# Differential Sandwich Theorems for Atangana–Baleanu Fractional Integral Applied to Extended Multiplier Transformation

Authors: Alb Lupaş Alina; Bipan Hazarika
Journal: Mathematical Problems in Engineering
 (2022)

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## 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 classAζ∗ 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.


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## Body

## 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 [9–11]. 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 [12–14]. 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.ℋU represents the class of analytic functions in U=z∈ℂ:z<1. Two remarkable subclasses of ℋU are(1)ℋa,n=f∈ℋU:fz=a+anzn+an+1zn+1+…,z∈U,where n is a positive integer and a is a complex number, and(2)An=f∈ℋU:fz=z+an+1zn+1+…,z∈U,with A=A1.We expose below the notions of differential subordination and, respectively, differential superordination:The analytic functionf in U is subordinate to the analytic function g in U, denoted f≺g, if there exists an analytic Schwarz function w in U, with the properties wz<1, for all z∈U, w0=0 and fz=gwz, for all z∈U. When the function g is univalent in U, the differential subordination is equivalent to fU⊂gU and f0=g0.Considerh an univalent function in U and ψ:ℂ3×U⟶ℂ. If the analytic function p in U verifies the second order differential subordination(3)ψpz,zp′z,z2p″z;z≺hz,z∈U,then p is a solution of the differential subordination. A dominant of the solutions of the differential subordination is the univalent function q for which p≺q for all p satisfying (3). The best dominant of (3) is a dominant q˜ for which q˜≺q for all dominants q of (3).Considerh an analytic function in U and ψ:ℂ2×U⟶ℂ. If p and ψpz,zp′z,z2p″z;z are univalent functions and p verifies the second order differential superordination(4)hz≺ψpz,zp′z,z2p″z;z,z∈U,then p is a solution of the differential superordination (4). A subordinant is an analytic function q for which q≺p for all p satisfying (4). The best subordinant is an univalent subordinant q˜ for which q≺q˜ for all subordinants q of (4).Miller and Mocanu [16] obtained conditions for h, q and ψ such that the following implication(5)hz≺ψpz,zp′z,z2p″z;z⇒qz≺pz.holds.The Riemann–Liouville fractional integral ([17]) is introduce by the relation(6)IcvcRLfz=1Γν∫czz−wν−1fwdw,Reν>0.The extended Atangana–Baleanu integral ([18]) IzvcABfz is introduce by the relation(7)IzvcABfz=1−νBνfz+νBνIzνcRLfz,for any ν∈ℂ, z∈D/c, where c is a fixed complex number and f an analytic function on an open star-domain D centered at c.The multiplier transformation ([8]) Im,α,lfz is introduced by the relation(8)Im,α,lfz≔z+∑k=2∞1+αk−1+l1+lmakzk,for f∈A, m∈ℕ∪0, α,l≥0,We applied Atangana–Baleanu fractional integral forc=0 to multiplier transformation and we obtained a new operator ([6]):Definition 1 (See [6]).
The Atangana–Baleanu fractional integral related to the multiplier transformationIm,α,lf is defined by the following equation:(9)Izv0ABIm,α,lfz=1−νBνIm,α,lfz+νBνIzv0RLIm,α,lfz,where f∈A, ν∈ℂ, m∈ℕ∪0, α,l≥0, z∈D/0.
After a simple calculation, this operator has the following form:(10)Izv0ABIm,α,lfz=1−νBνz+νΓ2+νBνzν+1+1−νBν∑k=2∞1+αk−1+ll+1makzk+νBν∑k=2∞1+αk−1+ll+1mΓk+1ν+k+1akzk+ν,for the function fz=z+∑k=2∞akzk∈A.
We will use the following known results to prove our subordination and superordination results from the next section.Definition 2.
Reference [15] Q is the set of all analytic and injective functions f on U¯/Ef, with Ef=ζ∈∂U:limz⟶ζfz=∞, with the property f′ζ≠0 for ζ∈∂U/Ef.Lemma 1 (See [15]).
Consider the univalent functionq in U and θ,ϕ analytic functions in a domain D⊃qU with ϕw≠0 for w∈qU. Let Qz=zq′zϕqz and hz=θqz+Qz. Q is supposed to be starlike univalent in U with the property Rezh′z/Qz>0 for z∈U.
