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

Authors: Najla M. Alarifi; Rabha W. Ibrahim
Journal: Journal of Function Spaces
 (2021)

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## 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.


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## Body

## 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∪≔z∈ℂ:z<1 be the open unit disk. Two analytic functions g1, g2 in ∪ are called subordinated denoting by g1≺g2org1z≺g2z,z∈U, if there exits an analytic function ω,w≤z<1 having the formula
(1)g1z=g2wz,z∈∪.
g1 is majorized by g2 denoting by g1≪g2 if and only if
(2)g1z=wzg2z,z∈∪;equivalently, the coefficient inequality is held an≤bn, respectively.There is a deep construction between subordination and majorization [9] in ∪ for selected distinct classes comprising the convex class C:
(3)1+Rzg″zg′z>0,z∈∪,and starlike functions S∗(4)Rzg′gz>0,z∈∪.Definition 2.
We present a class of analytic functions by(5)fz=z+∑n=2∞anzn,z∈∪.
This class is denoted byΛ and known as the class of univalent functions which is normalized by f0=f′0−1=0.
Associated with the termsS∗ and C, we present the term P of all analytic functions p in ∪ with a positive real part in ∪ and p0=1.
Two analytic functionsf,g are called convoluted, denoting by f∗g if and only if
(6)f∗gz=∑n=0∞anzn∗∑n=0∞gnzn=∑n−0∞angnzn.Definition 3.
The generalized Mittag-Leffler function is defined by [10–12]
(7)Ξv,μ℘z=∑n=0∞℘nΓvn+μznn!,where ϑn represents the Pochhammer symbol and
(8)Ξv,μ1z∶=Ξv,μz=∑n=0∞znΓvn+μ℘0=1,℘n=℘℘+1⋯℘+n−1.
Note thatΞv,u℘z is an ultimate traditional generalization of the function ez, where Ξ1,11z=ez.Moreover, it can be formulated by the Fox-Write hypergeometric function, as follows:(9)Ξv,μ℘z=1Γ℘Ψ11℘,1v,μ;z.


