Generalization of k -Uniformly Starlike and Convex Functions Using q -Difference Operator

: In this article we have deﬁned two new subclasses of analytic functions k − S q [ A , B ] and k − K q [ A , B ] by using q -difference operator in an open unit disk. Furthermore, the necessary and sufﬁcient conditions along with certain other useful properties of these newly deﬁned subclasses have been calculated by using q -difference operator.


Introduction
Assume that H(U ) represents the analytic functions class in an open unit disk U ; U = {z : z ∈ C with |z| < 1}.
Here C denotes the complex numbers set.
Similarly we consider the class A of those analytic functions that satisfies f (z) = z + ∞ ∑ n=2 a n z n (for all z ∈ U ).
The class A is normalized by In the literature, the univalent functions class in U is expressed by S . According to [1], the starlike functions class in U is represented by S * , that include f ∈ A with given condition Furthermore, the convex functions class in U is represented by K, that consists the functions f ∈ A with given condition It can be deduced from conditions defined in Equations (2) and (3) (see [2]) that f (z) ∈ K ⇐⇒ z f (z) ∈ S * .
The analytic functions of the form p(z) = 1 + ∞ ∑ n=1 p n z n , (4) are denoted by the class P, for which (p(z)) > 0 (for all z ∈ U ).
Subordination between any two analytic functions f and g in U may be represented as In case of Schwarz function w in U , if w is analytic and satisfies |w(z)| < 1 & w(0) = 0, then f (z) = g(w(z)).
Similarly, if g satisfies condition of univalent function in U . The equivalence transformed into f (z) ≺ g(z) (z ∈ U ).
The conic domain was introduced by Kanas et al., described in [3], denoted by Ω k having the form The extremal functions, p k (z), family of conic domain Ω k , having p k (0) = 1 with p k (0) > 0 are normalized univalent functions in the form where and we choose t ∈ (0, 1) such that Here R(t) is Legendre's complete elliptic integral of the first kind and R (t) given by is the complementary integral of R(t).

Definition 1 ([4]).
Assume that h be an analytic function with h(0) = 1, The class of analytic functions represented by P [A, B] was initially introduced by Janowski in 1973 (see [4]). He demonstrated that if a function p ∈ P exists then h(z) ∈ P [A, B]. Mathematically, it takes the form

Definition 2.
Assume that q ∈ (0, 1) define the q-number [λ] q , then Definition 3. According to [5,6], the q-derivative of a function f in a subset of C is defined by It provided the existence f (0). Similarly from Definition 3, it is noticed that which is differentiable in the subset of C. We also deduced from Equations (1) and (7) that [n] q a n z n−1 .
Recently, the usage of the q-derivative operator is quite significant due to its applications in many diverse areas of mathematics, physics and other sciences. According to Srivastava et al. [7], the q-difference operator (D q ) in the context of Geometric Function Theory (GFT) was first utilized by Ismail et al., described in [8]. They studied a q-extension of starlike functions in U (see Definition 4 below). Afterwards many mathematicians continued their research highlighting the fundamental role in GFT. Mahmood et al. in [9] presents a detail description of the q-starlike functions class in conic domain, whereas in [10], the authors provided the class of q-starlike functions associated with Janowski functions. Moreover, the problems related to upper bound of third Hankel determinant (H 3 (1)) for the class of q-starlike functions have been investigated, available in [11]. Later on, Srivastava et al. [12] have investigated the Hankel and Toeplitz determinants of a subclass of q-starlike functions. Many other authors have studied and investigated a number of other new subclasses of q-starlike, q-convex and q-close-to-convex functions. They obtained a number of useful results like, coefficient inequalities, sufficient conditions, partial sums results and results related to radius problems (see for example [13][14][15][16]).

Definition 4 ([8])
. Let a function f ∈ S. Then f ∈ S * q if and As q → 1 − , it is clearly noticed that Similarly the class of q-starlike functions denoted by S * q decreases to the known class S * . Likewise, using subordination principle among analytic functions, the conditions described in Equations (9) and (10), may be revised as follows (see also [17]) Remark 1. For function f ∈ A, the idea of Alexander's theorem [1] was used by Baricz and Swaminathan [18] to define the class C q of q-convex function in the following way In order to utilize the q-difference operator (D q f (z)), this study has introduced two new subclasses of A, i.e., k − S q [A, B] and k − K q [A, B].
where p k (z) is given by Equation (6).

lim
Hence, from Equations (12) and (13) we can write As far as we know, there is minimal work on q-calculus related with conic domain in the literatures. The major objective of this work is to define a new subclass of q-starlike functions associated with the conic type domain. We find a number of useful results for our define function class and present some special cases of our results, in form of corollaries and remarks.