Ifp is an analytic function such that pU⊆D, p0=q0 and(11)zp′zϕpz+θpz≺zq′zϕqz+θqz.
thenpz≺qz and q is the best dominant.Lemma 2 (See [19]).
Consider the convex univalent functionq in U and θ, ϕ analytic functions in a domain D⊃qU . Let ψz=zq′zϕqz starlike univalent in U and Reθ′qz/ϕqz>0 for z∈U.
Ifpz∈ℋq0,1∩Q, with pU⊆D and zp′zϕpz+θpz is an univalent function in U and(12)zq′zϕqz+θqz≺zp′zϕpz+θpz.
thenqz≺pz and q is the best subordinant.


## 3. Differential Subordination and Superordination

The first subordination result involving Atangana–Baleanu fractional integral applied to multiplier transformation given in Definition1 is the following theorem.Theorem 1.
Consider the analytic and univalent functionq in U with qz≠0, for all z∈U and Izv0ABIm,α,lfz/zn∈ℋU, f∈A, m∈ℕ∪0, α,l≥0, n>0, ν∈ℂ, z∈U/0. Let zq′z/qz a starlike univalent function in U,(13)Re2adqz2+bdqz+1−zq′zqz+zq″zq′z>0,for a,b,c,d∈ℂ, d≠0, z∈U and(14)ψm,α,lνn,a,b,c;z≔aIzv0ABIm,α,lfz/zn∈ℋUIm,α,lfzz2n+bIzv0ABIm,α,lfzzn+c+dnzIzv0ABIm,α,lfz′Izv0ABIm,α,lfz−1.
If the differential subordination(15)ψm,α,lνn,a,b,c;z≺aqz2+bqz+c+dzq′zqz,is satisfied by q, for a,b,c,d∈ℂ, d≠0, then we obtain the following differential subordination(16)Izv0ABIm,α,lfzzn≺qz,z∈U,and the best dominant is q.Proof.
Setpz:=Izv0ABIm,α,lfz/zn, z∈U, z≠0 and differentiating it, we obtain(17)p′z=nIzv0ABIm,α,lfzzn−1Izv0ABIm,α,lfz′z−Izv0ABIm,α,lfzz2=nIzv0ABIm,α,lfzzn−1Izv0ABIm,α,lfz′z−nzpz.
We obtainzp′z/pz=nzIzv0ABIm,α,lfz′/Izv0ABIm,α,lfz−1.
Letθw:=aw2+bw+c and Qw:=d/w, it is easy to verify that θ is analytic in ℂ, ϕ is analytic in ℂ/0 and ϕw≠0,w∈ℂ/0.
Also, by settingQz=zq′zϕqz=dzq′z/qz and hz=θqz+Qz=aqz2+bqz+c+dzq′z/qz, we get that Qz is a starlike univalent function in U.
We obtain(18)h′z=2aqzq′z+bq′z+dq′z+zq″zqz−zq′z2qz2,zh′zQz=zh′zdzq′z/qz=2adqz2+bdqz+1−zq′zqz+zq″zq′z.
We obtain that(19)Rezh′zQz=Re2adqz2+bdqz+1−zq′zqz+zq″zq′z>0.
In this condition we can write(20)apz2+bpz+c+dzp′zpz=aIzv0ABIm,α,lfzz2n+bIzv0ABIm,α,lfzzn+c+dnzIzv0ABIm,α,lfz′Izv0ABIm,α,lfz−1,and by using (15), we obtain(21)apz2+bpz+c+dzp′zpz≺aqz2+bqz+c+dzq′zqz.
Applying Lemma1, we get pz≺qz, z∈U, equivalently with Izv0ABIm,α,lfz/zn≺qz, z∈U and the best dominant is q.Corollary 1.
Suppose that (13) holds. If(22)ψm,α,lνn,a,b,c;z≺a1+Az1+Bz2+b1+Az1+Bz+c+dA−Bz1+Az1+Bz,
form∈ℕ∪0, α,l≥0, n>0, ν,a,b,c,d∈ℂ, d≠0, −1≤B<A≤1, with ψm,α,lνn,a,b,c;z defined in (14), then(23)Izv0ABIm,α,lfzzn≺1+Az1+Bz,z∈U,and the best dominant is 1+Az/1+Bz.Proof.
Consideringqz=1+Az/1+Bz, when −1≤B<A≤1 in Theorem 1, we obtain the corollary.Corollary 2.