### 2.2. Complex Prabhakar Operator (CPO)

The Prabhakar integral operator is defined for analytic function(10)ψz∈H0,1=ψz=ψ1z+ψ2z2+⋯,z∈∪by the formula [13, 14]
(11)Pα,βγ,ωψz=∫0zz−ζβ−1Ξα,βγωz−ζαψζdζ=ψ·ϱα,βγ,ωzα,β,γ,ω∈ℂ,z∈∪.Moreover [13, 14],
(12)ϱα,βγ,ωz≔zβ−1Ξα,βγωzα,Ξα,βγχ=∑n=0∞Γγ+nΓγΓαn+βχnn!.For example, letψz=zς−1, then (see [15], Corollary 2.3)
(13)Pα,βγ,ωzς−1=∫0zz−ζβ−1Ξα,βγωz−ζαζς−1dζ=Γςzβ+ς−1Ξα,β+ςγωzα.The Prabhakar derivative can be computed by the formula [13]
(14)Dkα,βγ,ωfχ=dkdχkPα,k−β−γ,ωfχ.Definition 4.
Letψ∈Λ. Then the complex Prabhakar differential operator (CPFDO) of (13) is formulated in terms of the Riemann-Liouville derivative, as follows:
(15)DkRα,βγ,ωψz=dkdzk∫0zz−ζk−β−1Ξα,k−β−γωz−ζαψζdζ=dkdzkPα,k−β−γ,ωψz,and in terms of the Caputo derivative, as follows:
(16)DkCα,βγ,ωψz=∫0zz−ζk−β−1Ξα,k−β−γωz−ζαdkdζkψζdζ=Pα,k−β−γ,ωdkdzkψz.Note that(17)DkCα,βγ,ωψz=DkRα,βγ,ωψz−∑m=0k−1zm−βΞα,m−β−γωzαψm0.For example, letψz=zε,ε≥1, then in virtue of [15] (Corollary 2.3), we conclude that
(18)Dα,βγ,ωzε=∫0zz−ζ1−β−1Ξα,1−β−γωz−ζαddζψζdζ∶=∫0Zz−ζμ−1Ξα,μ−γωz−ζαddζζεdζ=ε∫0zζε−1z−ζμ−1Ξα,μ−γωz−ζαdζ=Γε+1zμ+ε−1Ξα,μ+ε−γωzα,μ∶=1−β.1CIn general, we have(19)DkCα,βγ,ωzε=∫0zz−ζk−β−1Ξα,k−β−γωz−ζαdkdζkζεdζ=∫0zz−ζk−β−1Ξα,k−β−γωz−ζαddζζεdζ=1−k+εk∫0zζε−kz−ζk−β−1Ξα,k−β−γωz−ζαdζ=1−k+εk∫0zζε−k+1−1z−ζk−β−1Ξα,k−β−γωz−ζαdζ∶=vk∫0zζv−1z−ζμ−1Ξα,μ−γωz−ζαdζ=vkΓvzv+μ−1Ξα,μ+v−γωzα,where μ≔k−β, ν≔ε−k+1, and vk=Γ1+ε/Γ1+ε−k. Hence, we obtain
(20)Dα,βγ,ωzε=Γ1+εzv+μ−1Ξα,μ+v−γωzαkC=Γk+vzv+μ−1Ξα,μ+v−γωzα.We have the following property.Proposition 5.
Letψ∈Λ. Define a functional Δα,βγ,ωkC:∪⟶∪ by
(21)ΔkCα,βγ,ω≔zβΞα,2−β−γwzαDCα,βγ,ω.
ThenΔα,βγ,ωkCψ=Δα,βγ,ωkC∗ψ∈Λα,β,γ,ω∈ℂ,z∈∪.Proof.
Letψ∈Λ. Then a computation implies
(22)Δα,βγ,ωψz=zβΞα,2−β−γωzαDα,βγ,ωψzkCkC=zβΞα,2−β−γωzαDα,βγ,ωz+∑n=2∞ψnznkC=zβΞα,2−β−γωzαDα,βzγ,ω+∑n=2∞ψnDkCα,βγ,ωznkC=zβΞα,2−β−γωzαΞα,2−β−γωzαz1−β+∑n=2∞ψnΓn+1zn−βΞα,n+1−β−γωzα=z+∑n=2∞ψnΓn+1Ξα,n+1−β−γωzαΞα,2−β−γωzαzn∶=z+∑n=2∞ψnδnzn=z+∑n=2∞δnzn∗z+∑n=2∞ψnzn=Δα,βγ,ω∗kCψz,where δn∶=Γn+1Ξα,n+1−β−γωzα/Ξα,2−β−γωzα. This indicates that Δα,βγ,ωψ∈ΛkC.☐We callΔα,βγ,ωkC the normalized complex Prabhakar operator (NCPO) in the open unit disk. Since Δα,βγ,ωkC∈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 thatδn 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 operatorkCΔαγβω is expanded by the series
(23)ϒkCα,βγ,ωψz=z+∑n=2∞ψnΞα,2−β−γωzαΓn+1Ξα,n+1−β−γωzαzn.
It is clear that(24)ϒkCα,βγ,ω∗ΔkCα,βγ,ωψz=ΔkCα,βγ,ω∗ϒkCα,βγ,ωψz=ψz.
The linear convex combination of the operatorsϒkCα,βγ,ω and Δα,βγ,ωkC can be recognized by the formula
(25)∑α,βγ,ωψzk∁=∁ΔkCα,βγ,ωψz+1−∁ϒkCα,βγ,ωψz,where∁∈0,1. Clearly, ∑α,βγ,ωψz∈Λk∁, whereψ∈Λ.


