Coefficient Estimates for a Subclass of Meromorphic Multivalent q-Close-to-Convex Functions

By making use of the concept of basic (or q-) calculus, many subclasses of analytic and symmetric q-starlike functions have been defined and studied from different viewpoints and perspectives. In this article, we introduce a new class of meromorphic multivalent close-to-convex functions with the help of a q-differential operator. Furthermore, we investigate some useful properties such as sufficiency criteria, coefficient estimates, distortion theorem, growth theorem, radius of starlikeness, and radius of convexity for this new subclass.


Introduction, Definitions and Motivation
Calculus without notion of limits is called q-calculus or quantum calculus. This relatively new and advanced field of study became a core of attractions for many well-known mathematicians and physicists due to its various applications in applied mathematics, physics and various engineering areas, see details in [1,2]. The start of this field can be connected with the research of Jackson [3,4], who gave various applications of q-calculus and introduced the q-analogue of derivative and integral, while the most recent and trending work can be viewed in the study of Aral and Gupta [1,2,5]. Later, Aral and Anastassiu [6][7][8][9] gave the q-generalization of complex operators, which are known as q-Picard and q-Gauss-Weierstrass singular integral operators.
The aforementioned works of Srivastava [10,11] motivated a number of mathematicians to give their findings. In the above-cited work by Srivastava [11], many well-known convolution and fractional q-operators were surveyed. For example, in their published article [12], by use of the concept of convolution, a q-analogue of Ruscheweyh differential operator was defined by Kanas and Rȃducanu. More applications of this operator can be seen in the paper [13]. Very recently, by using q-Deference operator, Srivastava et al. [14] studied a certain subclass of analytic function with symmetric points. Several other authors (see, for example, [15][16][17][18][19][20][21][22]) have studied and generalized the classes of symmetric and other q-starlike functions from different viewpoints and perspectives. For some more recent investigation about q-calculus, we may refer the interested reader to [23,24]. In this article, we introduce a new family of meromorphic multivalent functions associated with Janowski domain using a differential operator and study some of its properties.
By the notation A p , we denote the family of all meromorphic multivalent functions that are analytic and invariant (or symmetric) under rotations, in the punctured disc D = {z ∈ C : 0 < |z| < 1} and with the following normalization conditions Furthermore, a function f is said to be in the class MS * Next, a function f is said to be in the class Clearly, we see that MS * p (0) = MS * p , the class of meromorphic p-valent starlike functions. Similarly MK * p (α) denote the class of meromorphic p-valent close-to-convex functions and defined as where g(z) ∈ MS * p . The q-derivative (or q-difference) operator of a function f , where 0 < q < 1, is defined by It can easily be seen that [n, q]a n z n−1 , q l , and [0, q] = 0.
For any non-negative integer n, the q-number shift factorial is defined by [3, q] · · · [n, q], n ∈ N.
Furthermore, the q-generalized Pochhammer symbol for x ∈ R is given by The differential operator D µ,q : A p → A p is defined in [25] as where µ ≥ 0. Now, using Equation (1), one can easily find that In this article, we are essentially motivated by the recently published paper of Hu et al. in Symmetry (see [26]) and some other related works as discussed above (see for example [27][28][29][30][31]), we now define a subclass MK µ,q (p, m, A, B) of A p by using the operator D m µ,q as follows.

Definition 1.
A function f ∈ A p is said to be in the functions class MK µ,q (p, m, A, B), if the following condition is satisfied where g(z) ∈ MS * p , −1 ≤ B < A ≤ 1 and 0 < q < 1, here the notation "≺" stands for the familiar concept of subordinations.
For particular values to parameters m, p, µ, A, B and q, we have some known and new consequences as follows.

1.
If we put m = 0 and let , is the functions class of Janowski-type meromorphic multivalent close-to-convex functions.

2.
If we put A = 1, B = −1 and m = 0 we have MK * p,q , where MK * p,q , is the class of meromorphic multivalent q-close-toconvex functions.

3.
By putting A = 1, B = −1 and m = 0 and let q → 1 − , we have MK * p , where MK * p , is functions class of meromorphic multivalent close-to-convex functions.

