Q-MEROMORPHIC CLOSE-TO-CONVEX FUNCTIONS RELATED WITH JANOWSKI FUNCTION

In the present paper, we introduce and explore certain new classes of meromorphic functions related to closed-to-convexity and q-calculus. Such results as coefficient estimates, grow the property and partial sums are derived. It is important to mention that our results are generalization of number of existing results in literature.

In Geometric Function Theory, several subclasses of the meromorphic functions have already been examined and investigated through many perceptions, see ( [9,10, 12, 18, 21, 22]). Ismail et al. [8] were the first to use the q-derivative operator ∆ q in order to study a certain q-analogue of the class T * of starlike functions in U . Certain basic properties of the q-close-to-convex functions were studied by Raghavendar and Swaminathan [28], Aral et al. [2] successfully studied the applications of the q-calculus in operator theory. In fact, they found significant application of the q-calculus mainly in the Geometric Function Theory. Moreover, the generalized q-hypergeometric function was first introduced by Srivastava [26], see also ( [1, 3, 5, 6, 14, 16, 20]).
A function f ∈ 1 is said to be meromorphic starlike of order α defined as: A related class of meromorphic convex function M C (α) is defined as: By M K (α), we mean f ∈ 1 and the class of all close-to-convex functions which satisfies the condition Re ωf The study of operators plays main role in the theory of geometric functions. Many differential and integral operators can be written in terms of convolution of certain holomorphic functions.
b t ω t ∈ 1 and f given in (1). The Convolution (Hadamard product) is denoted by f * g and defined as: A function h analytic in U and of the form A given function Ψ with Ψ(0) = 1 is said to belong to the class S * [A, B] if and only if This class was presented and studied by Janowski [11]. By taking A = 1 and B = −1, we obtain the class P of functions with a positive real part. It is important to mention that Ψ (ω) ∈ S * [A, B] if and only if there exists r ∈ P such that Motivated by the works of Srivastava et al. see ( [7, 17, 19, 23, 25, 27])also see( [4,13,15,24,29]). In this paper, we shall consider new subfamilies of q meromorphic close-to-convex functions with respect to Janowski functions.
Throughout in this paper, we assume unless otherwise mentioned.
Definition 1. (see [9] and [10] ) The q-derivative (q-difference) ∆ q of a function f is defined in a given subset of C by where 0 < q < 1. This implies the following.
The function ∆ q f has Maclaurin's series representation where q ∈ (0, 1) and define the q-number [γ] q by For more details about q-derivatives, we refer the reader to (see [6]).
Definition 2. For f ∈ 1 , let the q-derivative operator (q-difference operator) be defined by Similarly Definition 3. A function f ∈ 1 is said to belong to the class f ∈ T * (q,η) [A, B] if and only if Where g ∈ M S (α), It is easily observed that is the well-known function of meromorphic close-to-convex function.

Coefficient estimates.
Theorem 1. A function f ∈ 1 of the form given by (1) where (10) and Proof. Assuming that (8) holds, it suffices to show that Consider we have (6) and (7) in above equation.
The last expression become This complete the proof of Theorem 2.1. □ with equality for each t, we define the function of the form where Λ(t, B, q), Λ(t, η, A, q) and γ(η, A, B, q) are given by (9), (10) and (11) respectively.

Distortion inequalities.
where equality holds for the function . Then in view of Theorem (2.1), we have where equality holds for the function Proof. Let f ∈ T * (q,η) [A, B]. Then in view of theorem (2.1), we have Differentiate (14) and (15), we get (17) Comparing (16) and (17). We have thus completed the proof of Theorem 2.4. □

Partial sums.
In this section, we examine the ratio of a function of the form (1) to its sequence of partial sums when the coefficients of f are sufficiently small to satisfy condition (8). We will determine sharp lower bounds for Theorem 4. If f of the form (1) satisfies condition (8), then and Re where Proof. In order to prove inequality (18), we set Finally, to prove the inequality in (18), we get The proof of inequality in (18) is now completed. Similarly, we set We have completed the proof of (19), which complete the proof of Theorem 2.5. □ Theorem 5. If f of the form (1) satisfies condition (8), then and Re where κ ν is given by (20). The proof of Theorem 2.6, is similar to that of Theorem 2.5.

Radius of starlikeness.
In the next theorem we find the radius of q-starlikeness for the class T * , Proof. In order to prove above result, we must show that Since the appropriate condition for a function f to be in the class M S (α) is given by The inequality in (25) can be written as: With the aid of (8), inequality (26) is true if Solving (27) for |ω|, we have In view of (28) the proof of our theorem is now completed. □ We call T * (q,η,1) [A, B] the class of q close-to-convex function of Type 1 related with the Janowski functions.

Main Results and Their Demonstration.
We first derive the presence results for the succeeding generalized q-starlike functions: , which are associated with the Janowski functions.
Then, by Definition 2.10, we have By using the triangle inequality and equation (29), we find that The last expression in (30) now implies that f ∈ T * (q.η,2) [A, B], that is, that As we know This last equation now shows that f ∈ T * (q,η,1) [A, B], that is, that . We have thus completed the proof of Theorem 2.11. □
Using Definition 2.9 of the class T * (q,η,2) [A, B] associated with the Janowski functions.
We have thus completed the proof of Theorem 2.12.

Conclusion
In our current investigation, we have presented and studied thoroughly some new subclasses of q meromorphic close-to-convex functions, which is connected with the Janowski functions. Then we discussed some interesting properties and characteristics of these new subclasses, including distortion theorem, radius problem and partial sum. Some special cases have been discussed as applications of our main results. The technique and ideas of this paper may stimulate further research in this dynamic field.
Author Contribution Statements All authors contributed equally to design and implementation of the research. They jointly analyzed the results and wrote the manuscriprt. They read and approved the final manuscript.
Declaration of Competing Interests The authors declare that they have no competing interest.