On Certain Analogues of Noor Integral Operators Associated with Fractional Integrals

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Introduction
The theory of quantum calculus and its applications has been applied in several branches of mathematics, engineering sciences, and physics.Hence, many researchers have used q-calculus to study discrete dynamical systems, discrete stochastic processes, q-deformed super algebras, q-transform analysis, and so on.In literature, differentiation and integration of function are formulated by using the quantum theory of calculus (or q-calculus) [1][2][3].The Jackson q-calculus is also involved in various areas of science including fractional q-calculus, optimal control, nonlinear integrodifferential equations, q-difference, and q-integral equations [4][5][6][7].Ismail et al. [8] are the first to employ the theory of q-calculus for investigating the geometric function theory.Srivastava in [9] points out some comprehensive reviews and applications in the geometric function theory of q-calculus and discusses many important applications of starlike functions.Arif et al. in [10,11] derive some properties of multivalent functions by using q-calculus.Authors in [12] discuss q-calculus and the Salagean operator to obtain differential subordination results.Aouf and Mostafa [13] used differen-tial subordination to define a new subclass of analytic functions with q-analogue fractional differential operator.Mahmood and Sokół [14] apply properties of the Ruscheweyh q-differential operator for a subclass of analytic functions and study some of its applications.Kanas and Raducanu [15] investigate q-analogues of the Ruscheweyh operator by using the Hadamard product; see, for some details, [14,16] and [11,17].
Let f be a real or complex value and D be unit disc D = z, z < 1 , 0 < q < 1.Then, the q-difference operator is defined by [1] The q-differentiation rules may be wrote as D q f z g z = g z D q f z + f qz D q g z , 2 D q f z g z = g z D q f z − f z D q g z g z g qz 3 Let A a, n consist of analytic functions in the unit disc D = z, z < 1 of the form f z = a + z n + ∑ ∞ k=2 a k+n z k+n .For a = 0 and n = 1, we use A = A 0, 1 .Therefore, the function f ∈ A has the expansion of the form Note that every function f ∈ A is normalized by f 0 = 0 and f ′ 0 = 1.The class of univalent functions in A is denoted by S. In particular, S * is the class of starlike functions, CV is the class of convex functions, and K is the class of close-to-convex function [18,19].Recently, authors in [20] used the convolution to introduce three new subclasses of starlike functions, convex functions, and closeto-convex functions with the novel Borel distribution operator.
Let f ∈ A be given by ( 4) and Then, the convolution of f and g is denoted by f * g, which is a function in A given by We say that the function f is subordinate to g in D and write f z ≺ g z for z ∈ D if there exists a Schwartz analytic function w in D such that w 0 = 0 and w z < 1 z ∈ D and f z = g w z [21].In particular, if the function g is univalent in D, then f z ≺ g z if and only if Ma and Minda [22] studied the class of starlike and convex functions by using the principle of differential subordination.Those differential subordinations provide interesting results when they are used to study new sets of univalent functions [23][24][25].
Making use of equations ( 4) and (1), we can easily obtain that where It is clear that 1 q = 1.For k ∈ ℤ + , the q-factorial is given by [3] In addition, with t > 0, the q-Pochhammer symbol has the form [3] Note that t q,k = t k when q ⟶ 1 − .For t > 0, the q-analogue of the gamma function is presented as Now, we define a q-analogue Noor integral operator I n q A ⟶ A as follows: where F n q is defined by the relation Hence, f is of the form (4). Therefore, This, by taking q ⟶ 1 − , shows that the operator I n q defined in ( 14) reduces to the familiar Noor integral operator of [26,27].Now, by using the idea of Cho and Aouf [28], we introduce the q-fractional Riemann-Liouville integral of order λ (λ > 0) as follows.
Definition 1 (see [28]).The q-fractional Riemann-Liouville integral of order λ (λ > 0) is defined for a function f by where f is an analytic function in D.
Many other useful studies are introduced in the field of analytic functions including the fractional integral operator and its applications (see [29][30][31]).
In this paper, we introduce the q-fractional integral operator by using the q-Noor integral operator.This operator is based on the q-fractional Riemann-Liouville integral of order λ (λ > 0).Also, by using a newly defined q-fractional integral operator, we introduce a subclass S q n λ, δ of analytic functions and prove that S q n λ, δ is a convex set.
2 Journal of Function Spaces Furthermore, several exciting subordination results of the q-fractional integral operator are obtained.

