The Properties of Meromorphic Multivalent q -Starlike Functions in the Janowski Domain

: Many researchers have deﬁned the q -analogous of differential and integral operators for analytic functions using the concept of quantum calculus in the geometric function theory. In this study, we conduct a comprehensive investigation to identify the uses of the S˘al˘agean q - differential operator for meromorphic multivalent functions. Many features of functions that belong to geometrically deﬁned classes have been extensively studied using differential operators based on q -calculus operator theory. In this research, we extended the idea of the q -analogous of the S˘al˘agean differential operator for meromorphic multivalent functions using the fundamental ideas of q -calculus. With the help of this operator, we extend the family of Janowski functions by adding two new subclasses of meromorphic q -starlike and meromorphic multivalent q -starlike functions. We discover signiﬁcant ﬁndings for these new classes, including the radius of starlikeness, partial sums, distortion theorems, and coefﬁcient estimates.


Introduction and Definitions
Currently, researchers have given more attention to the study of q-calculus due to its applications in the fields of physics and mathematics.Before Ismail et al. [1] looked into the q-extension of the class of starlike functions, Jackson [2,3] was the first to consider some applications of q-calculus and define the q-analogue of the derivative and integral.After that, several scholars carried out great studies in geometric function theory (GFT).The q-Mittag-Leffler functions were specifically researched by Srivastava and others, and the authors of [4] also studied the class of q-starlike functions and looked into a third Hankel determinant.A recent survey-cum-expository review conducted by Srivastava [5] is also beneficial for researchers studying these subjects.In this review study, Srivastava [5] discussed applications of the fractional q-derivative operator in geometric function theory and provided some mathematical justifications.In their paper [6], Arif et al. defined and explored the q-derivative operator for multivalent functions, and [7] Zang et al. defined a generalized conic domain and then investigated a novel subclass of q-starlike functions using the definition of subordination and q-calculus operator theory.Recently, many well-known mathematicians have used q-calculus and studied some subclasses of analytic functions and their properties (see, for example, [8,9]).Recently, several authors published a series of studies [10][11][12] focusing on the classes of q-starlike functions connected to Janowski functions [13] from various angles.
The above works serve as the main inspiration for this article, which will first define a new q-analog of the Sȃlȃgean differential operator for meromorphic multivalent functions.
By taking this operator into consideration, a new subclass of meromorphic multivalent functions related to Janowski functions is defined and studied, along with its geometric properties such as sufficient coefficient estimates, partial sums, distortion theorems, and the radius of starlikeness.
The set M(p) contains all meromorphic multivalent functions h that are analytic in the punctured open unit disk U * = {ς : ς ∈ C and 0 < |ς| < 1}, and have the following series of representation: In particular, if p = 1, then In other words, we have A function h ∈ MS * (p, α) is called a meromorphic multivalent starlike functions of the order α (0 In particular, we have MS * (p, 0) = MS * (p).
The basic ideas of these classes started in 1959 when Cluin [14] studied meromorphic schlicht functions.In 1963, Pommerenke [15] defined a class of meromorphic starlike functions and investigated coefficient estimates, and in [16], Royste studied meromorphic starlike multivalent functions for the first time and also found the same type of coefficient problems for the class of meromorphic starlike multivalent functions.In 1970, Miller [17] defined a class of meromorphic convex functions and investigated some generalized coefficient problems and other useful characteristics of meromorphic convex functions.
Cho and Owa [18] examined the partial sum for meromorphic p-valent functions, while Aouf et al. [19] determined a class of meromorphic p-valent functions and investigated the partial sums for meromorphic p-valent functions.In 2004, Srivastava [20] suggested some new classes of meromorphic multivalent functions and described some helpful features of meromorphic functions.Frasin and Maslina [21] investigated positive coefficients for a class of meromorphic functions.
A function ϕ(z) is said to be in the class P[F, K] if it is analytic in U * with ϕ(z) = 1 and Recalling certain definitions of the q-calculus operator theory would be helpful because they are essential for understanding this article.Unless otherwise stated, we assume the following throughout the article: q ∈ (0, 1), −1 ≤ K < F ≤ 1, and p ∈ N. Definition 1 ([22]).The q-number [ζ] q is defined by and for any non-negative integer i, we have Definition 2 ([2,3]).Let A be the set of all analytic functions h in the open unit disk U = {ς : ς ∈ C and |ς| < 1} and have the following series representation.
The q-derivative (or q-difference) D q is defined by , (ς = 0). (3) For h ∈ A, and from Equation (3), we have Definition 3 ([23]).The Sȃlȃgean q-differential operator for h ∈ A is defined by Mahmood et al. extended the concept of the q-difference operator for h ∈ M and constructed a new subclass MS * q [F, K] of meromorphic functions using the analogue of Definition 2: Definition 4 ( [24]).For h ∈ M, the q-derivative (or q-difference) D q is defined by For h ∈ M, and from Equation ( 4), we have Using Equations (1) and (4), we extend the idea of the Sȃlȃgean q-differential operator for meromorphic functions as follows: Definition 5. Let h ∈ M.Then, the Sȃlȃgean q-differential operator for a meromorphic function is given by Definition 6.Let h be a meromorphic multivalent function given by Equation (1).Then, the Sȃlȃgean q-differential operator is given by Remark 1.By taking p = 1 in Equation ( 7), then we have the Sȃlȃgean q-differential operator for h ∈ M, which is given by Equation ( 6).
In the case of the recently introduced Sȃlȃgean q-differential operator h ∈ M, we introduce a novel subclass of meromorphic q-starlike functions connected to Janowski functions.
We provide a novel subclass of meromorphic q-starlike functions connected to Janowski functions in the context of the recently introduced Sȃlȃgean q-differential operator h ∈ M(p).
Remark 2. It can be easily observed that which was introduced and studied by Mahmood et al. [24].
which was introduced and studied by Ali et al. [25].
where MS * denotes the class of meromorphic starlike function.
The sufficient condition for h ∈ MS * q,p [m, F, K] is examined in Theorem 1, which can be used as a supporting result to research further findings.We will also look into the relationship between a function h of the type (Equation (1)) and the partial sums of its series when the coefficients are sufficiently small.