### 2.3. Subclasses of NCPO

In terms of the NCPO, we formulate the next classes.Definition 7.
A functionψ∈Λ is considered to be in the class kcSα,β∗γ,ωσ if and only if
(26)kcSα,β∗γ,ωσ=ψ∈Λ:zkCΔα,βγ,ωψz′kCΔα,βγ,ωψz≺σz,σ0=1.We shall deal with the conditions of a functionψ to be in kcSα,β∗γ,ωσ whenever σ∈C is convex as well as nonconvex.Definition 8.
A functionψ∈Λ is considered to be in the class kcJα,βγ,ωA,B,♭ if and only if
(27)kcJα,βγ,ωA,B,♭=ψ∈Λ:1+1♭2kCΔα,βγ,ωψzkCΔα,βγ,ωψz−kCΔα,βγ,ωψ−z≺1+Az1+Bz.We request the next result, which can be located in [6].Lemma 9.
Define the class of analytic functions as follows: forϱ∈ℂ and a positive integer n(28)ℍψ,n=ψ:ψz=ϱ+ϱnzn+ϱn+1zn+1+⋯.(i)
Letℓ∈ℝ. Then Rψz+ℓzψ′z>0⟶Rψz>0. In addition, if ℓ>0 and ψ∈ℍ1,n, then there are constants ℘>0 and κ>0 such that κ=κℓ,℘,n and
(29)ψz+ℓzψ′z≺1+z1−zκ⟶ψz≺1+z1−z℘(ii)
Letc∈0,1 and ψ∈ℍ1,n. Then there exists a fixed real number ℓ>0 so that
(30)Rψ2z+2ψz⋅zψ′z>c⟶Rψz>ℓ(iii)
Letψ∈ℍψ,n with Rψ>0. Then
(31)Rψz+zψ′z+z2ψ″z>0or for ℵ:∪⟶R such that
(32)Rψz+ℵzzψ′zψz>0.
ThenRψz>0.


## 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∪≔z∈ℂ:z<1 be the open unit disk. Two analytic functions g1, g2 in ∪ are called subordinated denoting by g1≺g2org1z≺g2z,z∈U, if there exits an analytic function ω,w≤z<1 having the formula
(1)g1z=g2wz,z∈∪.
g1 is majorized by g2 denoting by g1≪g2 if and only if
(2)g1z=wzg2z,z∈∪;equivalently, the coefficient inequality is held an≤bn, respectively.There is a deep construction between subordination and majorization [9] in ∪ for selected distinct classes comprising the convex class C:
(3)1+Rzg″zg′z>0,z∈∪,and starlike functions S∗(4)Rzg′gz>0,z∈∪.Definition 2.
We present a class of analytic functions by(5)fz=z+∑n=2∞anzn,z∈∪.
This class is denoted byΛ and known as the class of univalent functions which is normalized by f0=f′0−1=0.
Associated with the termsS∗ and C, we present the term P of all analytic functions p in ∪ with a positive real part in ∪ and p0=1.
Two analytic functionsf,g are called convoluted, denoting by f∗g if and only if
(6)f∗gz=∑n=0∞anzn∗∑n=0∞gnzn=∑n−0∞angnzn.Definition 3.
The generalized Mittag-Leffler function is defined by [10–12]
(7)Ξv,μ℘z=∑n=0∞℘nΓvn+μznn!,where ϑn represents the Pochhammer symbol and
(8)Ξv,μ1z∶=Ξv,μz=∑n=0∞znΓvn+μ℘0=1,℘n=℘℘+1⋯℘+n−1.
Note thatΞv,u℘z is an ultimate traditional generalization of the function ez, where Ξ1,11z=ez.Moreover, it can be formulated by the Fox-Write hypergeometric function, as follows:(9)Ξv,μ℘z=1Γ℘Ψ11℘,1v,μ;z.