Definition and Coefficient Bounds
We introduce a q-fractional integral of the operator I n q f z .
Definition 5. Let 0 < q ≤ 1, λ ≥ 0, and n ∈ ℕ.The q-fractional integral of the operator I n q f is defined by the following: We note that D −λ q,z I n q f z ∈ A 0, λ + 1 and D 0 q,z I n q f z = I n q f z .Now, by taking q ⟶ 1 − , we have Now, we study the subclass of analytic function by using the new operator.Definition 6.Let the function f ∈ A, 0 < q < 1, δ ∈ 0, 1 , and λ ≥ 0. We define the subclass S q n λ, δ of functions f which satisfy the inequality By allowing q ⟶ 1 − in Definition 6, the class S q n λ, δ is denoted by S n λ, δ .
Example 1.In this example, we show that the set S q n 0, δ is nonempty. Since By using assertion (21) for λ = 0, we have So, we get From ( 24) and ( 26) and using the series expansion of This, indeed, implies that Therefore, we obtain that the function which is a member of the set S q n 0, δ .
Proof.Consider two functions f 1 and f 2 from S q n λ, δ which are given in the form It is sufficient to show that the function By using the q-difference operator, we obtain Thus, the desired results are obtained.
Taking q ⟶ 1 − into Theorem 7 leads to the following corollary.
q λ, δ , h z = s z + q c / c q zD q s z , and c ∈ ℕ such that Proof.The differential equation h z = s z + q c c q zD q s z 40 has a unique solution (38).Since r, s ∈ A, then, in view of (4), we have So, the differential subordination (37) can be rewritten in the form Applying Lemma 2 gives This proves (39).The proof of Lemma 9 is completed.
Note that when q ⟶ 1, then Λ c q z ⟶ Λ c z .For example, we have Remark 10.Plots of the suggested functions Λ 1 z , Λ 2 z , Λ 3 z , and Λ 4 z in the unit disc D are illustrated in Figures 1 and 2. These plots show that these suggested functions are convex in the unit disc D.
An analytic function r in (37) is said to be a solution of the differential subordination.The analytic function φ is a dominant of the solution of the differential subordination (37), if r ≺ φ for all r satisfying (37).
A dominant φ is said to be the best dominant of (37) if it satisfies φ ≺ φ for all dominants φ of (37).The best dominant is unique up to the rotation of D. Theorem 11.Let 0 < q < 1, r, h ∈ S q n λ, δ with h z = s z + q c z/ c q D q s z , and c ∈ ℕ such that Λ c q z defined in (36) is a convex function.If z c F z = c q z 0 t c−1 f t d q t, z ∈ D, the following differential subordination Proof.In view of the definition F, we have D q z c F z = c q z c−1 h z , z ∈ D. Since the q-derivative rule in (2) holds, we get Now, by using the D −λ q,z I n q , we for z ∈ D can obtain the differential equation as follows: From (45) and (48), we have By using the notation we obtain r z ∈ A. This implies that r z + q c c q zD q r z ≺ s z + q c c q zD q s z , z ∈ D 51 Now, applying Lemma 9, we obtain r z ≺ s z , z ∈ D 52 By using Remark 10, we can obtain the following corollary.
In the next theorem, we derive an exciting inclusion for the class S q n λ, δ .and c ∈ ℕ such that Λ c q z , defined in (36), is a convex function.If s z = I c q r z = c q /z c z 0 t c−1 h t d q t is a convex function, then we have the following inclusion: Proof.In view of Theorem 11, we obtain the following: r z + q c c q zD q r z ≺ h z , 56 where r z = D −λ q,z F z , z ∈ D. Now, by applying Lemma 9, we obtain r z ≺ s z .Indeed, where From the convexity of s z and using the fact that s D is symmetric with respect to the real axis, we have This completes the proof of Theorem 13.
By using Remark 10, we can obtain the following corollaries.

