Applications of a New q-Difference Operator in Janowski-Type Meromorphic Convex Functions

Bakhtiar Ahmad , Muhammad Ghaffar Khan , Mohamed Kamal Aouf, Wali Khan Mashwani , Zabidin Salleh , and Huo Tang Govt. Degree College Mardan, 23200 Mardan, Pakistan Institute of Numerical Sciences, Kohat University of Science and Technology, Kohat, Pakistan Department of Mathematics, Faculty of Science, Mansoura 35516, Egypt Department of Mathematics, Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu, 21030 Kuala Nerus, Terengganu, Malaysia School of Mathematics and Computer Sciences, Chifeng University, Chifeng 024000, China


Introduction and Definitions
Let A p denote the family of all meromorphic p-valent functions f that are analytic in the punctured disc D = fz ∈ ℂ : 0 < jzj < 1g and obeying the normalization f z ð Þ = 1 z p + 〠 ∞ n=p+1 a n z n , z ∈ D: Also, let MC p ðαÞ denote the well-known family of meromorphic p-valent convex functions of order αð0 ≤ α < pÞ and defined as For 0 < q < 1, the q-difference operator or q-derivative of a function f is defined by It can easily be seen that for n ∈ ℕ, where ℕ stands for the set of natural numbers and z ∈ D, n, q ½ a n z n−1 , where n, q ½ = 1 − q n 1 − q = 1 + 〠 n l=1 q l , 0, q ½ = 0: ð5Þ For any nonnegative integer n, the q-number shift factorial is defined by n, q ½ ! = 1, n = 0, 1, q ½ 2, q ½ 3, q ½ ⋯ n, q ½ , n ∈ ℕ: ( ð6Þ Also, the q-generalized Pochhammer symbol for x ∈ ℝ is given by x, q ½ n = 1, n = 0, In (3), if q → 1 − , then this operator becomes the conventional derivative in the classical calculus, so the limits can be generalized by introducing the parameter q, with condition 0 < q < 1, and all such concepts, which have been developed thus, are known as quantum calculus (q -calculus). Many physical phenomena are better explained using this generalized operator, and as a result, this field attracted a lot of the researchers due to its various applications in the branches of mathematics and physics (see details in [1,2]). Jackson [3,4] was the pioneer of this field, who gave some applications of q-calculus and introduced the q-analogues of derivative and integral. Aral and Gupta [1,2,5] defined an operator, which is known as q -Baskakov Durrmeyer operator by using q-beta functions. The generalization of complex operators known as q -Picard and q-Gauss-Weierstrass singular integral operators was discussed by Aral and Anastassiu in [6][7][8]. Later, Kanas and Rȃducanu [9] introduced the q-analogue of a Ruscheweyh differential operator and studied its various properties. More applications of this operator can be seen in the paper [10]. Huda and Darus [11] utilized the q -analogue of a Liu-Srivastava operator and defined an integral operator. In somewhat similar way, Mohammed and Darus [12] introduced a generalized operator along with investigating a class of functions relating to q -hypergeometric functions. Later, Seoudy [13] estimated coefficient bounds for some q-starlike and q-convex functions of complex order. Recently, Arif and Ahmad defined a new q-differential operator for meromorphic multivalent functions and investigated classes related to q-meromorphic starlike and convex functions in their articles [14,15].
In this article, we introduce a new q-differential operator for meromorphic functions and use this operator to define and study some properties of a new family of meromorphic multivalent functions associated with circular domain.
We now define the differential operator D μ,q : A p → A p by where μ ≥ 0: Using (1), we can easily obtain Þ a n z n : We take In a similar way, for m ∈ ℕ ∪ f0g, we get Þ m a n z n : ð11Þ From (8) and (11), we get the following identity: We now define a subfamily MC μ,q ðp, m, A, BÞ of A p by using the operator D m μ,q as follows.
Here, the relation symbol "≺" is used for the subordinations.
We see that for particular values of p, m, A, B, μ, and q, we get some of the well-known classes few of which are listed below: (1) For m = 0 and q → 1 − , we get the class of meromorphic multivalent convex functions associated with Janowski functions denoted by MC * p ½A, B (2) For A = 1,B = −1, and m = 0, we get MC * p,q , the class of meromorphic multivalent convex functions in q -analogue It can easily be verified that a function f ∈ A p will be in the class MC μ,q ðp, m, A, BÞ, if and only if The following lemma is used in our main results.
Lemma 2 (see [16]). Let hðzÞ be analytic in D and have the form and kðzÞ is analytic and convex in D with series representation So if hðzÞ ≺ kðzÞ, then jd n j ≤ jk 1 j, for n ∈ ℕ = f1, 2, ⋯g:

Main Results and Their Consequences
In this section, we start with sufficiency criteria for this newly defined class and then, we give the coefficient estimates for the functions belonging to this class. The following lemma is proved which will be used in this section.

Lemma 3.
Suppose that the sequence fA p+n g ∞ n=1 is defined by Then, Proof. From (17), we have Thus, we obtain that From (20), we find that Thus, In conjunction with (17), we complete the proof of Lemma 3.

Journal of Function Spaces
Proof. For f ∈MC μ,q ðp, m, A, BÞ, we need to prove the inequality (14). For this, consider By using (8) and with the help of (3) and (11), Now, if we use the inequality (23), then and this completes the direct part of the proof.
Conversely, let f ∈ MC μ,q ðp, m, A, BÞ and be of the form (1); then, from (14), we have for z ∈ D, Re

Journal of Function Spaces
Now if the values of z are chosen on the real axis, then ðq p ∂ q ðz∂ q D m μ,q f ðzÞÞÞ/ð½p, q∂ q D m μ,q f ðzÞÞ is real. Using some calculations in the inequality (28) and letting z → 1 − through real values, we finally get (23).
Theorem 5. If f ∈ MC μ,q ðp, m, A, BÞ and is of the form (1), then where ϕ n ð Þ ≔ q p p + n, q Proof. If f ∈ A p is in the class MC μ,q ðp, m, A, BÞ, then it satisfies Now, let Since so hðzÞ is in the class P with its representation which is given by Now, But 1 + Az Now, using Lemma 2, we get now putting the series expansions of hðzÞ and f ðzÞ in (34), simplifying and comparing the coefficients of z n+p on both sides which implies that Now, by taking absolute on both sides with using the triangle inequality and using (39), we obtain Using the notation (31) and (32) implies that Now, we define the sequence fA p+n g ∞ n=1 as follows: In order to prove that we use the principle of mathematical induction. It is easy to verify that

Journal of Function Spaces
Thus, assuming that we find from (44) and (48) that Therefore, by the principle of mathematical induction, we have By means of Lemma 3 and (45), we know that Combining (50) and (51), we readily get the coefficient estimates (30).