Suppose that (13) holds. If(24)ψm,α,lνn,a,b,c;z≺a1+z1−z2γ+b1+z1−zγ+c+2dγz1−z2,for m∈ℕ∪0, α,l≥0, n>0, ν,a,b,c,d∈ℂ, d≠0, 0<γ≤1, where ψm,α,lνn,a,b,c;z is defined in (14), then(25)Izv0ABIm,α,lfzzn≺1+z1−zγ,z∈U,and the best dominant is 1+z/1−zγ.Proof.
Corollary follows by takingqz=1+z/1−zγ, 0<γ≤1, in Theorem 1.
The first superordination result is the following theorem:Theorem 2.
Consider an analytic and univalent functionq in U with the properties qz≠0 and zq′z/qz is a starlike univalent function in U. Suppose that(26)Re2adqz2+bqz>0,fora,b,d∈ℂ,d≠0.IfIzv0ABIm,α,lfz/zn∈ℋ0,n∩Q and ψm,α,lνn,a,b,c;z is a univalent function in U, with ψm,α,lνn,a,b,c;z defined in (14), then the differential superorodination(27)aqz2+bqz+c+dzq′zqz≺ψm,α,lνn,a,b,c;z.implies the following differential superordination(28)qz≺Izv0ABIm,α,lfzzn,z∈U,and the best subordinant is q.Proof.
Considerpz:=Izv0ABIm,α,lfz/zn, z∈U, z≠0.
Letθw:=aw2+bw+c and ϕw:=d/w, it is easy to verify that θ is an analytic function in ℂ, ϕ is an analytic function in ℂ/0 with ϕw≠0, w∈ℂ/0.
Whenθ′qz/ϕqz=q′z2aqz+bqz/d, it yields that Reθ′qz/ϕqz=Re2a/dqz2+b/dqz>0, for a,b,d∈ℂ, d≠0.
We get(29)aqz2+bqz+c+dzq′zqz≺apz2+bpz+c+dzp′zpz.
Applying Lemma2, we get(30)qz≺pz=Izv0ABIm,α,lfzzn,z∈U,and the best subordinant is q.Corollary 3.
Suppose that (26) holds. When Izv0ABIm,α,lfz/zn∈ℋ0,n∩Q and(31)a1+Az1+Bz2+b1+Az1+Bz+c+dA−Bz1+Az1+Bz≺ψm,α,lνn,a,b,c;z,for m∈ℕ∪0, α,l≥0, n>0, ν,a,b,c,d∈ℂ, d≠0, −1≤B<A≤1, with ψm,α,lνn,a,b,c;z defined in (14), then(32)1+Az1+Bz≺Izv0ABIm,α,lfzzn,z∈U,and the best subordinant is 1+Az/1+Bz.Proof.
Takingqz=1+Az/1+Bz, −1≤B<A≤1 in Theorem 2, we obtain the corollary.Corollary 4.
Suppose that (26) holds. If Izv0ABIm,α,lfz/zn∈ℋ0,n∩Q and(33)a1+z1−z2γ+b1+z1−zγ+c+2dγz1−z2≺ψm,α,lνn,a,b,c;z,for m∈ℕ∪0, α,l≥0, n>0, ν,a,b,c,d∈ℂ, d≠0, 0<γ≤1, with ψm,α,lνn,a,b,c;z introduced in (14), then(34)1+z1−zγ≺Izv0ABIm,α,lfzzn,z∈U,and the best subordinant is 1+z/1−zγ.Proof.
Corollary follows takingqz=1+z/1−zγ, 0<γ≤1, in Theorem 2.
Theorem1 combined with Theorem 2 give the following Sandwich theorem.Theorem 3.
Considerq1, q2 analytic and univalent functions in U with the properties q1z≠0, q2z≠0, for all z∈U, and zq1′z/q1z, zq2′z/q2z are starlike univalent functions. Consider q1 satisfies (3.1) and q2 satisfies (3.5). When Izv0ABIm,α,lfz/zn∈ℋ0,n∩Q and ψm,α,lνn,a,b,c;z introduced in (14) is an univalent function in U, then(35)aq1z2+bq1z+c+dzq1′zq1z≺ψm,α,lνn,a,b,c;z,≺aq2z2+bq2z+c+dzq2′zq2z,for m∈ℕ∪0, α,l≥0, n>0, ν,a,b,c,d∈ℂ, d≠0, implies(36)q1z≺Izv0ABIm,α,lfzzn≺q2z,z∈U,and the best subordinant is q1 and the best dominant is q2.
