Duality on q -Starlike Functions Associated with Fractional q -Integral Operators and Applications

: In this paper, we make use of the Riemann–Liouville fractional q -integral operator to discuss the class S ∗ q , δ ( α ) of univalent functions for δ > 0, α ∈ C − { 0 } , and 0 < | q | < 1. Then, we develop convolution results for the given class of univalent functions by utilizing a concept of the fractional q -difference operator. Moreover, we derive the normalized classes P ζ δ , q ( β , γ ) and P δ , q ( β ) (0 < | q | < 1, δ ≥ 0,0 ≤ β ≤ 1, ζ > 0 ) of analytic functions on a unit disc and provide conditions for the parameters q , δ , ζ , β , and γ so that P ζδ , q ( β , γ ) ⊂ S ∗ q , δ ( α ) and P δ , q ( β ) ⊂ S ∗ q , δ ( α ) for α ∈ C − { 0 } . Finally, we also propose an application to symmetric q -analogues and Ruscheweh’s duality theory.


Introduction
In recent decades, the theory of q-calculus has been applied to various areas of science and computational mathematics. The concept of q-calculus was used in quantum groups, q-deformed super algebras, q-transform analysis, q-integral calculus, optimal control, and many other fields, to mention but a few [1][2][3][4]. Soon after the concept of q-calculus was furnished, many basic q-hypergeometric functions, q-hypergeometric symmetric functions, and q-hypergeometric and hypergeometric symmetric function polynomials were discussed in geometric function theory [5]. Jackson [6] was the first to introduce and analyze the q-derivative and the q-integral operator. Later, various researchers applied the concept of the q-derivative to various sub-collections of univalent functions. Srivastava [7] used the q-derivative operator to describe some properties of a subclass of univalent functions. Agrawal et al. [8] extended a class of q-starlike functions to certain subclasses of q-starlike functions. Kanas et al. [9] used convolutions to define a q-analogue of the Ruscheweyh operator and studied some useful applications of their operator. Srivastava et al. [10] defined the q-Noor integral operator by following the concept of convolution. Purohit [11] introduced a subclass of univalent functions by using a certain operator of a fractional q-derivative. Aouf et al. [12] employed subordination results to discuss analytic functions associated with a new fractional q-analogue of certain operators. However, many extensions of different operators can be found in [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29] and the references cited therein.
Here, we will make use of definitions and notations used in the literature [30,31]. For a, q ∈ C, the q-analogue of the Pochhammer symbol is defined by and, hence, it is very natural to write (a; q) k = (a;q) ∞ (aq k ,q) ∞ , (k ∈ N ∪ {∞}). The extension of the Pochhammer symbol to a real number δ is given as Therefore, for any real number δ > 0, the q-analogue of the gamma function is defined by The q-analogue of the natural number n and the multiple q-shifted factorial for complex numbers a 1 , · · · , a k are, respectively, defined by Let a 1 , ..., a r , b 1 , ...b s be complex numbers; then, the q-hypergeometric series r φ s is denoted as r φ s a 1 , · · · , a r b 1 , · · · , b s ; q, z = ∞ ∑ n=0 (a 1 , · · · , a r ; q)n (b 1 , · · · , b s ; q)n z n −q It is clear that the series representation of the function r φ s converges absolutely for all z ∈ C if r ≤ s and converges only for |z| < 1 if r = s + 1. Now, let A be the collection of all analytic functions in the open unit disc U = {z ∈ C; |z| < 1} expressed in the normalized form and let A 0 be a collection comprising all functions g such that zg ∈ A and g(0) = 1, z ∈ C. Then, the sub-collection of A of functions that are univalent in U is denoted by S. However, in geometric function theory, a variety of sub-collections of univalent functions have been discussed. See the monographs published by [32,33] for details. Let us consider the Riemann-Liouville fractional q-integral operator of a non-integer of order δ defined by [34] Then, I δ q f −→ I q when δ −→ 1, where I q f is the q-Jackson integral defined by [6] With the concept of the Riemann-Liouville fractional q-integral of the non-integer order δ, we recall some rules associated with I δ q by (3): Agarwal [34] defined the q-analogue difference operator of a non-integer order δ as follows: Note that D δ q f −→ D q f when δ −→ 1. D q f is the q-derivative of the function f introduced in [6] in the subsequent form: Thus, for n ∈ N, through simple computations, we obtain Let 0 < |q| < 1, δ ≥ 0 ζ > 0, 0 ≤ β ≤ 1, and 0 < γ ≤ 1. By the definition of the q-analogue difference operator with the non-integer order δ, the following rules of D δ q hold: We define P ζ δ,q (β, γ) as the class of all functions f ∈ A satisfying the following condition: For 0 < |q| < 1, δ ≥ 0, and 0 ≤ β ≤ 1, the class P δ,q (β) consists of functions satisfying the following condition:

Now, for two functions
we recall the convolution (or the Hadamard product) of f and g, denoted by f * g, which is given by For a set V ⊆ A 0 , the dual set V * is defined by However, the second dual of V is defined as V * * = (V * ) * . However, V ⊆ V * * . For basic reference to this theory, we may refer to the book by Ruscheweyh [35] (see also [36][37][38]).

