Applications of Fractional Differential Operator to Subclasses of Uniformly q -Starlike Functions

: In this paper, we use the concept of quantum (or q -) calculus and deﬁne a q -analogous of a fractional differential operator and discuss some of its applications. We consider this operator to deﬁne new subclasses of uniformly q -starlike and q -convex functions associated with a new generalized conic domain, Λ β , q , γ . To begin establishing our key conclusions, we explore several novel lemmas. Furthermore, we employ these lemmas to explore some important features of these two classes, for example, inclusion relations, coefﬁcient bounds, Fekete–Szego problem, and subordination results. We also highlight many known and brand-new speciﬁc corollaries of our ﬁndings.


Introduction and Motivation
Perhaps the most intriguing part of the complex function theory is the interaction between geometry and analysis.Such connections between analytic structure and geometric behavior are at the heart of the theory of univalent functions.The domains of these functions characterize a wide variety of appealing geometric structures and canonical types.As an example, the image of an open unit disk under a normalized analytic and univalent function, ξ, contains a disk, |w| < δ.Moreover, the images of a few of the ranges of some of them specify starlike, convex, close-to-convex, some in certain directions, some uniformly convex (starlike), and so on.The ranges of these geometric functions are characteristic of certain geometries.Additionally, this area of study is also known as geometric function theory (GFT).
In GFT, researchers have shown particular interest in linear operators.What makes this research so important is that we are looking at the characteristics of many classes of functions under a certain linear operator at the same time.In 1915, Alexander [1] developed the first integral operator, which he effectively used in his study of analytical functions.This subfield of the analytic function theory of complex analysis, which includes derivative and fractional derivative operators, is the subject of active investigation.Recent works such as [2] demonstrate the relevance of differential and integral fractional operators to the scientific community.Intriguing new results have emerged from studies of differential and integral operators from a number of perspectives, including quantum (or q-) calculus, that have implications for other areas of physics and mathematics.Further investigation may reveal that such operators have a role in providing solutions to partial differential equations, given their significance in the study of differential equations via functional analysis and operator theory.In his survey-cum-expository review work, Srivastava [3] highlights the exciting operator applications that are emerging from such a methodology.
Many of these applications of the basic (or q-) calculus and the fractional basic (or q-) calculus in GFT have inspired the present work, in which we introduce and analyze new subclasses of uniformly q-starlike and q-convex functions associated with a new generalized conic domain.
Let the set of all functions ξ be denoted by A and of the form These are normalized analytic functions in the unit disk E = {τ : |τ| < 1}.Let us denote by S the collection of all functions in A that are univalent in E ( see [4]) .
For any two analytic functions, ξ and g, in E. We state as ξ(τ) is subordinate to g(τ), denoted by ξ(τ) ≺ g(τ) if there exists a Schwartz function w(τ) with w(0) = 0 and |w(τ For the analytic functions ξ and g, where Let P stand for the famous class of functions p that are analytic in E, and have the series form: In relation to a point a in E, a domain E is starlike if and only if every line segment connecting the point a to every other point in E entirely encircles the domain.Simply said, starlike refers to a domain that is starlike in relation to its origin.A domain E is convex if and only if it is starlike with regard to each and every point in E or if every line segment connecting between two points in E lies fully inside E. If a function ξ maps E onto a star-shaped (convex) domain, it is said to be a starlike (convex) function.All starlike and convex function classes are abbreviated as S * and C, respectively.These classes are distinguished analytically by the inequalities: For 0 ≤ γ < 1, let C(γ) and S * (γ) be the subclasses of S composed of convex and starlike functions of order γ, respectively.Analytically we can write It has been widely accepted that for a convex (starlike) function ξ, its image under E and any circles within E centered at the origin are convex (starlike) arcs.Nevertheless, argumentation is needed to determine whether the characteristic generally applies to circles with the center at any other point, let us say ζ.Goodman [5] provided the answer to this issue by defining uniformly convex and starlike functions.After much deliberation, Ronning [6] and Ma and Minda [7] proposed the one variable characterization of these functions, defining them as follows: The set of uniformly starlike functions, denoted by ST, includes the function ξ ∈A if and only if The class of uniformly convex functions, denoted by UCV, includes the function ξ ∈A if and only if Later in [8], Kanas and Wisniowska investigated the classes β − ST and β − UCV defined as To map the intersection of E and any disk center ζ, |ζ| ≤ β onto a convex domain, it was proven mathematically in [8] that the class β − UCV, for β ≥ 0, is a subclass of univalent functions.Thus, the concept of β-uniform convexity extends the definition of convexity and ζ is the origin and β = 0; then β − UCV = C (see [4]), and for β = 1, then β − UCV = UCV.This class was first described by Goodman [5] and has been extensively investigated by Ronning [9] and Ma and Minda [7].It should be pointed out that the β − UCV class really began much earlier in [10] with some extra criteria but without the geometric meaning.
In the previous section, we said that Kanas and Wisniowska (see [8]) proposed and analyzed the class β-UCV and subsequently the corresponding class β-ST.These classes in the conic domain Λ β , (β ≥ 0) were then defined by Kanas and Wisniowska (see [8]) as follows: Remark 1.For β = 0, this domain is the right half plane; for 0 < β < 1, it is a hyperbola; for β = 1, it is a parabola; and for β > 1, it is an ellipse.
For these conic regions, the functions p β (τ) (see [8]) play the role of extremal functions.
In [11], Al-Oboudi et al. used the idea of conic domain Λ β,γ and defined new subclasses of starlike and convex functions where From elementary computations, ∂Λ β,γ represents the conic sections symmetric about the real axis.
The following functions serve as extremal functions in various conic domains: ) where t is chosen such that K(t) is Legendre's complete elliptic integral of the first kind, while [9,12]), we obtain For 0 < β < 1, (see [7]), we obtain where B = 2 π arccos β.Finally, for β > 1, we have For details, see [7,9,12].From (3), we have where The motivation and use of the q calculus may be seen in the fact that it is used to study many families of analytic functions with wide-ranging applications in mathematics and related subjects.The quantum (or q-) calculus is also extensively employed in the context of approximation theory, especially for a number of operators, such as the convergence of operators to functions in the real and complex domains.Jackson (see [13]) was the first scholar to define the q-analogue of the classical derivative and integral and explain some of its applications.The q-beta function was subsequently used by Aral and Gupta to create the q-Baskakov-Durrmeyer operator (see [14]), and the q-Picard and q-Gauss-Weierstrass singular integral operators were investigated in [15].In addition, a Ruscheweyh q-differential operator was initially presented by Kanas and Raducanu (see [16]), and its applications for multivalent functions were studied by Arif et al. (see [17]).In the meanwhile, [18] explored q-calculus via the lens of convolution.In recent years, several researchers have defined and explored several q-analogous differential operators for analytic, multivalent, and meromorphic functions, and discussed applications of these operators in various contexts; for more information, see [3,19,20].Now, for your convenience, we provided the most basic definitions of quantum (or q-) calculus.

