A study of multivalent q-starlike functions connected with circular domain

In the present article, our aim is to examine some useful problems including the convolution problem, sufficiency criteria, coefficient estimates and Fekete-Szego type inequalities for a new subfamily of analytic and multivalent functions associated with circular domain. In addition, we also define and study a Bernardi integral operator in its $q$-extension for multivalent functions.


Introduction
The study of the q-extension of calculus or the q-analysis attracted and motivated many researchers becauuse of its applications in different parts of mathematical sciences. Jackson [15,14]) was one of the main contributer among all the mathematicians who initiated and established the theory of q-calculus. As an interesting sequel to [13], in which use was made of the q-derivative operator for the first time for studying the geometry of q-starlike functions, a firm footing of the usage of the q-calculus in the context of Geometric Function Theory was actually provided and the basic (or q-) hypergeometric functions were first used in Geometric Function Theory in a book chapter by Srivastava (see, for details, [29, pp. 347 et seq.]). The theory of q-starlike functions was later extended to various families of q-starlike functions by (for example) Agrawal and Sahoo [1] (see also the recent investigations on this subject by Srivastava et al. [32,33,34,35,36,37]). Motivated by these q-developments in Geometric Function Theory, many authors such as like Srivastava and Bansal [29] were added their contributions in this direction which has made this research area much more attractive.
In 2014, Kanas and Rȃducanu [17] used the familiar Hadamad product to define a q-extension of the Ruscheweyh operator and discussed important applications of this operator. Moreover, the extensive study of this q-Ruscheweyh operator was further made by Mohammad and Darus [5] and Mahmood and Sokó l [20]. Recently, a new idea was presented by Darus [21] and introduced a new differential operator called generalized q-differential operator with the help of q-hypergeometric functions where they studied some useful applications of this operator. For the recent extension of different operators in q-analogue, see the references [2,9,8]. The operator defined in [17] was extended further for multivalent functions by Arif et al. [10] in which they investigated its important applications. The aim of this paper is to define a family of multivalent q-starlike functions associated with circular domain and to study some of its useful properties.
Let A p (p ∈ N = {0, 1, 2, . . .}) contains all multivalent functions say f that are holomorphic or analytic in a subset D = {z : |z| < 1} of a complex plane C and having the series form: (1.1) For two analytic functions f and g in D, then f is subordinate to g, symbolically presented as f ≺ g or f (z) ≺ g (z) , if we can find an analytic function w with the properties w (0) = 0 & |w (z)| < 1 such that f (z) = g(w(z)) (z ∈ D) . Also, if g is univalent in D, then we have: For given q ∈ (0, 1), the derivative in q-analogue of f is given by Making (1.1) and (1.2) , we easily get that for n ∈ N and z ∈ D For n ∈ Z * := Z\ {−1, −2, . . .} , the q-number shift factorial is given as: Also, with x > 0, the q-analogue of the Pochhammer symbol has the form: and, for x > 0, the Gamma function in q-analogue is presented as We now consider a function Also we note that Now when q → 1 − , the operator defined in (1.6) becomes to the familiar differential operator investigated in [12] and further, setting p = 1, we get the most familiar operator known as Ruscheweyh operator [24] (see also [3,22]). Also, for different types of operators in q-analogue, see the works [2,4,6,7,9,21].

A Set of Lemmas
.

1)
and These results are best possible.
For the first and second part see the reference [18] and [28] respectively.

The Main Results and Their Consequences
Theorem 3.1. Let f ∈ A p has the series form (1.1) and satisfing the inequality given by Proof. To show f ∈ S * p (q, µ, A, B) , we just need to show the relation (1.8). For this we consider Using (1.6), and then with the help of (3.1) and (1. where we have used the inequality (3.1) and this completes the proof.
Varying the parameters µ, b, A and B in the last Theorem, we get the following known results discussed earlier in [26]. Then the function f ∈ S * q [A, B]. By choosing q → 1 − in the last corollary, we get the known result proved by Ahuja [3] and furthermore for A = 1 − α and B = −1, we obtain the result for the family S * (ξ) which was proved by Silverman [27].
Proof. If f ∈ S * p (q, µ, A, B) , then by definition we have Let us put Then by Lemma 2.1, we get |d n | ≦ A − B.
(3.6) Now, from (3.5) and (1.6), we can write Equating coefficients of z n+p on both sides Taking absolute on both sides and then using (3.6) , we have and this further implies where ψ n is given by (3.4) . So for n = 1, we have from (3.8) and this shows that (3.2) holds for n = 1. To prove (3.3) we apply mathematical induction. Therefore for n = 2, we have from (3.2)

using (3.2), we have
which clearly shows that (3.3) holds for n = 2. Let us assume that (3.3) is true for n ≦ m − 1, that is, Consider and this implies that the given result is true for n = m. Hence, using mathematical induction, we achived the inequality (3.3) .
Theorem 3.4. Let f ∈ S * p (q, µ, A, B) and be given by where υ is given by Proof. Let f ∈ S * p (q, µ, A, B) and consider the right hand side of (3.5) we have and after simple computations, we can rewrite Now, left hand side of (3.5), we have From (3.10) and (3.11) , we have

Now consider
using Lemma 2.2, we have where υ is given by This completes the proof.
Proof. From the relations (3.10) and (3.11) , we have equivalently, we have where we have used (2.1) and (2.2) .This completes the proof.
Theorem 3.6. Let f ∈ A p be given by (1.1) . Then the function f is in the class S * p (q, µ, A, B) , if and only if 12) for all and also for N = 0, L = 1.

Proof.
Since the function f ∈ S * p (q, µ, A, B) is analytic in D, it implies that L µ+p−1 q f (z) = 0 for all z ∈ D * = D\{0}; that is and this is equivalent to (3.12) for N = 0 and L = 1. From (1.7) , according to the definition of the subordination, there exists an analytic function w with the property that w (0) = 0 and |w(z)| < 1 such that which is equalent for z ∈ D, 0 ≦ θ < 2π 14) and further written in more simplified form Now using the following convolution properties in (3.15) and then simple computation gives which is the required direct part.
Assume that (3.1) holds true for L θ − 1 = N θ = 0, it follows that Thus the function h (z) = is analytic in D and h (0) = 1. Since we have shown that (3.15) and (3.1) are equivalent, therefore we have We now define an integral operator for the function f ∈ A p as follows; Definition 3.7. Let f ∈ A p . Then L : A p → A p is called the q-analogue of Benardi integral operator for multivalent functions defined by L (f ) = F η,p with η > −p, where F η,p is given by We easily obtain that the series defined in (3.18) is converges absolutely in D. Now If q → 1, then the operator F η,p reduces to the integral operator studied in [31] and further by taking p = 1, we obtain the familiar Bernardi integral operator introduced in [11].
Theorem 3.8. If f is of the form (1.1) belongs to the family S * p (q, µ, A, B) and where F η,p is the integral operator given by where υ is given by (3.20) and we have used Theorem 3.4 to complete the proof.

Conflicts of Interest
The authors agree with the contents of the manuscript and there are no conflicts of interest among the authors.