Preliminary Lemmas
The following lemmas are very useful in our investigation.

Lemma 1.
(Duality principle; see [35] ) Let V ⊆ A 0 be compact; it has the following property: where for all continuous linear functionals ϕ on A, and whereco stands for the closed convex hull of a set. then We see that the set V β,γ in (7) does not satisfy the property (6), i.e., if f ∈ V β,γ , then f (xz) ∈ V β,γ for all |x| ≤ 1, as is required in the Duality Principle. However, the Duality Principle can be stated with a slightly weaker but more complicated condition that V β,γ can be seen to satisfy (see [35] for more details).

Main Results
Definition 1. Let f ∈ A, δ > 0, and α ∈ C − {0}. Then, a function f is said to be in the class S * q,δ (α) if it satisfies the following inequality: where the operators D δ q and I δ q are given by (4) and (3), respectively.
Putting δ = 0 into Definition 1 leads to the following definition.

Definition 2.
The function f ∈ A is said to be in the class of q-starlike functions of order α, S * q (α), if it satisfies the following inequality: where D q f (z) is given by (5).

Proof.
Since By following simple computations, we can rewrite this as Since the function f satisfies (2), we obtain . (8) is equivalent to

Now, as Equation
it simplifies to Hence, the required result has been proven.
Putting δ = 0 into Theorem 1, we get the following corollary. where

Proof.
Let the function f be in the class P ζ q,δ (β, γ), |z| < R ≤ 1. If we denote then we have g ∈ V * * β,γ . If f satisfies 2, then we obtain [n] q a n z n−1 + [n] q a n 1 [n] q a n ζγ Therefore, We now obtain a one-to-one correspondence between P ζ q,δ (β, γ) and V * * β,γ . Thus, by For z ∈ U , consider the continuous linear functional λ z : A 0 −→ C such that By the Duality Principle, we have λ z (V) = λ z (V * * β,γ ). Therefore, (12) holds if and only if Using the properties of convolution, we obtain Hence, we have where z ∈ U , |x| = 1. The equality on the right side of Equation (13) takes its value on the line Rew = − (1−q) δ ζγ (1−β) , and so (13) is equivalent to (10).

Remark 1.
Under the hypothesis of Theorem 3.5, the inequality (10) can be written in the form .
Therefore, for more clarification, we can see that this satisfies the inequality when . (14) Assume that the function ψ is given by Then, inequality (14) can be written in the form Hence, ψ(z) ∈ S * q,δ (α) if and only if (15) is satisfied.
Putting δ = 0 into Theorem 2 leads to the following corollary.
Similarly, from Theorem 2, we get the following theorem.

Remark 2.
The function F 1 (x, z) can be represented in terms of a q-hypergeometric function as follows: Proof. From the definition of F 1 (x, z) introduced in (11), we infer that Since [n + 1] q = (q 2 ;q) n (q;q) n , we have (q; q) n (q; q) n (q 2 ; q) n (q 2 ; q) n z n − 2(1 − q).
Hence, by using the definition of r φ s from (1), the proof of the corollary is complete.
Putting δ = 0 into Remark 2 leads to the following corollary.
Corollary 4. The function F 2 (x, z) can be expressed in terms of the q-hypergeometric function as follows: We now consider the Riemann-Liouville fractional q-integral and obtain the following corollary.
Remark 3. The function F 1 (x, z) can be expressed in terms of the Riemann-Liouville fractional q-integral as follows: Proof. Since Equation (18) is satisfied, we have v n t n z n d q vd q t.
This completes the proof of the corollary.
Putting δ = 0 into Remark 3 leads to the following corollary.
Proof. Let ζ > 0 and γ > 0; we define In view of these representations, we can write Let f ∈ P q,δ (β). Then, by using Lemma 2, we may restrict our attention to the function f ∈ P ζ (β, γ) for which Thus, we obtain Hence, Equation (20) is equivalent to where Therefore, where K q is defined by (19).
In addition, since Re{D δ+1 q I δ q f (z)} > 0, we have f ∈ P q,δ (0). This completes the proof of the theorem.

Conclusions
In this article, a new class of univalent functions was introduced by using Riemann-Liouville fractional q-integrals and q-difference operators of non-integer orders. Then, some convolution results for such a class of univalent functions were obtained. In addition, two classes of normalized analytic functions in the unit disc were derived ,and some conditions on q, δ, ζ, β, and γ were given so that the new classes satisfied P ζ δ,q (β, γ) ⊂ S * q,δ (α) and P δ,q (β) ⊂ S * q,δ (α). The result obtained during this research can be further used for writing fractional differential and integral operators in order to extend the results of analytic functions.