Sufficient Condition
Theorem 1.If a function h ∈ M(p) of the form in Equation (1) satisfies the following condition, then h ∈ MS * q,p [m, F, K]: Proof.Supposing that Equation (9) is satisfied, then it is enough to prove that Now, we have The inequality in Equation (10 Thus, this completes the proof of Theorem 1. Corollary 1.If a function h ∈ M(p) of the form in Equation (1) belongs to the class MS * q,p [m, F, K], then This equality will satisfy the function Theorem 2. If a function h ∈ M of the form given in Equation ( 2) satisfies the following condition, then h ∈ MS * q [m, F, K]: By taking p = 1 and m = 1 in Theorem 1, then we have following known result, which was introduced in [24]: Corollary 2 ([24]).If a function h ∈ M of the form in Equation (1) satisfies the following condition, then h ∈ MS * q [F, K]: where

Distortion Inequalities
This equality holds for the function Similarly, we have Thus, this completes the proof of Theorem 3.

Theorem 4. If a function h of the form in Equation
(2) belongs to the class MS * q [m, F, K], then This equality holds for the function Proof.Here, we omit the proof of Theorem 4. It is similar to that of the proof of Theorem 3.
For p = 1 and m = 1 in Theorem 3, then we have the known corollary given in [24]: This equality holds for the function , (|ς| = r).
Proof.Here, we omit the proof of Theorem 5. Its proof is similar to that of the proof Theorem 3.
For p = 1 and m = 1, we have a known corollary introduced in [24]: 2.3.Partial Sums for the Function Class MS * q,p [m, F, K] In this section, we study the ratio of a function of the form in Equation (1) to its sequence of partial sums when the coefficients of h are sufficiently small to satisfy the condition in Equation (9).We will investigate the sharp lower bounds for and Re S m q,p h k (ς) S m q,p h(ς) .
The sequence of partial sums of h k is denoted by and where Proof.For the proof of the inequality in Equation (13), we set then after some simplification, we obtain .
We find that Finally, to prove Equation (13), it is enough to show that the L.H.S. of Equation ( 16) is bounded above by Hence, the proof of the inequality in Equation (13) is complete.
For the proof of the inequality in Equation (14), we fix where The inequality in Equation ( 18) Finally, we can find that the L.H.S. in Equation ( 19) is bounded above by and thus we have completed the inequality in Equation (14).Hence, the proof of Theorem 6 is complete.