Takingq1z=1+A1z/1+B1z, q2z=1+A2z/1+B2z, where −1≤B2<B1<A1<A2≤1, we get the following corollary.Corollary 5.
Suppose that (13) and (26) hold. If Izv0ABIm,α,lfz/zn∈ℋ0,n∩Q and(37)a1+A1z1+B1z2+b1+A1z1+B1z+c+dA1−B1z1+A1z1+B1z≺ψm,α,lνn,a,b,c;z≺a1+A2z1+B2z2+b1+A2z1+B2z+c+dA2−B2z1+A2z1+B2z,for m∈ℕ∪0, α,l≥0, n>0, ν,a,b,c,d∈ℂ, d≠0, −1≤B2≤B1<A1≤A2≤1, with ψm,α,lνn,a,b,c;z defined in (14), then(38)1+A1z1+B1z≺Izv0ABIm,α,lfzzn≺1+A2z1+B2z,hence the best subordinant is 1+A1z/1+B1z and the best dominant is 1+A2z/1+B2z.Corollary 6.
Suppose that (13) and (26) hold. When Izv0ABIm,α,lfz/zn∈ℋ0,n∩Q and(39)a1+z1−z2γ1+b1+z1−zγ1+c+2dγ1z1−z2≺ψm,α,lνn,a,b,c;z≺a1+z1−z2γ2+b1+z1−zγ2+c+2dγ2z1−z2,for m∈ℕ∪0, α,l≥0, n>0, ν,a,b,c,d∈ℂ, d≠0, 0<γ1,γ2≤1, where ψm,α,lνn,a,b,c;z is introduced in (14), then(40)1+z1−zγ1≺Izv0ABIm,α,lfzzn≺1+z1−zγ2,hence the best subordinant is 1+z/1−zγ1 and the best dominant is 1+z/1−zγ2.


## 4. Strong Differential Subordination and Superordination-Background

Consider the classℋU×U¯ of analytic functions in U×U¯, where U=z∈ℂ:z<1 and U¯=z∈ℂ:z≤1.In [20] the following classes were introduced connected to the theory of strong differential subordination:(41)Anζ∗=f∈ℋU×U¯:fz,ζ=z+an+1ζzn+1+…,z∈U,ζ∈U¯,with akζ the holomorphic functions in U¯ for k≥2, for n=1 we denote this class with Aζ∗, and(42)ℋ∗a,n,ζ=f∈ℋU×U¯:fz,ζ=a+anζzn+an+1ζzn+1+…,z∈U,ζ∈U¯,for a∈ℂ and n∈ℕ, akζ the holomorphic functions in U¯ for k≥n.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 functionfz,ζ is strongly subordinate to the analytic function Hz,ζ if there exists an analytic function w in U, with wz<1, w0=0 and fz,ζ=Hwz,ζ for all ζ∈U¯. It is denoted fz,ζ≺≺Hz,ζ, z∈U,ζ∈U¯.Remark 1.
Reference [22] (i) When fz,ζ is analytic in U×U¯ and univalent in U, for all ζ∈U¯, Definition 3 is equivalent to fU×U¯⊂HU×U¯ and f0,ζ=H0,ζ, for all ζ∈U¯.
(ii) Whenfz,ζ≡fz and Hz,ζ≡Hz, 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 functionq in U×U¯ and θ, ϕ analytic functions in a domain D⊃qU×U¯ with the property ϕw≠0 for w∈qU×U¯. Let Qz,ζ=zqz′z,ζϕqz,ζ starlike univalent function in U×U¯ and hz,ζ=θqz,ζ+Qz,ζ with Rezhz′z,ζ/Qz,ζ>0 for z∈U, ζ∈U¯.
Whenp is an analytic function such that pU×U¯⊆D, p0,ζ=q0,ζ and(43)zpz′z,ζϕpz,ζ+θpz,ζ≺≺zqz′z,ζϕqz,ζ+θqz,ζ.then pz,ζ≺≺qz,ζ and the best dominant is q.The notion of strong differential superordinations was introduced in [24] as the dual notion of strong differential subordination.Definition 4 (See [24]).
The analytic functionfz,ζ is strongly superordinate to the analytic function Hz,ζ if there exists an analytic function w in U, with the properties wz<1, w0=0 and Hz,ζ=fwz,ζ, for all ζ∈U¯. It is denoted Hz,ζ≺≺fz,ζ, z∈U, ζ∈U¯.Remark 2.
