A Certain Subclass of Multivalent Analytic Functions Defined by the q-Difference Operator Related to the Janowski Functions

A class of p-valent analytic functions is introduced using the q-difference operator and the familiar Janowski functions. Several properties of functions in the class, such as the Fekete–Szegö inequality, coefficient estimates, necessary and sufficient conditions, distortion and growth theorems, radii of convexity and starlikeness, closure theorems and partial sums, are discussed in this paper.


Introduction
The q-calculus is classical calculus without the concept of limit. In recent years, qcalculus has attracted great attention of scholars on account of its applications in the research field of physics and mathematics as, for example, in the study of quantum groups, q-deformed superalgebras, fractals and multifractal measures, optimal control problems and in chaotic dynamical systems. The application of q-calculus involving q-derivatives and q-integrals was initiated by Jackson [1,2]. Later, the q-derivative operator (or q-difference operator) was used to investigate the geometry of q-starlike functions for the first time in [3]. Moreover, Aral [4] and Anastassiou and Gal [5,6] generalized some complex operators which are known as the q-Picad and the q-Gauss-Weierstrass singular integral operators. Recently, Srivastava et al. [7] have written a series of articles [8][9][10] in which they combined the q-difference operator and the Janowski functions to define new function classes and studied their useful properties from different viewpoints. In addition, we choose to refer the interested reader to further developments on q-theory in [11][12][13][14][15][16][17][18]. In paricular, in his recent survey-cum-expository review article, Srivastava [18] exposed the trivial and inconsequential developments in the literature in which known q-results are being routinely translated into the corresponding (p, q)-results by forcing an obviously redundant or superfluous parameter p into the known q-results.
Let A p denote the class of p-valent analytic functions f (z) given by the following Taylor-Maclaurin series expansion: in the open unit disk D = {z : |z| < 1}. For p = 1, we write A := A 1 . In the whole paper, we let N, C and R be the sets of positive integers, complex numbers and real numbers, respectively.
A function f (z) ∈ A p is called to be a p-valent starlike function of order δ and is written as f (z) ∈ S * p (δ), if it satisfies the following inequality: for all z ∈ D.
From (1) and (2), we have the following equivalence: Definition 1. Let 0 < q < 1 and introduce the q-number [λ] q by Definition 2 (See [1,2]). Let 0 < q < 1. Then the q-difference operator D q of a function f (z) is given by One can observe from Definition 2 that provided that f (z) is a differentiable function in a set of C. Furthermore, for f (z) = z p + ∑ ∞ n=1 a p+n z p+n , one can see that A function f (z) belonging to A p is called to be a p-valent q-starlike function of order σ and is written as f (z) ∈ S * p,q (σ), if it meets the condition: Re for all z ∈ D.
A function f (z) belonging to A p is referred to as a p-valent q-convex function of order σ and is written as f (z) ∈ C p,q (σ), if it meets the condition: for all z ∈ D.
From (3) and (4), it is not difficult to verify that For analytic functions h(z) and p(z) (z ∈ D), the function p(z) is said to subordinate to the function h(z) and written p(z) ≺ h(z) (z ∈ D) , if there exists an analytic function w(z)(z ∈ D) with w(0) = 0 and |w(z)| < 1 so that p(z) = h(w(z)) (z ∈ D). Suppose that h(z) is analytic univalent in D, then the following equivalence holds true: In q-calculus concept, we now define the following subclasses of A p in connection with the q-difference operator D q .
A well-known question in GFT is to discuss the functional composed of combinations of certain coefficients of functions. The class A 1 is made up of functions of the form f (z) = z + a 2 z 2 + a 3 z 3 + · · · (z ∈ D). The Fekete-Szegö functional describes a specific relationship between coefficient a 2 and a 3 , i.e., |a 3 − λa 2 2 |, 0 λ 1. Fekete and Szegö [19] found that |a 3 − λa 2 2 | is bounded by 1 + 2 exp(−λ/(1 − λ)) for 0 λ < 1 and f (z) ∈ A 1 and the bound is sharp for every λ. In particular, if we let f (z) ∈ A 1 and λ = 1, then |a 3 − a 2 2 | 1. More recently, Srivastava et al. researched the Fekete-Szegö inequalities for several classes of q-convex and q-starlike functions in [20].
Let Ω denote the family of functions of the form: in D with |w(z)| < 1.
To derive the main results, we recall the following lemmas.
In this paper, we shall study some geometric properties of functions belonging to I p,q (α, A, B) such as Fekete-Szegö inequality, necessary and sufficient conditions, distortion and growth theorems, coefficient estimates, radii of convexity and starlikeness, closure theorems and partial sums.

Main Results
If f (z) ∈ I p,q (α, A, B) and µ ∈ C, then and where we have Therefore, we obtain where By applying Lemmas 1 and 2 to (8) and (9), respectively, we obtain (6) and (7). Now the proof of the Theorem is completed.
Furthermore, let

Theorem 2. Let
Then the function f (z) belongs to I p,q (α, A, B) if and only if Proof of Theorem 2. Assuming that the inequality (10) holds true, we need to show the inequality (5). Now we have which shows that the function f (z) belongs to I p,q (α, A, B).
On the other hand, we let the function f (z) = z p − ∑ ∞ n=1 |a p+n |z p+n ∈ I p,q (α, A, B). Then from (5), one can see that The inequality (11) is correct for z ∈ D. By choosing z = Rez → 1, we obtain (10). Thus, the Theorem is proved.
The result is best possible for f (z) defined as |a p+n |z p+n ∈ I p,q (α, A, B), then, for |z| = r < 1, The bounds are best possible for f (z) given as

Proof of Theorem 3.
Let |a p+n |z p+n . Then, by applying the triangle inequality, we have Since |z| = r < 1, we can see that r p+n r p+1 . Thus, we have and Considering f (z) ∈ I p,q (α, A, B), we know from Theorem 2 that Since the sequence (1 q is increasing regarding n (n 1), we have Hence by transitivity we obtain Substituting (14) into (12) and (13), we obtain the required results. The proof of Theorem 3 is completed.
n=1 |a p+n |z p+n ∈ I p,q (α, A, B), then, for |z| = r < 1, we have The results are best possible for the following function Proof of Theorem 4. Let |a p+n |z p+n . Then, from Definition 2, we can write By applying the triangle inequality, we obtain and Because of the function f (z) belonging to the class I p,q (α, A, B), we find from Theorem 2 that As we know that is an increasing sequence regarding n (n 1), so Thus, by transitivity, we have Now, by putting (17) in (15) and (16), we complete the proof of Theorem 4.
Additionally, let 0 δ < p. If then, for 0 < |z| < r 1 , f (z) is p-valent starlike function of order δ, where Proof of Theorem 5. Let f (z) ∈ I p,q (α, A, B). In order to prove f (z) ∈ S * p (δ), we need to show that The subordination above is equivalent to After some calculations and simplifications, we obtain From the inequality (10), we can obviously find that Inequality (18) can be seen to be true if it satisfies the following inequality: The above inequality indicates that then we obtain the required result. The proof of Theorem 5 is completed. With the aid of the method in the proof of Theorem 5, we also obtain the following theorems for the classes C p (δ), S * p,q (σ) and C p,q (σ), respectively.
The remaining part of the proof is similar to that of (19) and we omit it. Now we complete The proof of Theorem 9.
This indicates that the function H λ (z) belongs to I p,q (α, A, B). Now we complete the proof of Theorem 10.
Proof of Theorem 11. Theorem 11 is proved using the similar arguments as in Theorem 10.