Definition 2.
Below is a precise expression for (γ, q) j in (7) in terms of the q-gamma function: and q-factorial [j] q !defined by Definition 3 ([21]).For q ∈ (0, 1), we have the following definition of the q-number: Definition 4. For q ∈ (0, 1), we have the following definition of [n] q !: Definition 5.The notation [x] n,q , x ∈ C for the q-generalized Pochhammer is given by (n ∈ N) .
Additionally, the q-gamma function can be characterized as Definition 6 ([13]).For ξ ∈A, the q-derivative operator (q-difference operator) can be written as From ( 1) and ( 12), we have For n ∈ N and τ ∈ E, we have We can observe that lim Using the definition of the Pochhammer symbol in terms of Gamma functions by Note that for the derivative of negative order, it is the integral defined below: Definition 7.For α > 0, the fractional q-integral operator (see [22]) defined by where the q-binomial function (τ − tq) α−1 is defined by The representation of series 1 Φ 0 is given by The last equality is called q-binomial theorem (see [23]).For further details, see [3].
Note that the integral defined above is the derivative of negative order.
Definition 8. Let the smallest possible integer be m.Ð α q is the extended fractional q-derivative of order α, and it can be defined as We find from ( 14) that Remark 2. The case of −∞ < α < 0, Ð α q denotes a fractional q-integral of order α.
Remark 4. When q → 1 − , then we have the Owa and Srivastava operator defined in [25].
Geometrically, all values of the function p(τ) ∈ β − P q,γ take from the domain Λ β,q,γ which we will describe as where The domain Λ β,q,γ is denoted by the generalized conic domain.
Definition 12.The class SP n,q α,λ (β, γ) is defined as the set of all functions ξ ∈A satisfying the condition
Definition 13.The class UCV n,q α,λ (β, γ) is defined as the set of all functions ξ ∈A satisfying the condition Remark 11.When q → 1 − , then we have a known class of analytic functions defined by Al-Oboudi and Al-Amoudi in [11].

