A New Class of Analytic Normalized Functions Structured by a Fractional Differential Operator

Newly, the field of fractional differential operators has engaged with many other fields in science, technology, and engineering studies. The class of fractional differential and integral operators is considered for a real variable. In this work, we have investigated the most applicable fractional differential operator called the Prabhakar fractional differential operator into a complex domain. We express the operator in observation of a class of normalized analytic functions. We deal with its geometric performance in the open unit disk.


Introduction
The class of complex fractional operators (differential and integral) is investigated geometrically by Srivastava et al. [1] and generalized into two-dimensional fractional parameters by Ibrahim for a class of analytic functions in the open unit disk [2]. These operators are consumed to express different classes of analytic functions, fractional analytic functions [3] and differential equations of a complex variable, which are called fractional algebraic differential equations studding the Ulam stability [4,5].
We carry on our investigation in the field of complex fractional differential operators. In this investigation, we formulate an arrangement of the fractional differential operator in the open unit disk refining the well-known Prabhakar fractional differential operator. We apply the recommended operator to describe new generalized classes of fractional analytic functions including the Briot-Bouquet types. Consequently, we study the classes in terms of the geometric function theory.

Methods
Our methods are divided into two subsections, as follows.

Geometric Methods.
In this place, we clarify selected notions in the geometric function theory, which are situated in [6][7][8].
There is a deep construction between subordination and majorization [9] in ∪ for selected distinct classes comprising the convex class ðCÞ: and starlike functions ðS * Þ Definition 2. We present a class of analytic functions by This class is denoted by Λ and known as the class of univalent functions which is normalized by f ð0Þ = f ′ð0Þ − 1 = 0.
Associated with the terms S * and C, we present the term P of all analytic functions p in ∪ with a positive real part in ∪ and pð0Þ = 1.
Two analytic functions f , g are called convoluted, denoting by f * g if and only if Definition 3. The generalized Mittag-Leffler function is defined by [10][11][12] where ðϑÞ n represents the Pochhammer symbol and Note that Ξ ℘ v,u ðzÞ is an ultimate traditional generalization of the function e z , where Ξ 1 1,1 ðzÞ = e z .
Proof. Let ψ ∈ Λ. Then a computation implies where α,β ∈ Λ, then we can study it in view of the geometric function theory.
Our aim is to formulate it in terms of some well-known classes of analytic functions. It is clear that δ n is a complex connection (coefficient) of the operator and it is a constant when α = 0. Remark 6. The integral operator corresponding to the fractional differential operator C k Δ γ α ω β is expanded by the series It is clear that The linear convex combination of the operators C k ϒ γ,ω α,β and C k Δ γ,ω α,β can be recognized by the formula where ∁∈½0, 1.
We shall deal with the conditions of a function ψ to be in c k S * γ,ω α,β ðσÞ whenever σ ∈ C is convex as well as nonconvex.
We request the next result, which can be located in [6].

Results
Our results are as follows.
Proof. Define a function ρ as follows: Then a computation implies that In virtue of the first inequality, we have that C k Δ γ,ω α,β ψðzÞ is of bounded turning function, which leads to Rðzρ ′ ðzÞ + ρðzÞÞ > 0. Therefore, Lemma 9(i) indicates that RðρðzÞÞ > 0 which gives the first part of the theorem. Consequently, the second part is confirmed. In virtue of Lemma 9(i), we have a fixed real number ℓ > 0 such that κ = κðℓÞ and This implies that Journal of Function Spaces Suppose that According to Lemma 9(ii), there exists a fixed real number ℓ > 0 satisfying RðρðzÞÞ > ℓ and It follows from (37) that Rð C k Δ γ,ω α,β ψðzÞÞ ′ Þ > 0; consequently, by Noshiro-Warschawski and Kaplan theorems, Hence, Lemma 9(ii) implies Rð C k Δ γ,ω α,β ψðzÞ/zÞ > 0. The logarithmic differentiation of (33) yields Hence, Lemma 9(iii) implies, where ℵðzÞ = 1, Theorem 11. Suppose that ψ ∈ C k S * γ,ω α,β ðσÞ, where σðzÞ is convex in ∪. Then where ΨðzÞ is analytic in ∪, with Ψð0Þ = 0 and jΨðzÞj < 1. Also, for jzj = ξ, C k Δ γ,ω α,β ψðzÞ satisfies the inequality Proof. By the hypothesis, we receive the following conclusion: This gives the occurrence of a Schwarz function with Ψð0Þ = 0 and jΨðzÞj < 1 such that That is, Integrating the above equality, we get Consequently, we get By the definition of subordination, we arrive at the following inequality Note that the function σðzÞ plots the disk 0 < jzj < ξ < 1 onto a reign, which is convex and symmetric with respect to the real axis. That is, then we have the inequalities 5 Journal of Function Spaces consequently, we get In view of Equation (48), we obtain the general loginequality that is, Hence, we have ☐ Proceeding, we illustrate the sufficient condition of ψ to be in the class C k S * γ,ω α,β ψðσÞ, where σ is convex univalent satisfying σð0Þ = 1.

Journal of Function Spaces
It is well known that the function σðzÞ = e ∈z , 1 < j∈j ≤ π /2 is not convex in ∪, where the domain σð∪Þ is lima-bean (see [6] (P123)). One can obtain the same result of Theorem 12 as follows.
Proof. Let Then a computation implies This implies that [6] (P123) Proof. Let ψ∈ c k J γ,ω α,β ðA, B, ♭Þ. Then there occurs a function JðzÞ such that This confirms that However, J satisfies which is univalent, then we get

Conclusion
The Prabhakar fractional differential operator in the complex plane is formulated for a class of normalized function in the open unit disk. We formulated the modified operator in two classes of analytic functions to investigate its geometric behavior. Differential inequalities are formulated to include them. Examples showed the behavior of solutions and the formula. The suggested operators can be utilized to formulate some classes of analytic functions or to generalize other types of differential operators such a conformable, quantum, or fractal operators.

Data Availability
No data were used to support this study.