Symmetric Conformable Fractional Derivative of Complex Variables

It is well known that the conformable and the symmetric differential operators have formulas in terms of the first derivative. In this document, we combine the two definitions to get the symmetric conformable derivative operator (SCDO). The purpose of this effort is to provide a study of SCDO connected with the geometric function theory. These differential operators indicate a generalization of well known differential operator including the Sàlàgean differential operator. Our contribution is to impose two classes of symmetric differential operators in the open unit disk and to describe the further development of these operators by introducing convex linear symmetric operators. In addition, by acting these SCDOs on the class of univalent functions, we display a set of sub-classes of analytic functions having geometric representation, such as starlikeness and convexity properties. Investigations in this direction lead to some applications in the univalent function theory of well known formulas, by defining and studying some sub-classes of analytic functions type Janowski function and convolution structures. Moreover, by using the SCDO, we introduce a generalized class of Briot–Bouquet differential equations to introduce, what is called the symmetric conformable Briot–Bouquet differential equations. We shall show that the upper bound of this class is symmetric in the open unit disk.


Introduction
The term Symmetry from Greek means arrangement and organization in measurements. In free language, it mentions a concept of harmonious and attractive proportion and equilibrium. In mathematics, it discusses an object that is invariant via certain transformation or rotation or scaling. In geometry, the object has symmetry if there is an operator or transformation that maps the object onto itself [1,2]. Sàlàgean (1983) presented a differential operator for a class of analytic functions (see [3]). Many sub-classes of analytic functions are studied using this operator. Al-Oboudi [4] generalized this operator. These operators are studied widely in the last decade (see [5][6][7][8][9][10] for recent works). Our investigation is to study classes of analytic functions by using the symmetric differential operator in a complex domain. Recently, Ibrahim and Jahangiri [7] defined a special type of differential operators, which is called a complex conformable differential operator. This operator is an extension of the Anderson-Ulness operator [11].
A conformable calculus (CC) is a branch of the fractional calculus. It develops the term χ 1−℘ f (χ). While the complex conformable calculus (CCC) indicates the term ξ ϕ (ξ), where ξ is a complex variable and ϕ is a complex valued analytic function. In this work, we present a new SCDO in the open unit disk. We formulate it in some sub-classes of univalent functions. As applications, we generalize a class of Briot-Bouquet differential equations by using SCDO.

The Operator SCDO
This sections deals with definition of the SCDO as follows: Definition 1. Let (ξ) ∈ , and let ν ∈ [0, 1] be a constant then the SCDO keeps the following operating (2) The value ν = 0 indicates the Sàlàgean operator S k (ξ) = ξ + ∑ ∞ n=2 n k n ξ n . We proceed to impose a linear differential operator having the SCDO and the Ruscheweyh derivative. For ∈ , the Ruscheweyh derivative is defined as follows: where k k+n−1 are the combination terms.
The linear combination operator joining R k (ξ) and S k ν (ξ) is given by the formal Remark 1.
The class J ν (A, B, k) is a generalization of the class of the Janowski starlike functions [15] Note that S 0 (ν, ) = S * , S 1 (0, ) = C

The Outcomes
In this section, we study some properties of the SCDO.
Therefore, according to Lemma 1-i, we attain (σ(ξ)) > 0 which gets the first term of the theorem. According to second inequality, we indicate the pursuing subordination inequality Now, by employing Lemma 1-i, there occurs a fixed constant a > 0 with b = b(a) with the pursuing property S k Consequently, we indicate that (S k ν (ξ)/ξ) > , for values of ∈ [0, 1). Lastly, agree with the third relation to get According to Lemma 1-ii, there occurs a positive fixed number λ > 0 achieving the real inequality (σ(ξ)) > λ, and yielding for a few value in ∈ [0, 1). It indicates from (5) that S k ν (ξ)) > 0; thus, according to Noshiro-Warschawski and Kaplan Lemmas, this leads to S k ν (ξ) is univalent and of bounded turning in ∪. Now, via the differentiating (4) and concluding the real case, we indicate that Thus, by the conclusion of Lemma 1-ii, we have Taking the logarithmic differentiation (4) and indicating the real, we arrive at the following conclusion: A direct application of Lemma 1-iii, we get the positive real i.e., ( S k ν (ξ) ξ ) > 0. This completes the proof.

Theorem 3.
Suppose that ∈ B k (ν, α, ), and the convex analytic function g satisfies the integral equation implies the subordination and the outcome is sharp.
Proof. Here, we aim to utilize the result of Lemma 2. By the conclusion of F(ξ), we acquire Following the conditions of the theorem, we get By assuming We have According to Lemma 2, we obtain and g is the best dominant.

Theorem 4.
Let g be convex such that g(0) = 1. If , and this result is sharp.
Hence, we conclude that , and g is the best dominant.
By the definition of G(ξ), a computation yields that (w 1 a n + w 2 b n )ξ n then under the formal C k ν,α , we obtain By considering the derivative, we have (C k ν,α G(ξ))

Applications
A set of complex differential equations is an assembly of differential equations with complex variables. The most important study in this direction is to establish the existence and uniqueness results. There are diffident types of techniques including the utility of majors and minors (or subordination and superordination concepts) (see [12]). Investigation of ODEs in the complex domain suggests the detection of novel transcendental special functions, which currently called a Briot-Bouquet differential equation (BBDE) In this place, we shall generalize the BBDE into a symmetric BBDE by using SCDO. Numerous presentations of these comparisons in the geometric function model have recently achieved in [12].
Needham and McAllister [16] presented a two-dimensional complex holomorphic dynamical system, pleasing the 2-D form ξ t = Θ(ξ, ω); ω t = Θ(ξ, w), ξ, ω ∈ ∪ and t is in any real interval. Development application of the BBDE seemed newly, with different approaches (see [17]) to solve the equation of electronic nano-shells (see [18]). Controlled by the situation effort of traditional shell theory, the transposition fields of the nano-shell take the dynamic system where θ is the angles between ξ and ω and their conjugates.
Our purpose is to generalize this class of equation by utilizing the SCDO and establish its properties by applying the subordination concept. In view of (2), we have the generalized BBDE The subordination settings and alteration bounds for a session of SCDO specified in the following formula. A trivial resolution of (8) is given when ω = 1. Consequently, our vision is to carry out the situation, ∈ and ω = 0. We proceed to present the behavior of the solution of (8).
A calculation brings the next subordination relation