On q-analogue of meromorphic multivalent functions in lemniscate of Bernoulli domain

1 Abdul Wali Khan University Mardan, 23200 Mardan, Pakistan 2 Institute of Numerical Sciences, Kohat University of Science and Technology, Kohat, Pakistan 3 Faculty of Science, Department of Mathematics, AL AL-Bayt University Mafraq, Jordon 4 Department of Mathematics, Faculty of Science, Mansoura 35516, Egypt 5 Department of Mathematics and General Sciences, Prince Sultan University, Riyadh, Saudi Arabia 6 Department of Medical Research, China Medical University Taichung 40402, Taiwan 7 Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan


Background of the research
The q-calculus (Quantum Calculus) is a branch of mathematics related to calculus in which the concept of limit is replaced by the parameter q. This field of study has motivated the researchers in the recent past with its numerous applications in applied sciences like Physics and Mathematics, e.g., optimal control problems, the field of ordinary fractional calculus, q-transform analysis, q-difference and q-integral equations. The applications of q-generalization in special functions and quantum physics are of high value which makes the study pertinent and interesting in these fields. While the q-difference operator has a vital importance in the theory of special functions and quantum theory, number theory, statistical mechanics, etc. The q-generalization of the concepts of differentiation and integrations were introduced and studied by Jackson [1]. Similarly, Aral and Gupta [2,3] used some what similar concept by introducing the q-analogue of operator of Baskakov Durrmeyer by using q-beta function. Later, Aral and Anastassiu et. al. in [4,5] generalized some complex operators, q-Gauss-Weierstrass singular integral and q-Picard operators. For more details on the topic one can see, for example [6][7][8][9][10][11][12][13][14][15][16][17]. Some of latest inovations in the field can be seen in the work of Arif et al. [18] in which they investigated the q-generalization of Harmonic starlike functions. While Srivastava with his co-authors in [19,20] investigated some general families in q-analogue related to Janowski functions and obatained some interesting results. Later, Shafiq et al. [21] extended this idea of generalization to close to convex functions. Recently, more research seem to have diversified this field with the introduction of operator theory. Some of the details of such work can be seen in the work of Shi and co-authors [22]. Also some new domains have been explored such as Sine domain in the recent work [23]. Motivated from the discussion above we utilize the concepts of q-calculus and introduce a subclass of p-valent meromorphic functions and investigate some of their nice geometric properties.

Preliminaries
Before going into our main results we give some basic concepts relating to our work. Let M p represents the class of meromorphic multivalent functions which are analytic in Let f (z) and g (z) be analytic in D = {z ∈ C : |z| < 1}. Then the function f (z) is subordinated to g (z) in D, written as f (z) ≺ g (z), z ∈ D, if there exist a Schwarz function ω(z) such that f (z) = g (ω(z)), where ω(z) is analytic in D, with w(0) = 0 and w(z) < 1, z ∈ D.
Let P denote the class of analytic function l(z) normalized by such that Re(z) > 0. We now consider a class of functions in the domain of lemniscate of Bernoulli. All functions l(z) will belong to such a class if it satisfy; These functions lie in the right-half of the lemniscate of Bernoulli and with this geometrical representation is the reason behind this name. With simple calculations the above can be written as Similarly SL * , in parallel comparison to starlike functions, for analytic functions is where A represents the class of analytic functions and z ∈ D. Alternatively Sokol and Stankiewicz [24] introduced this alongwith some properties. Further study on this was made by different authors in [25][26][27]. Upper bounds for the coefficients of this class are evaluated in [28]. An important problem in the field of analytic functions is to study a functional a 3 − va 2 2 called the Fekete-Szegö functional. Where a 2 and a 3 the coefficients of the original function with a parameter v over which the extremal value of the functional is evaluated. The problem of obtaining the upper bound of this functional for subclasses of normalized functions is called the Fekete-Szegö problem or inequality. M. Fekete and G. Szegö [29], were the first to estimate this classical functional for the class S. While Pfluger [30] utilized Jenkin's method to prove that this result holds for complex µ such that Re µ 1−µ ≥ 0. For other related material on the topic reader is reffered to [31][32][33].
Similarly the class of Janowski functions is defined for the function equivalently the functions of this class satisfies more details on Janowski functions can be seen in [34]. The q-derivative, also known as the q-difference operator, for a function is with z 0 and 0 < q < 1. With simple calculations for n ∈ N and z ∈ D * , one can see that [n] q a n z n−1 , Now we define our new class and we discuss the problem of Fekete-Szegö for this class. Some geometric properties of this class related to subordinations are discussed in connection with Janowski functions.

The class MSL * p,q
We introduce MSL * p,q , a family of meromorphic multivalent functions associated with the domain of lemniscate of Bernoulli in q-analogue as: If f (z) ∈ M p , then it will be in the class MSL * p,q if the following holds In this research article we investigate some properties of meromorphic multivalent functions in association with lemniscate of Bernoulli in q-analogue. The important inequality of Fekete-Szegö is evaluated in the beginning of main results. Then we evaluate some bounds of ξ which associate 1 + ξ [ p ] q ( f (z)) 2 and 1 + ξ Utilizing these theorems along with some conditions we prove that a function may be a member of MSL * p,q .

Sets of Lemma
The following Lemmas are important as they help in our main results.

Main results
In this section we start with Fekete-Szegö problem in the first two theorems. Then some important results relating to subordination are proved using q-Jack's Lemma and with the help of these results the functions are shown to be in the class of MSL * p,q in the form of some corollaries.

Theorem
Let f ∈ MSL * p,q and are of the form (1.1) , then Proof. Let f ∈ MSL * p,q , then we have Thus for we have l(z) is in P and So from (3.1) , we get By comparing of coefficients of z k+p , we get 3) with µ is defined as above.

Proof. From (3.3) and (3.4) it follows that
using above notations, we get where v ∈ [0, 1] using Lemma 2.2, we get the second inequality.

Theorem
and if holds, then z p f (z) ≺ √ 1 + z.
Proof. Suppose that From (3.10) it follows that , Using the condition (3.9) ,we have
If at some z = z 0 , ω (z) attains its maximum value i.e. |ω (z 0 )| = 1. Then using Lemma 2.3, we have which shows that the increasing nature of φ (m) and so its minimum value will be at m = 1 thus which contradicts (3.12) , therefore |ω (z)| < 1 and so the desired result.

Theorem
Proof. Here we define a function using some simplification we obtain that and so . Now if ω (z) attains, at some z = z 0 , its maximum value which is |ω (z 0 )| = 1. Then by Lemma 2.3, with m ≥ 1 we have, ω (z 0 ) = e iθ and z 0 D q ω (z 0 ) = mω (z 0 ) , with θ ∈ [−π, π] so this shows φ (m) an increasing function which implies that at m = 1 it will have its minimum value and this is a contradiction as J (z) ≺ 1+Az 1+Bz , thus |ω (z)| < 1 and so the result.
Now if z p f (z) = 1 + ω (z), with simple calculations we can easily obtain and so