Reference [24] (i) When fz,ζ is analytic in U×U¯, and univalent in U, for all ζ∈U¯, Definition 4 is equivalent to HU×U¯⊂fU×U¯ and H0,ζ=f0,ζ, for all ζ∈U¯.
(ii) Whenfz,ζ≡fz and Hz,ζ≡Hz, the strong superordination is the usual superordination.Definition 5.
Reference [25] Q∗ is the set of analytic and injective functions on U¯×U¯/Ef,ζ, with Ef,ζ=y∈∂U:limz⟶yfz,ζ=∞, with the property fz′y,ζ≠0 for y∈∂U×U¯/Ef,ζ. Q∗a is the subclass of Q∗ with f0,ζ=a.
To obtain the strong differential superordinations results we need the following lemma.Lemma 4 (See [23]).
Letq the convex univalent function in U×U¯ and ν,ϕ analytic functions in a domain D⊃qU×U¯. Let Reθz′qz,ζ/ϕqz,ζ>0 for z∈U, ζ∈U¯ and ψz,ζ=zqz′z,ζϕqz,ζ starlike univalent function in U×U¯.
Whenpz,ζ∈ℋ∗q0,ζ,1,ζ∩Q∗, such that pU×U¯⊆D and θpz,ζ+zpz′zϕpz,ζ is univalent function in U×U¯ and(44)zqz′z,ζϕqz,ζ+θqz,ζ≺≺zpz′z,ζϕpz,ζ+θpz,ζ,then qz,ζ≺≺pz,ζ and the best subordinant is q.The multiplier transformation was extended in [26] to the class of analytic functions Anζ∗ defined in [22].Definition 6 (See [26]).
The multiplier transformationIm,λ,l:Aζ∗⟶Aζ∗ is introduced by the following infinite series:(45)Im,λ,lfz,ζ≔=z+∑j=2∞1+λj−1+ll+1majζzj,for f∈Aζ∗, m∈ℕ∪0, λ,l≥0.
We extend the Riemann–Liouville fractional integral ([17]) for a function f∈Aζ∗ by the relation(46)IzvcRLfz,ζ=1Γν∫czz−wν−1fw,ζdw,Reν>0,and also the extended Atangana–Baleanu integral, for a function f∈Aζ∗, denoted by IzvcABfz,ζ, by the following equation:(47)IzvcABfz,ζ=1−νBνfz,ζ+νBνIzvcRLfz,ζ.
We extend the operator defined in [6] given in Definition 1 for a function f∈Aζ∗.Definition 7.
The Atangana–Baleanu fractional integral regarding to the extended multiplier transformationIm,α,lfz,ζ is defined by(48)Izv0ABIm,α,lfz,ζ=1−νBνIm,α,lfz,ζ+νBνIzv0RLIm,α,lfz,ζ,for f∈Aζ∗, m∈ℕ∪0, α,l≥0, ν∈ℂ, and any z∈D/0.
Making an easy computation, we obtain the following form for this extended operator:(49)Izv0ABIm,α,lfz,ζ=1−νBνz+νΓ2+νBνzν+1+1−νBν∑k=2∞1+αk−1+ll+1makζzk+νBν∑k=2∞1+αk−1+ll+1mΓk+1ν+k+1akζzk+ν,for the function fz,ζ=z+∑k=2∞akζzk∈Aζ∗.
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 Section3 we get the following results for the extended operator given in Definition 7:Theorem 4.
Consider the analytic and univalent functionqz,ζ in U×U¯ with qz,ζ≠0, for all z∈U/0, ζ∈U¯ and Izv0ABIm,α,lfz,ζ/zn∈ℋU×U¯, f∈Aζ∗, m∈ℕ∪0, α,l≥0, n>0, ν∈ℂ, z∈U/0, ζ∈U¯. Suppose that zqz′z,ζ/qz,ζ is starlike univalent in U×U¯. Let(50)Re2adqz,ζ2+bqz,ζ+1−zqz′z,ζqz,ζ+zqz2′′z,ζqz′z,ζ>0,for a,b,c,d∈ℂ, d≠0, z∈U/0, ζ∈U¯ and(51)ψm,α,lνn,a,b,c;z,ζ≔=aIzv0ABIm,α,lfz,ζz2n+bIzv0ABIm,α,lfz,ζzn+c+dnzIzv0ABIm,α,lfz,ζz′Izv0ABIm,α,lfz,ζ−1.