## 2.2. Complex Prabhakar Operator (CPO)

The Prabhakar integral operator is defined for analytic function(10)ψz∈H0,1=ψz=ψ1z+ψ2z2+⋯,z∈∪by the formula [13, 14]
(11)Pα,βγ,ωψz=∫0zz−ζβ−1Ξα,βγωz−ζαψζdζ=ψ·ϱα,βγ,ωzα,β,γ,ω∈ℂ,z∈∪.Moreover [13, 14],
(12)ϱα,βγ,ωz≔zβ−1Ξα,βγωzα,Ξα,βγχ=∑n=0∞Γγ+nΓγΓαn+βχnn!.For example, letψz=zς−1, then (see [15], Corollary 2.3)
(13)Pα,βγ,ωzς−1=∫0zz−ζβ−1Ξα,βγωz−ζαζς−1dζ=Γςzβ+ς−1Ξα,β+ςγωzα.The Prabhakar derivative can be computed by the formula [13]
(14)Dkα,βγ,ωfχ=dkdχkPα,k−β−γ,ωfχ.Definition 4.
Letψ∈Λ. Then the complex Prabhakar differential operator (CPFDO) of (13) is formulated in terms of the Riemann-Liouville derivative, as follows:
(15)DkRα,βγ,ωψz=dkdzk∫0zz−ζk−β−1Ξα,k−β−γωz−ζαψζdζ=dkdzkPα,k−β−γ,ωψz,and in terms of the Caputo derivative, as follows:
(16)DkCα,βγ,ωψz=∫0zz−ζk−β−1Ξα,k−β−γωz−ζαdkdζkψζdζ=Pα,k−β−γ,ωdkdzkψz.Note that(17)DkCα,βγ,ωψz=DkRα,βγ,ωψz−∑m=0k−1zm−βΞα,m−β−γωzαψm0.For example, letψz=zε,ε≥1, then in virtue of [15] (Corollary 2.3), we conclude that
(18)Dα,βγ,ωzε=∫0zz−ζ1−β−1Ξα,1−β−γωz−ζαddζψζdζ∶=∫0Zz−ζμ−1Ξα,μ−γωz−ζαddζζεdζ=ε∫0zζε−1z−ζμ−1Ξα,μ−γωz−ζαdζ=Γε+1zμ+ε−1Ξα,μ+ε−γωzα,μ∶=1−β.1CIn general, we have(19)DkCα,βγ,ωzε=∫0zz−ζk−β−1Ξα,k−β−γωz−ζαdkdζkζεdζ=∫0zz−ζk−β−1Ξα,k−β−γωz−ζαddζζεdζ=1−k+εk∫0zζε−kz−ζk−β−1Ξα,k−β−γωz−ζαdζ=1−k+εk∫0zζε−k+1−1z−ζk−β−1Ξα,k−β−γωz−ζαdζ∶=vk∫0zζv−1z−ζμ−1Ξα,μ−γωz−ζαdζ=vkΓvzv+μ−1Ξα,μ+v−γωzα,where μ≔k−β, ν≔ε−k+1, and vk=Γ1+ε/Γ1+ε−k. Hence, we obtain
(20)Dα,βγ,ωzε=Γ1+εzv+μ−1Ξα,μ+v−γωzαkC=Γk+vzv+μ−1Ξα,μ+v−γωzα.We have the following property.Proposition 5.
Letψ∈Λ. Define a functional Δα,βγ,ωkC:∪⟶∪ by
(21)ΔkCα,βγ,ω≔zβΞα,2−β−γwzαDCα,βγ,ω.
ThenΔα,βγ,ωkCψ=Δα,βγ,ωkC∗ψ∈Λα,β,γ,ω∈ℂ,z∈∪.Proof.
Letψ∈Λ. Then a computation implies
(22)Δα,βγ,ωψz=zβΞα,2−β−γωzαDα,βγ,ωψzkCkC=zβΞα,2−β−γωzαDα,βγ,ωz+∑n=2∞ψnznkC=zβΞα,2−β−γωzαDα,βzγ,ω+∑n=2∞ψnDkCα,βγ,ωznkC=zβΞα,2−β−γωzαΞα,2−β−γωzαz1−β+∑n=2∞ψnΓn+1zn−βΞα,n+1−β−γωzα=z+∑n=2∞ψnΓn+1Ξα,n+1−β−γωzαΞα,2−β−γωzαzn∶=z+∑n=2∞ψnδnzn=z+∑n=2∞δnzn∗z+∑n=2∞ψnzn=Δα,βγ,ω∗kCψz,where δn∶=Γn+1Ξα,n+1−β−γωzα/Ξα,2−β−γωzα. This indicates that Δα,βγ,ωψ∈ΛkC.☐We callΔα,βγ,ωkC the normalized complex Prabhakar operator (NCPO) in the open unit disk. Since Δα,βγ,ωkC∈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 thatδn 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 operatorkCΔαγβω is expanded by the series
(23)ϒkCα,βγ,ωψz=z+∑n=2∞ψnΞα,2−β−γωzαΓn+1Ξα,n+1−β−γωzαzn.
It is clear that(24)ϒkCα,βγ,ω∗ΔkCα,βγ,ωψz=ΔkCα,βγ,ω∗ϒkCα,βγ,ωψz=ψz.
The linear convex combination of the operatorsϒkCα,βγ,ω and Δα,βγ,ωkC can be recognized by the formula
(25)∑α,βγ,ωψzk∁=∁ΔkCα,βγ,ωψz+1−∁ϒkCα,βγ,ωψz,where∁∈0,1. Clearly, ∑α,βγ,ωψz∈Λk∁, whereψ∈Λ.