A Set of Lemmas
Here, we provide several lemmas that may be used to further explore the paper's key outcomes.

Lemma 4 ([30]
).Let ξ and g be univalent starlike functions of order 1 2 .Then, for every function ξ ∈A, we have where co denotes the closed convex hull.
New lemmas are explored here that will be useful in establishing the article's results.
Therefore, (36) becomes With this, the proof of Lemma 5 is finished.
By using (37) in (38), we have Now by using Lemma 5 on (39), we have With this, the proof of Lemma 6 is finished.
Proof.From (30), the result is obvious, where
Proof.If ξ(τ) ∈ SP n,q α,λ (β, γ), then by definition, there is a Schwartz function w with w(0) = 0 and |w(τ)| < 1, in such a way that Let h ∈ P, defined as This can be written as Similarly, By using ( 42) and ( 43) in (41) and comparing both sides, we obtain and After some simple calculation of (45), we obtain and where Now by using Lemma 1 on (47), we have This concludes the proof of Theorem 1.
When we take q → 1−, then Theorem 2 makes use of a well-established finding from [11].
Theorem 3. The class UCV n,q α,λ (β, γ) has the function ξ of the type (1) if Proof.Use the same technique of Theorem 2; we obtain the proof of Theorem 3.
When we take q → 1−, then from Theorem 3, we have a known result studied in [11].
Corollary 2. The class UCV n α,λ (β, γ) has the function ξ of the type (1) if where Theorem 4. The class SP n,q α,λ (β, γ) has the function ξ of the type (1).Then and Proof.Let ξ ∈ SP n,q α,λ (β, γ); then When we evaluate the τ-coefficient of each side, we obtain Taking the mod and applying Lemma 6, we have For n = 2 in (52), we obtain Therefore, the result is true for n = 2. Let n = 3 in (52); we have .
Therefore, the result is true for n = 3.Let n = 4 in (52); we have .

n-times
Additio ally, it is known that [33] ϕ Therefore, we obtain by repeatedly using Lemma 3 n-times ).
Remark 15.When we take q → 1−, then from Theorem 5, we have a known result studied in [11].
The result explored in [11] is obtained from Theorem 5 when q → 1 − .
Proof.The proof of Theorem 6 can be obtained by using the same method as that used to prove Theorem 5.
Remark 18.When we take q → 1−, then from Theorem 6, we have a known result studied in [11].

Conclusions
The operators of q-fractional calculus have been addressed and effectively implemented in a number of recent and continuing publications; see [36].From the fractional calculus q-Pochhammer symbol, several writers have expanded the notions of fractional q-integral and fractional q-derivative by proposing many different lower limits of integration.Many of these applications of the basic (or q-) calculus and the fractional basic (or q-) calculus in the geometric function theory of complex analysis have inspired the present work.
We were motivated to conduct this research after reading Srivastava's survey-cumexpository review essay [3], in which he describes the application of both the fundamental (or q-) calculus and the fractional (or q-) calculus to the study of geometric functions.In Section 1, we discussed some background ideas that are presented in this article and also used the q-calculus operator theory and successfully defined the q-analogous of a fractional differential operator for analytic functions.Considering this operator, we defined subclasses of q-starlike and q-convex functions.In Section 2, we mentioned some known lemmas, and also we proved some new lemmas.In Section 3, by utilizing these lemmas, we examined several useful properties, such as inclusion relations, coefficient bounds, 2 − α, 2 − µ; τ) * • • • * ϕ(2 − α, 2 − µ; τ)] *