If the strong differential subordination(52)ψm,α,lνn,a,b,c;z,ζ≺≺aqz,ζ2+bqz,ζ+c+dzqz′z,ζqz,ζ,is satisfied by q, for a,b,c,d∈ℂ, d≠0, then we get the strong differential subordination(53)Izv0ABIm,α,lfz,ζzn≺≺qz,ζ,z∈U,ζ∈U¯,and the best dominant is q.Proof.
Definepz,ζ:=Izv0ABIm,α,lfz,ζ/zn, z∈U, z≠0, ζ∈U¯. Differentiating it with respect to z, we get(54)pz′z,ζ=nIzv0ABIm,α,lfz,ζzn−1Izv0ABIm,α,lfz,ζz′z−Izv0ABIm,α,lfz,ζz2=nIzv0ABIm,α,lfz,ζzn−1Izv0ABIm,α,lfz,ζz′z−nzpz,ζ.
Then,zpz′z,ζ/pz,ζ=nzIzv0ABIm,α,lfz,ζz′/Izv0ABIm,α,lfz,ζ−1.
Letθw:=aw2+bw+c and Qw:=d/w, it is easy to verify that θ is analytic in ℂ, ϕ is analytic in ℂ/0 and ϕw≠0,w∈ℂ/0.
Also, by setting the starlike univalent functionQz,ζ=zqz′zϕqz,ζ=dzqz′z,ζ/qz,ζ in U×U¯ and hz,ζ=θqz,ζ+Qz,ζ=aqz,ζ2+bqz,ζ+c+dzqz′z,ζ/qz,ζ, we get hz′z,ζ=2aqz,ζqz′z,ζ+bqz′z,ζ+dqz′z,ζ+zqz2′′z,ζqz,ζ−zqz′z,ζ2/qz,ζ2 and zhz′z,ζ/Qz,ζ=zhz′z,ζ/dzqz′z,ζ/qz,ζ=2a/dqz,ζ2+b/dqz,ζ+1−zqz′z,ζ/qz,ζ+zqz2′′z,ζ/qz′z,ζ.
We obtain thatRezhz′z,ζ/Qz,ζ=Re2a/dqz,ζ2+b/dqz,ζ+1−zqz′z,ζ/qz,ζ+zqz2′′z,ζ/qz′z,ζ>0.
We can write.apz,ζ2+bpz,ζ+c+dzpz′z,ζ/pz,ζ=aIzv0ABIm,α,lfz,ζ/z2n+bIzv0ABIm,α,lfz,ζ/zn+c+dnzIzv0ABIm,α,lfz,ζz′/Izv0ABIm,α,lfz,ζ−1.
Taking account the strong differential subordination (5.3), we getapz,ζ2+bpz,ζ+c+dzpz′z,ζ/pz,ζ≺≺aqz,ζ2+bqz,ζ+c+dzqz′z,ζ/qz,ζ.
Applying Lemma3, we obtain pz,ζ≺≺qz,ζ, z∈U, ζ∈U¯, which is equivalently with Izv0ABIm,α,lfz,ζ/zn≺≺qz,ζ, z∈U, ζ∈U¯, and the best dominant is q.Corollary 7.
Suppose that (50) holds. When(55)ψm,α,lνn,a,b,c;z,ζ≺≺aζ+Azζ+Bz2+bζ+Azζ+Bz+c+dA−Bζzζ+Azζ+Bz,for m∈ℕ∪0, α,l≥0, n>0, ν,a,b,c,d∈ℂ, d≠0, −1≤B<A≤1, with ψm,α,lνn,a,b,c;z,ζ defined in (51), then(56)Izv0ABIm,α,lfz,ζzn≺≺ζ+Azζ+Bz,z∈U,ζ∈U¯,and the best dominant is ζ+Az/ζ+Bz.Proof.
Takingqz,ζ=ζ+Az/ζ+Bz, −1≤B<A≤1 in Theorem 4, we obtain the corollary.Corollary 8.
Suppose that (50) holds. When(57)ψm,α,lνn,a,b,c;z,ζ≺≺aζ+zζ−z2γ+bζ+zζ−zγ+c+2dγζzζ2−z2,for m∈ℕ∪0, α,l≥0, n>0, ν,a,b,c,d∈ℂ, d≠0, 0<γ≤1, with ψm,α,lνn,a,b,c;z,ζ defined in (51), then(58)Izv0ABIm,α,lfz,ζzn≺≺ζ+zζ−zγ,z∈U,ζ∈U¯,and the best dominant is ζ+z/ζ−zγ .Proof.