Theorem 7.
If h ∈ M(p) of the form in Equation (1) satisfies the condition in Equation (9), then Re S m q,p h(ς) S m q,p h p,k (ς) and Re S m q,p h p,k (ς) S m q,p h(ς) where χ k+p is given by Equation (15).
Proof.Here we omit the proof of Theorem 7. It is similar to that of Theorem 6.

Partial Sums for the Function Class MS *
q [m, F, K] We will study the ratio of a function of the form in Equation (1) to its sequence of partial sums when the coefficients of h are sufficiently small to satisfy the condition in Equation (9).We will investigate the sharp lower bounds for and Re S m q h k (ς) S m q h(ς) .
The sequence of partial sums of h k is denoted by Theorem 8.If we let h ∈ M of the form in Equation (2) satisfy the condition in Equation (12), where Proof.Here, we omit the proof for Theorem 8.It is similar to that of the proof for Theorem 7.
Theorem 9.If we let h ∈ M of the form in Equation (2) satisfy the condition in Equation ( 12), then Re S m q h(ς) S m q h k (ς) where χ k+1 is given by Equation (20).
Proof.Here, we omit the proof for Theorem 9.It is similar to that of the proof for Theorem 6.

Radius of Starlikeness
In the next result, we obtain the radius of starlikeness for the class MS * q,p [m, F, K]: Theorem 10.Let the function h with Equation (1) belong to the class MS * is positive, then the function h is p-valently meromorphically starlike to the order α in |ς| ≤ r.Proof.To prove the above result, we have to show that From the above inequality, we have Now, we can set the inequality in Equation (22) as follows: With the help of Equation (9), the inequality in Equation ( 23) By solving Equation (24) for |ς|, we have This completes the proof.

Discussion
This section serves as an introduction to the conclusions section, we will specifically highlight the relevance of our primary findings and their applications.With a primary motive to consolidate the study of the famous convex function with starlike and convex functions, Govindaraj and Sivasubramanian in [23] involved the q-calculus operator and defined the Sȃlȃgean q-differential operator for analytic functions.However, the meromorphic functions and meromorphic multivalent functions could not be defined with the other geometrically defined subclasses of M and M(p) using the same meromorphic q-analogue of the Sȃlȃgean differential operator.For the functions in M and M(p), we smartly established a Sȃlȃgean q-differential operator in this study so that normalization could be preserved.
When considering the Sȃlȃgean q-differential operator for h ∈ M, the family of functions MS * q [m, F, K] (see Definition 7) is defined to include q-starlike functions, and the other family of functions MS * q,p [m, F, K] (see Definition 8) is defined by using the Sȃlȃgean q-differential operator for h ∈ M(p).
Another notable difference from earlier research is the fact that we found criteria for the classes of MS * q [m, F, K] and MS * q,p [m, F, K] that are more broadly applicable.Hence, if we let p = 1 and m = 1, then some of our results in Section 2 will reduce to results for the class of q-starlike functions introduced in [24].The approach used by different authors in this paper in arriving at solutions to the challenges of the classes is the same.However, several novel and traditional results can be obtained as a special case of our main findings.

Conclusions
The extension and unification of various well-known classes of functions were the main objectives of this paper.In this article, we used the q-calculus operator theory, introduced the Sȃlȃgean q-differential operator for meromorphic multivalent functions and defined two new subclasses of meromorphic multivalent functions in the Janowski domain.We investigated some interesting properties, such as coefficient estimates, partial sums, distortion theorems, and the radius of starlikeness.The technique and ideas of this paper may stimulate further research in the theory of multivalent meromorphic functions and further generalized classes of meromorphic functions can be defined and investigated for several other useful properties such as Hankal determinants, Feketo-Sezego problems, coefficient inequalities, growth problems, and many others.