## 2.3. Subclasses of NCPO

In terms of the NCPO, we formulate the next classes.Definition 7.
A functionψ∈Λ is considered to be in the class kcSα,β∗γ,ωσ if and only if
(26)kcSα,β∗γ,ωσ=ψ∈Λ:zkCΔα,βγ,ωψz′kCΔα,βγ,ωψz≺σz,σ0=1.We shall deal with the conditions of a functionψ to be in kcSα,β∗γ,ωσ whenever σ∈C is convex as well as nonconvex.Definition 8.
A functionψ∈Λ is considered to be in the class kcJα,βγ,ωA,B,♭ if and only if
(27)kcJα,βγ,ωA,B,♭=ψ∈Λ:1+1♭2kCΔα,βγ,ωψzkCΔα,βγ,ωψz−kCΔα,βγ,ωψ−z≺1+Az1+Bz.We request the next result, which can be located in [6].Lemma 9.
Define the class of analytic functions as follows: forϱ∈ℂ and a positive integer n(28)ℍψ,n=ψ:ψz=ϱ+ϱnzn+ϱn+1zn+1+⋯.(i)
Letℓ∈ℝ. Then Rψz+ℓzψ′z>0⟶Rψz>0. In addition, if ℓ>0 and ψ∈ℍ1,n, then there are constants ℘>0 and κ>0 such that κ=κℓ,℘,n and
(29)ψz+ℓzψ′z≺1+z1−zκ⟶ψz≺1+z1−z℘(ii)
Letc∈0,1 and ψ∈ℍ1,n. Then there exists a fixed real number ℓ>0 so that
(30)Rψ2z+2ψz⋅zψ′z>c⟶Rψz>ℓ(iii)
Letψ∈ℍψ,n with Rψ>0. Then
(31)Rψz+zψ′z+z2ψ″z>0or for ℵ:∪⟶R such that
(32)Rψz+ℵzzψ′zψz>0.
ThenRψz>0.