Corollary follows by using Theorem4 for qz,ζ=ζ+z/ζ−zγ, 0<γ≤1.Theorem 5.
Consider the analytic and univalent functionq in U×U¯ with the properties qz,ζ≠0 and zqz′z,ζ/qz,ζ is starlike univalent function in U×U¯. Suppose that(59)Re2adqz,ζ2+bqz,ζ>0,fora,b,d∈ℂ,d≠0.
WhenIzv0ABIm,α,lfz,ζ/zn∈ℋ0,n,ζ∩Q∗ and ψm,α,lνn,a,b,c;z,ζ defined by (51) is an univalent function in U×U¯, then the strong differential superordination(60)aqz,ζ2+bqz,ζ+c+dzqz′z,ζqz,ζ≺≺ψm,α,lνn,a,b,c;z,ζ,implies the strong differential superordination(61)qz,ζ≺≺Izv0ABIm,α,lfz,ζzn,z∈U,ζ∈U¯,and the best subordinant is q.Proof.
Denotepz,ζ:=Izv0ABIm,α,lfz,ζ/zn, z∈U, z≠0, ζ∈U¯.
Setθw:=aw2+bw+c and ϕw:=d/w it is easy to verify that θ is an analytic function in ℂ, ϕ is an analytic function in ℂ/0 with ϕw≠0, for w∈ℂ/0.
Becauseθz′qz,ζ/ϕqz,ζ=qz′z,ζ2aqz,ζ+bqz,ζ/d, it follows that Reθz′qz,ζ/ϕqz,ζ=Re2a/dqz,ζ2+b/dqz,ζ>0, for a,b,d∈ℂ, d≠0.
It yields(62)aqz,ζ2+bqz,ζ+c+dzqz′z,ζqz,ζ≺≺apz,ζ2+bpz,ζ+c+dzpz′z,ζpz,ζ.
By Applying Lemma4, we get(63)qz,ζ≺≺pz,ζ=Izv0ABIm,α,lfz,ζzn,z∈U,ζ∈U¯,and the best subordinant is q.Corollary 9.
Suppose that (59) holds. If Izv0ABIm,α,lfz,ζ/zn∈ℋ0,n,ζ∩Q∗ and(64)aζ+Azζ+Bz2+bζ+Azζ+Bz+c+dA−Bζzζ+Azζ+Bz≺≺ψm,α,lνn,a,b,c;z,ζ,for m∈ℕ∪0, α,l≥0, n>0, ν,a,b,c,d∈ℂ, d≠0, −1≤B<A≤1, with ψm,α,lνn,a,b,c;z,ζ defined by (51), then(65)ζ+Azζ+Bz≺≺Izv0ABIm,α,lfz,ζzn,z∈U,ζ∈U¯,and the best subordinant is ζ+Az/ζ+Bz.Proof.
Takingqz,ζ=ζ+Az/ζ+Bz, −1≤B<A≤1 in Theorem 5, we obtain the corollary.Corollary 10.
Suppose that (59) holds. If Izv0ABIm,α,lfz,ζ/zn∈ℋ0,n,ζ∩Q∗ and(66)aζ+zζ−z2γ+bζ+zζ−zγ+c+2dγζzζ2−z2≺≺ψm,α,lνn,a,b,c;z,ζ,for m∈ℕ∪0, α,l≥0, n>0, ν,a,b,c,d∈ℂ, d≠0, 0<γ≤1, where ψm,α,lνn,a,b,c;z,ζ is defined in (51), then(67)ζ+zζ−zγ≺≺Izv0ABIm,α,lfz,ζzn,z∈U,ζ∈U¯,and the best subordinant is ζ+z/ζ−zγ.Proof.
Corollary follows takingqz,ζ=ζ+z/ζ−zγ, 0<γ≤1, in Theorem 5.
Theorem4 combined with Theorem 5 give the following Sandwich theorem for the extended operator.Theorem 6.
