Some Applications of Analytic Functions Associated with q -Fractional Operator

: This paper introduces a new fractional operator by using the concepts of fractional q - calculus and q -Mittag-Lefﬂer functions. With this fractional operator, Janowski functions are generalized and studied regarding their certain geometric characteristics. It also establishes the solution of the complex Briot–Bouquet differential equation by using the newly deﬁned operator.


Introduction and Preliminaries
Let the class of all analytic functions h in the open unit disk U = {η ∈ C : |η| < 1} be denoted by A and the Taylor series expansion of h ∈ A be given as The subset of A consisting of all univalent functions is denoted by S. The function h is subordinate to an analytic function ψ written h(η) ≺ ψ(η) if there exists a function u(η) (η ∈ U ) such that u(0) = 0 and |u(η)| < 1, such that h(η) = ψ(u(η)).
Let h(η) be majorized by ψ(η), written as It is noted that majorization is closely related to the concept of quasi-subordination between analytic functions.
The convolution of two analytic functions h and ψ (denoted by h * ψ) is defined as where h is given by (1) and The theory of basic and fractional quantum calculus plays an important role in many diverse areas of mathematical, physical and engineering sciences.The various types of fractional differential equations play an important role not only in mathematics but also in physics, control systems, dynamical systems and engineering to create the mathematical modeling of many physical phenomena (see, for example, [2][3][4][5][6]).
Fractional calculus is a vast and growing subject of interest for mathematicians and physicist.The theory of fractional calculus has been applied to the theory of analytic functions.Fractional differential equations are emerging as a new and famous branch of applied mathematics that is being used for many mathematical models in science and engineering.In fact, fractional differential equations are viewed as an alternative model to nonlinear differential equations (see, for example, [7][8][9][10]).
The operators are used to define different subclasses of analytic functions and to solve fractional algebraic differential equations (see [11,12]).In order to investigate several subclasses of class A, the q-calculus as well as the fractional q-calculus have been used as an important tool.A firm footing of the usage of q-calculus in the context of geometric function theory was provided by Srivastava in the book [13].Recently, in [14], Srivastava and Bansal studied a certain family of q-Mittag-Leffler functions and found sufficient conditions for a function to belong to the family of close-to-convex functions.In the survey-cum-expository article [15], Srivastava provided the operators of basic (or q-) calculus and fractional qcalculus and discussed their applications to the geometric function theory of complex analysis.For more recent work on analytic functions, see [16][17][18][19] and the references therein.Now, we give some basic definitions of fractional q-calculus, which help us to define new classes of functions.Definition 1.For a, q ∈ C, the q-shifted factorial (a, q) n is defined by If a = q −m , (m ∈ N 0 ), then we may define: when a = 0 and |q| ≥ 1, then (a, q) ∞ diverges.So, whenever we use (a, q) ∞ , then |q| < 1 is assumed.
Definition 2. The q-Gamma function is defined in terms of (a, q) n in (2) as follows: and q-factorial [n] q ! is defined by: Definition 3 ([20]).For 0 < q < 1 and h ∈ A, the q-derivative operator (D q ) is defined as From ( 1) and ( 5), we get the following series of the form For n ∈ N and η ∈ U , we have [n] q a n η n−1 .
Indeed, for a complex-valued function h(η), the fractional q-derivative (or the qdifference) operator D q is given by Definition 5 below, which is defined as follows (see, for example, Purohit and Raina [28]; see also Srivastava [15]).Definition 5.For analytic function h(η), the fractional q-derivative operator D q of order δ is defined by Definition 6.For m being the smallest integer.The extended fractional q-derivative D δ q,η of order δ is defined by We find from (9) that Note that: When −∞ < δ < 0, then D δ q represents a fractional q-integral of h(η) of order δ, and for 0 ≤ δ < 2, then D δ q represents a fractional q-derivative of h(η) of order δ.

Definition 7 ([30]
). Selvakumaran et al. defined the q-differintegral operator Ω δ q : A → A as follows: Now, by using the technique of convolution on ( 7) and ( 10), we define a new type of q-differintegral operator D q (α, β, δ) : A → A, as follows: Definition 8.For h ∈ A, the q-differintegral operator D q (α, β, δ) of a function h(η) is defined by where, Definition 9. Let h ∈ A, a new extended form of the linear multiplier fractional q-differintegral operator, be defined as It is seen from ( 11) that, for h(η) given in (1), we have and Q(q, δ, α, β) is given by (12).

A Set of Lemmas
The following lemma is necessary to prove our main results.

Lemma 1 ([40]
).Let G(h, n) be a class of analytic functions defined as where, ∈ C and a positive integer n.
So, using Lemma 1(i) implies that Hence, the first part of the theorem is completed.Consequently, the second part is confirmed.In light of Lemma 1(i), we fixed a real number l > 0, such that k = k(l), and Therefore, (24) implies that Suppose that According to Lemma 1(ii), there exists a fixed real number l > 0 that satisfies Now from (26), we find that Taking the derivative of ( 22), we have Hence, Lemma 1(ii) implies Again, taking the logarithmic differentiation of (22) yields Hence, from Lemma 1(iii) and φ(η) = 1, we have Upper bounds of the operator Λ β,δ α,λ (m, q)h(η).

This implies that
Corollary 1.Let the assumption of Theorem 3 hold.Then, . According to the Theorem 3, we have where, σ ∈ C.Then, by a result given in [41], we get p (η) σ (η).
; hence, we obtained the required result.
Theorem 5.If h ∈ J δ,m q,λ (α, β, L, M, b), then the function Proof.Let h ∈ J δ,m q,λ (α, β, L, M, b); then, the function J(η) can be written as However, J satisfies In addition, B(η) is starlike in U , which implies that Hence, the Schwarz function which leads us to A simple calculation yields Therefore, we get the following inequalities: . Thus, we have Hence, the proof is completed.
Then, the solution of

Applications
The solution of the complex Briot-Bouquet (BB) differential equation is established in [40].We produce a presentation of our results in complex BB differential equations, and the class of BB differential equations is a link of differential equations whose consequences are visible in the complex plane.Recently, the complex modelings of phenomena in nature and society have been the object of several investigations based on the methods originally developed in a physical context.Ibrahim [10] studied various types of fractional differential equations in the complex domain, such as the Cauchy equation, the diffusion equation and telegraph equations.The study of first ODEs specifies new transcendental special functions as follows: In [40], many new applications of these equations in Geometric Function Theory have been discussed.Now, we investigate (32) by using the operator Λ β,δ α,λ (m, q)h(η) and find its solutions by applying the subordination relations.The operator Λ β,δ α,λ (m, q)h(η) propagates the complex BB differential equation as follows: where, ϕ(0) = h(0), η ∈ U .
A trivial solution of ( 33) is given when ς = 1; our investigation concerns the case with h ∈ A and ς = 0.
Hence, in the light of Theorem 2, the result given in (34) is completed.

Conclusions
We considered fractional q-differential operator and q-Mittag-Leffler functions and defined a new operator Λ β,δ α,λ (m, q)h(η).We investigated new subclasses of univalent functions associated with the operator Λ β,δ α,λ (m, q)h(η) in the open unit disk and discussed some geometric properties of this newly defined operator.By using the BB equation and involving the newly defined operator Λ β,δ α,λ (m, q), we investigated its solution.