## 3. Results

Our results are as follows.Theorem 10.
Letψ∈Λ. If one of the next inequalities is considered,
(i)
ΔkCα,βγ,ωψz is of bounded turning function(ii)
ΔkCα,βγ,ωψz′≺1+z/1−zκ,κ>0,z∈∪(iii)
RΔkCα,βγ,ωψz′ΔkCα,βγ,ωψz/z>c/2,c∈0,1,z∈∪(iv)
RzΔkCα,βγ,ωψz″−ΔkCα,βγ,ωψz′+2ΔkCα,βγ,ωψz/z>0(v)
RzΔkCα,βγ,ωψz′/ΔkCα,βγ,ωψz+2ΔkCα,βγ,ωψz/z>1
thenΔkCα,βγ,ωψz/z∈Pλ for some λ∈0,1.Proof.
Define a functionρ as follows:
(33)ρ=ΔkCα,βγ,ωψzz,z∈∪.
Then a computation implies that(34)zρ′z+ρz=ΔkCα,βγ,ωψz′.
In virtue of the first inequality, we have thatΔkCα,βγ,ωψz is of bounded turning function, which leads to Rzρ′z+ρz>0. Therefore, Lemma 9(i) indicates that Rρz>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=κℓ and
(35)ΔkCα,βγ,ωψzz≺1+z1−zℓ.
This implies that(36)RΔkCα,βγ,ωψzz>λ,λ∈0,1.
Suppose that(37)Rρ2z+2ρz⋅zρ′z=2RΔkCα,βγ,ωψzzΔkCα,βγ,ωψz′−ΔkCα,βγ,ωψz2z>ς.
According to Lemma9(ii), there exists a fixed real number ℓ>0 satisfying Rρz>ℓ and
(38)ρz=ΔkCα,βγ,ωψzz∈Pλ,λ∈0,1.
It follows from (37) that RΔkCα,βγ,ωψz′>0; consequently, by Noshiro-Warschawski and Kaplan theorems, ΔkCα,βγ,ωψz is univalent and of bounded turning function in ∪. Taking the derivative (33), then we get
(39)Rρz+zρ′z+z2ρ″z=RzΔkCα,βγ,ωψz″−ΔkCα,βγ,ωψz′+2ΔkCα,βγ,ωψzz>0.
Hence, Lemma9(ii) implies RΔkCα,βγ,ωψz/z>0.
The logarithmic differentiation of (33) yields
(40)Rρz+zρ′zρz+z2ρ″z=RzΔkCα,βγ,ωψz′ΔkCα,βγ,ωψz+2ΔkCα,βγ,ωψzz−1>0.
Hence, Lemma9(iii) implies, where ℵz=1,
(41)RΔkCα,βγ,ωψzz>0.☐The next results show the upper bound of the operatorΔkCα,βγ,ωψz utilizing the exponential integral in the open unit disk provided that the function ψ∈SkCα,β∗γ,ωσ.Theorem 11.
Suppose thatψ∈SkCα,β∗γ,ωσ, where σz is convex in ∪. Then
(42)ΔkCα,βγ,ωψz≺zexp∫0zσΨω−1ωdω,where Ψz is analytic in ∪, with Ψ0=0 and Ψz<1. Also, for z=ξ, ΔkCα,βγ,ωψz satisfies the inequality
(43)exp∫01σΨξ−1ξdξ≤ΔkCα,βγ,ωψzz≤exp∫01σΨξ−1ξdξ.Proof.
By the hypothesis, we receive the following conclusion:(44)zΔkCα,βγ,ωψz′ΔkCα,βγ,ωψz≺σz,z∈∪.
This gives the occurrence of a Schwarz function withΨ0=0 and Ψz<1 such that