Consider the analytic and univalent functionsq1 and q2 in U×U¯ with the properties q1z,ζ≠0, q2z,ζ≠0, for all z∈U, ζ∈U¯, and zq1z′z,ζ/q1z,ζ, zq2z′z,ζ/q2z,ζ are starlike univalent functions in U×U¯. Consider that q1 satisfies (50) and (59) is satisfied by q2. When Izv0ABIm,α,lfz,ζ/zn∈ℋ0,n,ζ∩Q∗ and ψm,α,lνn,a,b,c;z,ζ defined in (51) is an univalent function in U×U¯, then(68)aq1z,ζ2+bq1z,ζ+c+dzq1z′z,ζq1z,ζ≺≺ψm,α,lνn,a,b,c;z,ζ≺≺aq2z,ζ2+bq2z,ζ+c+dzq2z′z,ζq2z,ζ,for m∈ℕ∪0, α,l≥0, n>0, ν,a,b,c,d∈ℂ, d≠0, implies(69)q1z,ζ≺≺Izv0ABIm,α,lfz,ζzn≺≺q2z,ζ,z∈U,ζ∈U¯,and the best subordinant is q1 and the best dominant is q2.
Forq1z,ζ=ζ+A1z/ζ+B1z, q2z,ζ=ζ+A2z/ζ+B2z, with −1≤B2<B1<A1<A2≤1, we obtain the following corollary.Corollary 11.
Suppose that (50) and (59) hold. If Izv0ABIm,α,lfz,ζ/zn∈ℋ0,n,ζ∩Q∗ and(70)aζ+A1zζ+B1z2+bζ+A1zζ+B1z+c+dA1−B1ζzζ+A1zζ+B1z≺≺ψm,α,lνn,a,b,c;z,ζ≺≺aζ+A2zζ+B2z2+bζ+A2zζ+B2z+c+dA2−B2ζzζ+A2zζ+B2z,for m∈ℕ∪0, α,l≥0, n>0, ν,a,b,c,d∈ℂ, d≠0, −1≤B2≤B1<A1≤A2≤1, with ψm,α,lνn,a,b,c;z,ζ defined by (51), then(71)ζ+A1zζ+B1z≺≺Izv0ABIm,α,lfz,ζzn≺≺ζ+A2zζ+B2z,hence the best subordinant is ζ+A1z/ζ+B1z and the best dominant is ζ+A2z/ζ+B2z.Corollary 12.
Suppose that (50) and (59) hold. If Izv0ABIm,α,lfz,ζ/zn∈ℋ0,n,ζ∩Q∗ and(72)aζ+zζ−z2γ1+bζ+zζ−zγ1+c+2dγ1ζzζ2−z2≺≺ψm,α,lνn,a,b,c;z,ζ≺≺aζ+zζ−z2γ2+bζ+zζ−zγ2+c+2dγ2ζzζ2−z2,for m∈ℕ∪0, α,l≥0, n>0, ν,a,b,c,d∈ℂ, d≠0, 0<γ1,γ2≤1, where ψm,α,lνn,a,b,c;z,ζ is defined in (51), then(73)ζ+zζ−zγ1≺≺Izv0ABIm,α,lfz,ζzn≺≺ζ+zζ−zγ2,hence the best subordinant is ζ+z/ζ−zγ1 and the best dominant is ζ+z/ζ−zγ2.


## 6. Conclusion

A previously introduced operator given in Definition1 as Atangana–Baleanu fractional integral applied to multiplier transformation is used for obtaining new differential subordinations and superordinations. The theorems stated and proved in Section 3 of the present paper concern differential subordinations and superodinations involving Atangana–Baleanu fractional integral applied to multiplier transformation for which the best dominant and best subordinant are given, respectively. Considering specific functions with known geometric properties as best dominant and best subordinant, respectively, corollaries are obtained for each of the new theorems. Further, in Section 4, Atangana–Baleanu fractional integral is applied to extended multiplier transformation and a new operator is given in Definition 7 considering special classes of functions considered in the study of strong differential subordination and superordination theories. Using this newly introduced operator, strong differential subordinations and superordinations are obtained in Section 5 and the best dominant and best subordinant are provided, respectively. Using particular functions in the positions of the best dominant and best subordinant, corollaries are stated for each theorem.As further directions of study in which the results presented in this paper could be used, we mention the possible adaptation of the operators to quantum calculus following the line of research proposed in [27, 28] and having in mind the comprehensive presentation of fractional q -calculus given in [29].Also, Atangana–Baleanu fractional integral applied to extended multiplier transformation could be used for introducing and studying new classes of univalent functions just as Atangana–Baleanu fractional integral applied to multiplier transformation was used in [6].



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