Corollary 14. Let δ ∈ 0, 1 and h z
0 f t dt, then we have the following inclusion: where 0 tf dt, then we have the where t dt, then we have the following inclusion: where t dt, then we have the following inclusion: then we have the following result c q D −λ q,z I n q f z z c ≺ s z 65 Proof.Denoted by r z = c q D −λ q,z I n q f z /z c , we obtain z c c q r z = D −λ q,z I n q f z 66 By applying q-derivative and using rule (2), we derive r z + q c z c q D q r z = z 1−c D q D −λ q,z I n q f z 67 By applying Lemma 9, we get r z ≺ s z 68 This implies the differential subordination (70).This completes the proof of Theorem 18.
By using Remark 10, we can obtain the following corollary.

Applications
In this section, we obtain some interesting applications involving the q-analogue differential subordination.
then we have the following result: Proof.Let r z = c q D −λ q,z I n q f z /z c , z ∈ D. By putting λ = c − 1 in the assertions (67) and using (82), we obtain By applying Theorem 18 and Lemma 9, we for z ∈ D derive By using the principle of differential subordination, we have

Journal of Function Spaces
Since Re w 1/n ≥ Re w 1/n , for Re w > 0 and n ≥ 1, we establish inequality (83).
To prove the sharpness of (83), we, for f ∈ A, z ∈ D, define For this function, we have When z ⟶ −1, we get This completes the proof of Theorem 20.
We can obtain the following corollaries by using Remark 10.
then we have the following result: then we have the following result: then we have the following result: Proof.Similar to the proof of Theorem 20, for r z = c q D 1−c q,z I n q f z /z c , the differential subordination (92) is equivalent to By using Remark 10, we can obtain the following corollaries.
then we have the following result: , 90 then we have the following result: then we have the following result:

94
then we have the following result: Then, by using the differential subordinations (67) for λ = c − 1 and (112), we infer By applying Theorem 18 and Lemma 9, we have Also, we define a function h i z , i = 1, 2 in the following form: By using the differential subordination (112), we write Similar to the proof of Theorem 20, we derive Therefore, by applying Lemma 3, we have By applying Lemma 4 and using the fact that 105 By using inequality (104), we derive By using assertions (106) and (107), we obtain This completes the proof of Theorem 27.
We can obtain the following corollaries by using Remark 10.

Conclusions
In this paper, the topics related to applications in the geometric function theory of q-calculus are presented.The proposed q-differential operator was applied to introduce a q-analogue of a fractional integral operator, and the geometric behavior of the operator is also investigated using the principle of differential subordination.Several interesting results of the q-analogue fractional integral operator are obtained here by following the differential subordination method.A new class of convex functions, S q n λ, δ , are defined, and an inclusion for the class S q n λ, δ is obtained.The fractional integral operator D −λ q,z I n q is defined on open unit disc D, and some properties of differential subordination are studied.Therefore, the results obtained in this research could be further used for writing the dual theory of differential subordination which is added to the study of the q-fractional integral operator.

9 1 + A i z 1 +Corollary 29 .
Journal of Function SpacesCorollary 28.Let −1 ≤ B i A i ≤ 1, i = 1, 2. If f ∈ A satisfies D −1 z I n f i z ≺ z B i z , i = 1, 2 , 109then we have the following result:Let −1 ≤ B i < A i ≤ 1, i = 1, 2. If f ∈ A satisfies D −3 z I n f i z ≺ z 3 12 1 + A i z 1 + B i z , i = 1, 2 , 112then we have the following result: