Abstract

In this paper we derive subordination and superordination results regarding the Atangana–Baleanu fractional integral applied to multiplier transformation and we give several differential sandwich-type theorems. Then, we apply the Atangana–Baleanu fractional integral to extended multiplier transformation on the class of normalized analytic functions and we derive strong differential subordination and strong differential superordination results regarding the extended new operator and we give the corresponding differential sandwich-type theorems from the first part of the paper.

1. Introduction

Atangana–Baleanu fractional integral operator presents particular importance due to its nonsingular Mittag–Leffler kernel which allows for this operator to be used in many branches of applied mathematics for development and study of mathematical models which involve it. Disadvantages of the traditional fractional-order derivatives incorporating power-law kernel or exponential kernel have been overcome by introducing the new fractional derivative and the associated fractional integral with Mittag–Leffler kernel. Mittag–Leffler function is more suitable in expressing natural phenomena than the power function or exponential function. It appears naturally in several physical problems and the field of science and engineering. Hence, Atangana–Baleanu fractional derivative and the associated Atangana–Baleanu fractional integral operator are involved in many applications such as modelling groundwater fractal flow, viscoelasticity, and probability theory.

There are many articles showing the importance of Mittag–Leffler function in fractional calculus. A variety of fractional evolution processes presented as being governed by equations of fractional order, whose solutions are related to Mittag–Leffler-type functions are presented in [1]. A comprehensive survey on the role of the Mittag–Leffler function and its generalizations in fractional analysis and fractional modelling as well as highlights on the history of the Mittag–Leffler Function can be read in [2]. Other aspects regarding the importance of the Mittag–Leffler function in the framework of the Fractional Calculus starting with the analytical properties of the classical Mittag–Leffler function and continuing with the main applications of the Mittag–Leffler function are presented in [3].

Caputo fractional derivative used in defining Caputo fractional integral operator has he disadvantage that the power kernel generates a singularity at the end point of the interval. To eliminate this problem, at first, Caputo and Fabrizio [4] introduced a new nonsingular fractional derivative with exponential kernel. Atangana and Baleanu improved the Caputo–Fabrizio fractional derivative with nonsingular kernel and defined Atangana–Baleanu fractional derivative with nonlocal and nonsingular kernel using the generalized Mittag–Leffler function. The Atangana–Baleanu fractional derivative is a generalization of the Caputo–Fabrizio derivative. Another extension of the Caputo fractional derivative involving the generalized hypergeometric type function is introduced in [5].

The investigation presented in the present paper is related to new applications of Atangana–Baleanu fractional integral applied to multiplier transformation introduced in [6]. In that paper, a new class of analytic functions was introduced and studied using the operator obtained as a combination of Atangana–Baleanu fractional integral [7] and multiplier transformation [8].

In this paper, a new approach is considered and the operator introduced in [6] is used for studies related to the theories of differential subordination and differential superordination. The operator was considered due to its properties already proved in studies regarding geometric function theory. It was previously applied for introducing new classes of analytic functions, so it is natural to consider the idea of using it in another direction of study in geometric function theory: obtaining new strong differential subordinations and superordinations. Classical differential subordinations and superordinations are first considered and sandwich-type results are obtained. This approach is seen in recent papers such as [911]. The basic definitions and notations related to those theories are presented in Section 2 of the paper and the original results are contained in Section 3.

Next, the special case of strong differential subordinations and superordinations is considered, inspired by recent outcome regarding those theories [1214]. The known results used in the study are presented in Section 4 of the paper. In Section 5, the Atangana–Baleanu fractional integral applied to extended multiplier transformation on some interesting classes defined particularly for strong differential subordinations and superordinations is used for establishing new results regarding the two theories and sandwich-type results are stated by combining them.

2. Differential Subordination and Superordination-Background

The usual definitions and notations regarding the theories of differential subordination [15] and differential superodination [16] are recalled in this section.

represents the class of analytic functions in . Two remarkable subclasses of arewhere is a positive integer and is a complex number, andwith .

We expose below the notions of differential subordination and, respectively, differential superordination:

The analytic function in is subordinate to the analytic function in , denoted , if there exists an analytic Schwarz function in , with the properties , for all , and , for all . When the function is univalent in , the differential subordination is equivalent to and .

Consider an univalent function in and . If the analytic function in verifies the second order differential subordinationthen is a solution of the differential subordination. A dominant of the solutions of the differential subordination is the univalent function for which for all satisfying (3). The best dominant of (3) is a dominant for which for all dominants of (3).

Consider an analytic function in and . If and are univalent functions and verifies the second order differential superordinationthen is a solution of the differential superordination (4). A subordinant is an analytic function for which for all satisfying (4). The best subordinant is an univalent subordinant for which for all subordinants of (4).

Miller and Mocanu [16] obtained conditions for , and such that the following implication

holds.

The Riemann–Liouville fractional integral ([17]) is introduce by the relation

The extended Atangana–Baleanu integral ([18]) is introduce by the relationfor any , , where is a fixed complex number and an analytic function on an open star-domain centered at .

The multiplier transformation ([8]) is introduced by the relationfor , , ,

We applied Atangana–Baleanu fractional integral for to multiplier transformation and we obtained a new operator ([6]):

Definition 1 (See [6]). The Atangana–Baleanu fractional integral related to the multiplier transformation is defined by the following equation:where , , , , .
After a simple calculation, this operator has the following form:for the function .
We will use the following known results to prove our subordination and superordination results from the next section.

Definition 2. Reference [15] is the set of all analytic and injective functions on , with , with the property for .

Lemma 1 (See [15]). Consider the univalent function in and analytic functions in a domain with for . Let and . is supposed to be starlike univalent in with the property for .
If is an analytic function such that , andthen and is the best dominant.

Lemma 2 (See [19]). Consider the convex univalent function in and , analytic functions in a domain . Let starlike univalent in and for .
If , with and is an univalent function in andthen and is the best subordinant.

3. Differential Subordination and Superordination

The first subordination result involving Atangana–Baleanu fractional integral applied to multiplier transformation given in Definition 1 is the following theorem.

Theorem 1. Consider the analytic and univalent function in with , for all and , , , , , , . Let a starlike univalent function in ,for , , andIf the differential subordinationis satisfied by , for , , then we obtain the following differential subordinationand the best dominant is .

Proof. Set , , and differentiating it, we obtainWe obtain .
Let and , it is easy to verify that is analytic in , is analytic in and .
Also, by setting and , we get that is a starlike univalent function in .
We obtainWe obtain thatIn this condition we can writeand by using (15), we obtainApplying Lemma 1, we get , , equivalently with , and the best dominant is .

Corollary 1. Suppose that (13) holds. Iffor , , , , , , with defined in (14), thenand the best dominant is .

Proof. Considering , when in Theorem 1, we obtain the corollary.

Corollary 2. Suppose that (13) holds. Iffor , , , , , , where is defined in (14), thenand the best dominant is .

Proof. Corollary follows by taking , , in Theorem 1.
The first superordination result is the following theorem:

Theorem 2. Consider an analytic and univalent function in with the properties and is a starlike univalent function in . Suppose that

If and is a univalent function in , with defined in (14), then the differential superorodination

implies the following differential superordinationand the best subordinant is .

Proof. Consider , , .
Let and , it is easy to verify that is an analytic function in , is an analytic function in with , .
When , it yields that , for , .
We getApplying Lemma 2, we getand the best subordinant is .

Corollary 3. Suppose that (26) holds. When andfor , , , , , , with defined in (14), thenand the best subordinant is .

Proof. Taking , in Theorem 2, we obtain the corollary.

Corollary 4. Suppose that (26) holds. If andfor , , , , , , with introduced in (14), thenand the best subordinant is .

Proof. Corollary follows taking , , in Theorem 2.
Theorem 1 combined with Theorem 2 give the following Sandwich theorem.

Theorem 3. Consider , analytic and univalent functions in with the properties , , for all , and , are starlike univalent functions. Consider satisfies (3.1) and satisfies (3.5). When and introduced in (14) is an univalent function in , thenfor , , , , , impliesand the best subordinant is and the best dominant is .
Taking , , where , we get the following corollary.

Corollary 5. Suppose that (13) and (26) hold. If andfor , , , , , , with defined in (14), thenhence the best subordinant is and the best dominant is .

Corollary 6. Suppose that (13) and (26) hold. When andfor , , , , , , where is introduced in (14), thenhence the best subordinant is and the best dominant is .

4. Strong Differential Subordination and Superordination-Background

Consider the class of analytic functions in , where and .

In [20] the following classes were introduced connected to the theory of strong differential subordination:with the holomorphic functions in for , for we denote this class with , andfor and , the holomorphic functions in for .

We remind the notion of strong differential subordinations defined by J. A. Antonino and S. Romaguera in [21] and developed by G. I. Oros and Gh. Oros in [22].

Definition 3 (See [22]). The analytic function is strongly subordinate to the analytic function if there exists an analytic function in , with , and for all . It is denoted , .

Remark 1. Reference [22] (i) When is analytic in and univalent in , for all , Definition 3 is equivalent to and , for all .
(ii) When and , the strong subordination is the usual subordination.
To obtain the strong differential subordinations results we need the following lemma:

Lemma 3 (See [23]). Consider the univalent function in and , analytic functions in a domain with the property for . Let starlike univalent function in and with for , .
When is an analytic function such that , andthen and the best dominant is .

The notion of strong differential superordinations was introduced in [24] as the dual notion of strong differential subordination.

Definition 4 (See [24]). The analytic function is strongly superordinate to the analytic function if there exists an analytic function in , with the properties , and , for all . It is denoted , , .

Remark 2. Reference [24] (i) When is analytic in , and univalent in , for all , Definition 4 is equivalent to and , for all .
(ii) When and , the strong superordination is the usual superordination.

Definition 5. Reference [25] is the set of analytic and injective functions on , with , with the property for . is the subclass of with .
To obtain the strong differential superordinations results we need the following lemma.

Lemma 4 (See [23]). Let the convex univalent function in and analytic functions in a domain . Let for , and starlike univalent function in .
When , such that and is univalent function in andthen and the best subordinant is .

The multiplier transformation was extended in [26] to the class of analytic functions defined in [22].

Definition 6 (See [26]). The multiplier transformation is introduced by the following infinite series:for , , .
We extend the Riemann–Liouville fractional integral ([17]) for a function by the relationand also the extended Atangana–Baleanu integral, for a function , denoted by , by the following equation:We extend the operator defined in [6] given in Definition 1 for a function .

Definition 7. The Atangana–Baleanu fractional integral regarding to the extended multiplier transformation is defined byfor , , , , and any .
Making an easy computation, we obtain the following form for this extended operator:for the function .
Using this operator, new strong differential subordinations and superordinations results are obtained in the next section.

5. Strong Differential Subordination and Superordination

Similar to the results from Section 3 we get the following results for the extended operator given in Definition 7:

Theorem 4. Consider the analytic and univalent function in with , for all , and , , , , , , , . Suppose that is starlike univalent in . Letfor , , , andIf the strong differential subordinationis satisfied by , for , , then we get the strong differential subordinationand the best dominant is .

Proof. Define , , , . Differentiating it with respect to , we get