(45)zΔkCα,βγ,ωψz′ΔkCα,βγ,ωψz=σΨz,z∈∪.
That is,(46)ΔkCα,βγ,ωψz′ΔkCα,βγ,ωψz−1z=σΨz−1z.
Integrating the above equality, we get(47)logΔkCα,βγ,ωψz−logz=∫0zσΨω−1ωdω.
Consequently, we get(48)logΔkCα,βγ,ωψzz=∫0zσΨω−1ωdω.
By the definition of subordination, we arrive at the following inequality(49)ΔkCα,βγ,ωψz≺zexp∫0zσΨω−1ωdω.
Note that the functionσz plots the disk 0<z<ξ<1 onto a reign, which is convex and symmetric with respect to the real axis. That is,
(50)σ−ξz≤RσΨξz≤σξz,ξ∈0,1,then we have the inequalities
(51)σ−ξ≤σ−ξz,σξz≤σξ;consequently, we get
(52)∫01σΨ−ξz−1ξdξ≤R∫01σΨξ−1ξdξ≤∫01σΨξz−1ξdξ.
In view of Equation (48), we obtain the general log-inequality
(53)∫01σΨ−ξz−1ξdξ≤logΔkCα,βγ,ωψzz≤∫01σΨξz−1ξdξ;that is,
(54)exp∫01σΨ−ξz−1ξdξ≤ΔkCα,βγ,ωψzz≤exp∫01σΨξz−1χdξ.
Hence, we have(55)exp∫01σΨ−ξ−1ξdξ≤ΔkCα,βγ,ωψzz≤exp∫01σΨξ−1ξdξ.☐Proceeding, we illustrate the sufficient condition ofψ to be in the class SkCα,β∗γ,ωψσ, where σ is convex univalent satisfying σ0=1.Theorem 12.
Ifψ∈Λ satisfies the inequality
(56)zΔCα,βγ,ωψz′ΔkCα,βγ,ωψz2+zΔCα,βγ,ωψz″ΔCα,βγ,ωψz′−zΔCα,βγ,ωψz′ΔkCα,βγ,ωψz≺σz,then ψ∈SkCα,β∗γ,ωσ.Proof.
The proof directly comes from [6] (Theorem 3.1a). Taking
(57)pz=zΔCα,βγ,ωψz′ΔkCα,βγ,ωψz,and Pz=1 in the inequality
(58)pz+Pz⋅zp′z≺σz,then we obtain
(59)pz+Pz⋅zp′z=zΔCα,βγ,ωψz′ΔkCα,βγ,ωψz×2+zΔCα,βγ,ωψz″ΔCα,βγ,ωψz′−zΔCα,βγ,ωψz′ΔkCα,βγ,ωψz≺σz.
This implies that(60)pz=zΔkCα,βγ,ωψz′ΔkCα,βγ,ωψz≺σz,σ∈C,that is ψ∈SkCα,β∗γ,ωσ.☐Corollary 13.
Let the assumption of Theorem12 hold. Then
(61)zΔCα,βγ,ωψz′ΔkCα,βγ,ωψz×1+zΔCα,βγ,ωψz″ΔCα,βγ,ωψz′−zΔCα,βγ,ωψz′ΔkCα,βγ,ωψz≪σ′z.Proof.
Let(62)pz=zΔCα,βγ,ωψz′ΔkCα,βγ,ωψz.
In view of Theorem12, we have
(63)zΔCα,βγ,ωψz′ΔkCα,βγ,ωψz≺σz,where σ∈C. Then by [9] (Theorem 3), we get p′z≪σ′z for some z∈∪, where
(64)p′z=zΔCα,βγ,ωψz′ΔkCα,βγ,ωψz1+zΔCα,βγ,ωψz″ΔCα,βγ,ωψz′−zΔCα,βγ,ωψz′ΔkCα,βγ,ωψz.☐It is well known that the functionσz=e∈z,1<∈≤π/2 is not convex in ∪, where the domain σ∪ is lima-bean (see [6] (P123)). One can obtain the same result of Theorem 12 as follows.Theorem 14.