4.
If we take A = 1 = p, B = −1 and m = 0 and let q → 1 − , we have MK * , where MK * , denotes the class of meromorphic close-to-convex functions.
The following equivalent condition for a function f ∈ MK µ,q (p, m, A, B) can be verified easily For our main result, we need the following important result.

Lemma 1 ([32]
). An analytic function h(z) having series representation and another function k(z) with the following series representation If h(z) ≺ k(z), then |d n | ≤ |k 1 |, for n ∈ N.

A Set of Main Results
In this section, we give our main results.

Theorem 1.
If a function f ∈ A p has a series representation given by Equation (1), then f ∈ MK µ,q (p, m, A, B) if it satisfies the following inequality it holds true.
Proof. For f to be in the class MK µ,q (p, m, A, B), we only need to prove the inequality (7). For this, we consider If the series representation of g(z) is given by Now, by using Equation (4), and then with the help of Equations (2) and (5), we find, after some simplification, that the above is equal to and using the inequality (11) we find that the above is less than 1. The direct part of the proof of our theorem is now completed.
Conversely, let f ∈ MK µ,q (p, m, A, B) and is given by Equation (1), then from Equation (7), we have for z ∈ D, Now, choose values of z on the real axis so that is real. Upon clearing the denominator in Equation (10) and letting z → 1 − through real values, we obtain Equation (11).
For an analytic case, the result is as follows: is holds true.
In the next theorem, we obtain the coefficient bounds of the functions of this class.
Proof. If f ∈ A and is in the class MK µ,q (p, m, A, B), then it satisfies −q p z∂ q D m µ,q f (z) [p, q]g(z) we now put the series expansions of h(z), g(z) and f (z) in Equation (13) we obtain the required result.
Corollary 2. If f ∈ MK µ,q (1, m, A, B) and is of the form (1). Then, Next, for our defined functions class, we give results related to growth and distortion.

Theorem 3.
If the function f ∈ MK µ,q (p, m, A, B). Then, for |z| = r, we have Proof. Since a p+n z p+n , a p+n r p+n Since for |z| = r < 1 we have r p+n < r p and Similarly, As by Equation (11) which can be written as Now, the required result can easily be obtained if we make use of Equations (16)- (18).

Corollary 3.
For a function f ∈ MK µ,q (p, 1, A, B). Then, for |z| = r, we have Theorem 4. If f ∈ MK µ,q (p, m, A, B) and is of the form (1). Then, for |z| = r Proof. With the help of Equation (2) and using Equation (3), we can easily find that , q] m+1 a p+n z p+n−m .
Since, for the parameters m ≤ n, n ≥ p + 1 and also for |z| = r < 1 implies that r n−m ≤ r p hence, we have Similarly, Now, by using Equation (11), we obtain the following inequality In the next two theorems for the functions class M µ,q (p, m, A, B), the radii of starlikeness and convexity is given. q (p, m, A, B). Then, f ∈ MC p (α) for |z| < r 1 , where Proof. Let f be in the class MK µ,q (p, m, A, B), to prove that f is in the class MC p (α), it will be enough if we show Using Equation (1) in conjunction with some elementary calculations, we have From Equation (11) (1, m, A, B). Then, f ∈ MC p (α) for |z| < r 1 , where q (p, m, A, B). Then, f ∈ MS * p (α) for |z| < r 2 , where

Proof.
We know that f is in the class MS * p (α), if and only if Now, make use of Equation (1) and, after some simple calculations, we have Now, from Equation (11)  The proof of our Theorem is now complete.

Concluding Remarks and Observations
Recently, the basic (or q-) calculus is a center of attraction for many well-known mathematicians, because of its diverse applications in many areas of Mathematics and Physics see for example [11,33]. In our present investigations, we were essentially motivated by the recent research going on in this field of study, and we have first introduced a new class of meromorphic multivalent q-close-to-convex function with the help of a q-differential operator. We next investigate some useful properties such as sufficiency criteria, coefficient estimates, distortion theorem, growth theorem, radius of starlikness and radius of convexity for this new subclass of meromorphic multivalent q-close-to-convex functions.