, we refer the reader to [3,4,27,32]. and Note that by the last inequality it is obvious that in the limit as q → 1-, we have This closed disk is merely in the right-half planem and the class S * q of q-starlike functions turns into the prominent class S * .
where the notation "≺" stands for the familiar notion of subordination. Equivalently, we can write condition (1.9) as Remark 1 First of all, it is easy to see that is the function class introduced and studied by Ali et al. [6]. Secondly, we have where MS * p,q is the class of meromorphic p-valent q-starlike functions. Thirdly, we have where MS * p is the well-known class of meromorphic p-valent starlike functions. Fourthly, we have where MS * is the class of meromorphic starlike functions. The class MS * and other similar classes have been studied by Pommerenke [30] and Clunie and Miller in [13,26], respectively, and by many others.
In this paper, with the help of a certain q-differential operator, we introduce a new subclass of meromorphic multivalent functions involving the Janowski functions. Further-more, we investigate some useful geometric and algebraic properties of these functions.
We discuss sufficiency criteria, distortion bounds, coefficient estimates, radius of starlikness, radius of convexity, inclusion property, and convex combinations via some examples, and for some particular cases of the parameters defined, we show the credibility of these results.

A set of lemmas
In our main results, we use the following important lemmas.

Main results
Proof For f to be in the class M μ,q (p, m, O 1 , O 2 ), we need to show inequality (1.10). For this, consider .
Using inequality (3.1), we can get the direct part of the proof. For the converse part, let f ∈ M μ,q (p, m, O 1 , O 2 ) be given by (1.1). Then from (1.10), for ζ ∈ D, we have Now choose values of ζ on the real axis such that is real. Clearing the denominator in (3.2) and letting ζ → 1through real values, we obtain (3.1).

Example 2 For the function
Thus f ∈ M μ,q (p, m, O 1 , O 2 ), and inequality (3.1) is sharp for this function.

Corollary 1 ([6]) If f is in the class
and has the form (1.1) in univalent form, then The result is sharp for function given by In the following, we discuss the growth and distortion theorems for our new class of functions.
Theorem 3 Let f ∈ M μ,q (p, m, O 1 , O 2 ) be of the form (1.1). Then for |ζ | = r, we have The result is sharp for the function given in (3.3) with k = p + 1.
Proof We have Since r k < r p for r < 1 and k ≥ p + 1, for |ζ | = r < 1, we have Similarly, we have which also can be written as Since r k-m ≤ r p for m ≤ k and k ≥ p + 1, for |ζ | = r < 1, we have Similarly, (3.7) Now by (3.1) we get the inequality We easily observe that . Now using this inequality in (3.6) and (3.7), we obtain the required result.
Corollary 2 If f ∈ MS * p is of the form (1.1), then In the next two theorems, we discuss the radii problems for the functions of the class .
. To prove f ∈ MC p (α), we only need to show Using (1.1), after some simple computation, we get and thus from which we get the desired condition.
Using (1.1), after simplification, we get is of the form However, Now using Lemma 2, we obtain Putting the series expansions of h(ζ ) and f (ζ ) into (3.11), simplifying, and comparing the coefficients at ζ k+p on both sides, we get Now for k = 1, 2, and 3, using the fact that |a p | = 1, we get the required result.
Using the notion of subordination, we get the next result on inclusion property of this class.