Ifψ∈Λ satisfies the inequality
(65)1+zΔCα,βγ,ωψz″ΔCα,βγ,ωψz′≺e∈z,1<∈≤π2,then ψ∈SkCα,β∗γ,ωe∈z.Proof.
Let(66)pz≔zΔCα,βγ,ωψz′ΔkCα,βγ,ωψz.
Then a computation implies(67)pz+zp′zpz=zΔCα,βγ,ωψz′ΔkCα,βγ,ωψz+zΔCα,βγ,ωψz′/ΔkCα,βγ,ωψz1+zΔCα,βγ,ωψz″/ΔCα,βγ,ωψz′−zΔCα,βγ,ωψz′/ΔkCα,βγ,ωψzzΔCα,βγ,ωψz′/ΔkCα,βγ,ωψz=1+zΔCα,βγ,ωψz″ΔCα,βγ,ωψz′≺e∈z.
This implies that [6] (P123)
(68)pz=zΔkCα,βγ,ωψz′ΔkCα,βγ,ωψz≺e∈z;that is, ψ∈SkCα,β∗γ,ωe∈z.☐Theorem 15.
Ifψ∈kcJα,βγ,ωA,B,♭ then the function
(69)Bz=12ψz−ψ−z,z∈∪,satisfies
(70)1+1♭Δk+1Cα,βγ,ωBzΔkCα,βγ,ωBz−1≺1+Az1+Bz,RzBz′Bz≥1−ð21+ð2,z=ð<1.Proof.
Letψ∈kcJα,βγ,ωA,B,♭. Then there occurs a function Jz such that
(71)♭Jz−1=2Δk+1Cα,βγ,ωψzΔkCα,βγ,ωψz−ΔkCα,βγ,ωψ−z,♭J−z−1=2Δk+1Cα,βγ,ωψ−zΔkCα,βγ,ωψ−z−ΔkCα,βγ,ωψz.
This confirms that(72)1+1♭Δk+1Cα,βγ,ωBzΔkCα,βγ,ωBz−1=Jz+J−z2.
However,J satisfies
(73)Jz≺1+Az1+Bz,which is univalent, then we get
(74)1+1♭Δk+1Cα,βγ,ωBzΔkCα,βγ,ωBz−1≺1+Az1+Bz.
Also,Bz is starlike in ∪ which implies that
(75)ℏz≔zBz′Bz≺1−z21+z2.
Hence, a Schwarz functionk∈∪,∣kz∣≤∣z∣<1,k0=0 gets
(76)ℏz≺1−kz21+kz2,which leads to
(77)k2ζ=1−ℏζ1+ℏζ,ζ∈∪,ζ,ζ=r<1.
A calculation yields(78)1−ℏζ1+ℏζ=kζ2≤ζ2.
Therefore, we get the following inequality: or(79)ℏζ−1+ζ41−ζ42≤4ζ41−ζ42(80)ℏζ−1+ζ41−ζ4≤2ζ21−ζ4.
Thus, we have(81)Rℏz≥1−ð21+ð2,ζ=ð<1.
This completes the assertion of the theorem.☐Example 16.
(i)
Let(82)zf′zfz≔zΔCα,βγ,ωψz′ΔkCα,βγ,ωψz,ΔkCα,βγ,ωψz=z1−z2,ψ∈Λ.
Then the solution ofzf′z/fz=1+z/1−z is formulated, as follows:
(83)ΔCα,βγ,ωψz=z1−z2,ψ∈Λ.
Moreover, the solution of the equation(84)fz+zf′zfz=1+z1−zis approximated to fz=z/1−z.
(ii)
The solution ofzf′z/fz=1+z/1−z0.25 is given in terms of the hypergeometric function, as follows (see Figure 1):
(85)fz=cexp1.8z+1z+11−z0.25F11.25;0.25,1;2.25;0.5z+0.5,z+1z2.25F11.25;0.25,1;2.25;0.5z+0.5,z+1+z+1F12.25;0.25,2;3.25;0.5z+0.5,z+1+0.125z+0.125F12.25;1.25,1;3.25;0.5z+0.5z+1.Figure 1
Plot of the solution forzfz/fz and fz+zfz/fz, respectively.


## 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.



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