Main Results
In this section, we assume The LHS of Equation (15) in U is holomorphic, which follows f (z) = 0 and z ∈ U * = U \{0}.
This means 1 z f (z) = 0, because Equation (14) holds for L = 1. From Equation (15) we can say that there must exist a function w(z) in U , which should be analytic having |w(z)| < 1 with w(0) = 0. This is because of the property of subordination between the two holomorphic functions, such that and is equivalent to Since, From Equation (18), we may write Equation (17) in the form which gives similar result as presented in Equation (14). This proves the necessary part of Theorem 1.
Conversely: As we know that for L = 1, Equation (14) holds, it obeys the condition that 1 z f (z) is not equal to zero for all z ∈ U . Therefore, the function ψ(z) = zD q f (z) f (z) is analytic in U . In the earlier part of our proof, it was shown that our supposition Equation (14) can also be written in the form of Equation (16). So If we write then Equation (19) shows that ψ(z) ∩ ζ(z) = ∅. Thus, the connected component of C\ζ(∂U ) includes the simply-connected domain ψ(U ). As we know that ψ(0) = ζ(0) along with univalence of function giving ψ(z) ≺ ζ(z), as mentioned in the subordination fact (Equation (15)), i.e., f ∈ k − S q [A, B], which gives the desired result. (1)

Corollary 3 ([20]). For any function f represented by Equation
By putting q → 1 − in corollary 2, we will get the desired corollary.

Theorem 2.
For any function f represented by Equation (1).

Proof. Let us consider
.
From identity zD q f (z) * g(z) = f (z) * zD q g(z). By using the relation Theorem 1 also gives this result. By setting k = 0 in Theorem 2, one may get the result obtained by Aouf and Seoudy, presented in [20].
By taking k = 0 and q → 1 − in corollary 5, we will get the desired corollary.

Theorem 3.
For any function f represented by Equation (1), f ∈ k − S q [A, B] has a necessary and sufficient condition Proof. Keeping in view Theorem 1, we could found Using Equation (21) we may write [n] q (L − 1) − L a n z n−1 , which completes the desired proof.

Corollary 7 ([20]).
For any function f represented by Equation (1), f ∈ S q [A, B] has a necessary and sufficient condition Substituting k = 0 in Theorem 3, one may get the desired corollary.

Corollary 8 ([20]).
For any function f represented by Equation (1), f ∈ S[A, B] has a necessary and sufficient condition

Corollary 9 ([20]).
For any function f represented by Equation (1), f ∈ S q (α) has a necessary and sufficient condition By taking k = 0 and q → 1 − in Theorem 3, one may get the desired corollary.

Theorem 4.
For any function f represented by Equation (1). The function f ∈ k − K q [A, B] has a necessary and sufficient condition Proof. From Theorem 2, we could found After simplification, the LHS of Equation (23) may takes the form [n] q (L − 1) − L a n z n−1 .
This proves our required result.
Substituting k = 0 in Theorem 4, one may get the desired corollary.
Corollary 11 ([20]). For any function f represented by Equation (1), f ∈ K[A, B] has a necessary and sufficient condition Corollary 12 ([20]). For any function f represented by Equation (1), f ∈ K q (α) has a necessary and sufficient condition As an application of Theorems 3 and 4, one may determine inclusion property and coefficient estimates for a function of the form Equation (1) (24) Proof. Assume that Equation (24) holds, it is sufficient that Now let us consider Corollary 13 ([20]). If any function f represented by Equation (1) satisfies Substituting k = 0 in Theorem 5, we will get the desired corollary.

Theorem 6.
If any function f represented by Equation (1) satisfies Theorem 5 and Equation (12) give the immediate proof of the desired theorem.

Conclusions
In this work, we have introduced two subclasses of analytic functions k − S q [A, B] and k − K q [A, B] associated with q-difference operator in an open unit disk. The necessary and sufficient conditions of these newly introduced subclasses are investigated, whereas certain results and properties are studied by applying the q-difference operator in detail. The obtained results are compared to the previous known work with corollaries.
In concluding our present investigation, we draw the attention of interested readers toward the prospect that the results presented in this paper can be obtain for other subclasses of analytic functions. One may attempt to produce the same results for the class of q-symmetric starlike functions involving the Janowski functions and conic domains.

Conflicts of Interest:
The authors declare no conflict of interest.