Abstract
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.
1. 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.
2. Methods
Our methods are divided into two subsections, as follows.
2.1. Geometric Methods
In this place, we clarify selected notions in the geometric function theory, which are situated in [6–8].
Definition 1. Let be the open unit disk. Two analytic functions , in are called subordinated denoting by , if there exits an analytic function having the formula is majorized by denoting by if and only if equivalently, the coefficient inequality is held , respectively.
There is a deep construction between subordination and majorization [9] in ∪ for selected distinct classes comprising the convex class : and starlike functions
Definition 2. We present a class of analytic functions by
This class is denoted by and known as the class of univalent functions which is normalized by .
Associated with the terms and , we present the term of all analytic functions in with a positive real part in and .
Two analytic functions are called convoluted, denoting by if and only if
Definition 3. The generalized Mittag-Leffler function is defined by [10–12] where represents the Pochhammer symbol and Note that is an ultimate traditional generalization of the function , where .
Moreover, it can be formulated by the Fox-Write hypergeometric function, as follows:
2.2. Complex Prabhakar Operator (CPO)
The Prabhakar integral operator is defined for analytic function by the formula [13, 14]
For example, let , then (see [15], Corollary 2.3)
The Prabhakar derivative can be computed by the formula [13]
Definition 4. Let . Then the complex Prabhakar differential operator (CPFDO) of (13) is formulated in terms of the Riemann-Liouville derivative, as follows: and in terms of the Caputo derivative, as follows:
Note that
For example, let , then in virtue of [15] (Corollary 2.3), we conclude that
In general, we have where , , and . Hence, we obtain
We have the following property.
Proposition 5. Let . Define a functional by Then .
Proof. Let . Then a computation implies where . This indicates that .☐
We call the normalized complex Prabhakar operator (NCPO) in the open unit disk. Since , 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 is a complex connection (coefficient) of the operator and it is a constant when .
Remark 6. The integral operator corresponding to the fractional differential operator is expanded by the series It is clear that The linear convex combination of the operators and can be recognized by the formula where. Clearly, , where.
2.3. Subclasses of NCPO
In terms of the NCPO, we formulate the next classes.
Definition 7. A function is considered to be in the class if and only if
We shall deal with the conditions of a function to be in whenever is convex as well as nonconvex.
Definition 8. A function is considered to be in the class if and only if
We request the next result, which can be located in [6].
Lemma 9. Define the class of analytic functions as follows: for and a positive integer (i)Let . Then . In addition, if and , then there are constants and such that and (ii)Let and . Then there exists a fixed real number so that (iii)Let with . Then or for such that Then .
3. Results
Our results are as follows.
Theorem 10. Let . If one of the next inequalities is considered, (i) is of bounded turning function(ii)(iii)(iv)(v)then for some .
Proof. Define a function as follows:
Then a computation implies that
In virtue of the first inequality, we have that is of bounded turning function, which leads to . Therefore, Lemma 9(i) indicates that 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 such that and
This implies that
Suppose that
According to Lemma 9(ii), there exists a fixed real number satisfying and
It follows from (37) that ; consequently, by Noshiro-Warschawski and Kaplan theorems, is univalent and of bounded turning function in ∪. Taking the derivative (33), then we get
Hence, Lemma 9(ii) implies .
The logarithmic differentiation of (33) yields
Hence, Lemma 9(iii) implies, where ,
☐
The next results show the upper bound of the operator utilizing the exponential integral in the open unit disk provided that the function .
Theorem 11. Suppose that , where is convex in . Then where is analytic in , with and . Also, for , satisfies the inequality
Proof. By the hypothesis, we receive the following conclusion: This gives the occurrence of a Schwarz function with and 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 plots the disk onto a reign, which is convex and symmetric with respect to the real axis. That is, then we have the inequalities consequently, we get In view of Equation (48), we obtain the general log-inequality that is, Hence, we have ☐
Proceeding, we illustrate the sufficient condition of to be in the class , where is convex univalent satisfying .
Theorem 12. If satisfies the inequality then .
Proof. The proof directly comes from [6] (Theorem 3.1a). Taking and in the inequality then we obtain This implies that that is .☐
Corollary 13. Let the assumption of Theorem 12 hold. Then
Proof. Let In view of Theorem 12, we have where . Then by [9] (Theorem 3), we get for some , where ☐
It is well known that the function is not convex in , where the domain is lima-bean (see [6] (P123)). One can obtain the same result of Theorem 12 as follows.
Theorem 14. If satisfies the inequality then .
Proof. Let Then a computation implies This implies that [6] (P123) that is, .☐
Theorem 15. If then the function satisfies
Proof. Let . Then there occurs a function such that This confirms that However, satisfies which is univalent, then we get Also, is starlike in which implies that Hence, a Schwarz function gets which leads to A calculation yields Therefore, we get the following inequality: or Thus, we have This completes the assertion of the theorem.☐
Example 16. (i)Let Then the solution of is formulated, as follows: Moreover, the solution of the equation is approximated to . (ii)The solution of is given in terms of the hypergeometric function, as follows (see Figure 1):
4. 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.
Conflicts of Interest
The authors declare no conflict of interest.
Authors’ Contributions
All authors contributed equally and significantly to writing this article. All authors read and agreed to the published version of the manuscript.