# Certain Analytic Formulas Linked to Locally Fractional Differential and Integral Operators

Authors: Ibtisam Aldawish; Rabha W. Ibrahim
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The present investigation is aimed at defining different classes of analytic functions and conformable differential operators in view of the concept of locally fractional differential and integral operators. We present a novel generalized class of analytic functions, which we call it locally fractional analytic functions in the open unit disk. For the suggested class, we look at conditions to get the starlikeness and convexity properties.


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## Body

## 1. Introduction

The idea of local fractional calculus (LFC) is an innovative differentiation and integration model for functions affecting on special fractal sets. Experts and scientists have been attracted by this theory. The LFC of a complex variable was established by Yang [1] (see [2] for additional material). The issue of explanation of fractal operators over analytic functions is glowered to be solved since the effort of Viswanathan and Navascues [3]. They presented a technique to define α-fractal operator on ℂkI, the space of all k−real-valued continuous functions defined on a compact interval I. In our work, we extend this idea into a complex domain, which is already compact in the z-plane. Therefore, in our opinion, the current study contributes to the theory of fractal functions and makes it easier for them to find new applications in a variety of domains, such as numerical analysis, functional analysis, and harmonic analysis; for example, in relation to PDEs. We anticipate that the current investigation will open the door to further research on shape-preserving fractal approximation in the different function spaces that are being taken into consideration. The reader is encouraged to see [4] for a fractal approximation that preserves shape in the space of differential functions.Letϑ∈0,1 be a fractional number and ξ=χ+iη be a complex number. Then, the fractal complex number ξϑ can be defined by [2]
(1)ξϑ≔χϑ+iνηϑ,χ,η∈ℝ.The local fractional derivative (LFD) at a random pointξ0 for a complex function υξ can be formulated as follows:
(2)Δϑυξ=Γ1+ϑlimξ⟶ξ0υξ−υξ0ξ−ξ0ϑ,ϑ∈0,1,ξ,ξ0∈ℂ,υ:ℂ⟶ℂ.Letυξ=ξnϑ, for example; then, the LFD can be calculated as follows:
(3)Δϑξnϑ=Γ1+nϑΓ1+n−1ϑξn−1ϑ.Obviously, whenϑ=1, the LFD reduces to the normal derivative ξn=nξn−1. Moreover, for n=1, the LFD becomes
(4)Δϑξϑ=Γ1+ϑ.On Cantor sets, the local fractional differential operator (LFDO) can be used to construct a variety of different transformations and summations (see, e.g., Yang et al. [5–7]).LetAϑ,ϑ∈0,1 be a class of locally fractional normalized functions in Ω≔ξ∈ℂ:ξ<1 such that
(5)υξϑ=ξϑ+∑n=2∞vnξnϑ,ξ∈Ω.Clearly, whenϑ=1, we attain the usual normalized class A:
(6)υξ=ξ+∑n=2∞vnξn,ξ∈Ω.Two functionsυ,ω∈Aϑ are convoluted if they accept
(7)υξϑ∗ωξϑ=ξϑ+∑n=2∞vnwnξϑn,ξ∈Ω,where
(8)ωξϑ=ξϑ+∑n=2∞wnξϑn.Moreover, they are subordinatedυ≺ω if they satisfy the equality υξ=ωλξ, where λ is analytic with λξ≤ξ<1 (see [8]). This definition can be extended to LDC by suggesting ξ⟶ξϑ.


### 1.1. Convoluted Operators

The local derivative suggests the subsequent functional operator: forn≥1.
(9)ΔνξϑnΓ1+ϑξnϑ−1−ϑ=Γ1+nϑ/Γ1+n−1ϑξn−1ϑΓ1+νξnϑ−1−ϑ=Γ1+nϑΓ1+ϑΓ1+n−1ϑξn.As a result, we define the following function for alln≥1,
(10)E∧ϑξ=ξ+∑n=2∞Γ1+nϑΓ1+ϑΓ1+n−1ϑξn.Obviously,E^∈A. By using the convolution operator, we define the the following LFDO: Dϑ:Ω⟶Ω, such that
(11)Dϑυξ≔E∧ϑ∗υξ=ξ+∑n=2∞Γ1+nϑΓ1+ϑΓ1+n−1ϑvnξn.Correspondingly, the fractional locally integral operator (FLIO) is defined as follows:(12)Jϑξ≔Ëϑ∗υξ=ξ+∑n=2∞Γ1+ϑΓ1+n−1ϑΓ1+nϑvnξn.We proceed to define thek−LFDO and k−LFIO, as follows:
(13)Dkϑυξ≔E∧ϑ∗⋯∗E∧ϑ⏟k−times∗υξ=ξ+∑n=2∞Γ1+nϑΓ1+ϑΓ1+n−1ϑkvnξn≔ξ+∑n=2∞E∧nkvnξn.Jkϑυξ≔Ëϑ∗⋯∗Ëϑ⏟k−times∗υξ=ξ+∑n=2∞Γ1+ϑΓ1+n−1ϑΓ1+nϑkvnξn≔ξ+∑n=2∞Ënkvnξn.It is obvious that we reach the well-known Salagean differential and integral operators, respectively, whenϑ=1.Definition 1.
Define two formulas of analytic functionsυ∈A as follows:
(14)Sυξ≔ξυ′ξυξ,ξ∈Ω,Kυξ≔1+ξυ′′ξυ′ξ,ξ∈Ω.
Consequently, there are two classes for this investigation, the starlike and convex classes with(15)RSυξ>0,RKυξ>0,ξ∈Ω,respectively.


### 1.2. Locally Fraction Conformable Operator

Definition 2.
Let℘ be a nonnegative number, such that ℘ be the integer part of ℘. By using the k−LFDO, we have the following locally fractional conformable differential operator:
(16)∧℘Dkϑυξ=∧℘−℘∧℘Dkϑυξ=p1℘−℘,ξp1℘−℘,ξ+p0℘−℘,ξ∧℘Dkϑυξ+p0℘−℘,ξp1℘−℘,ξ+p0℘−℘,ζξ∧℘Dkϑυξ,where for q≔℘−℘∈0,1,
(17)∧0Dkϑυξ=Dkϑυξ,∧qDkϑυξ=p1q,ξp1q,ξ+p0q,ξDkϑυξ+p0q,ξp1q,ξ+p0q,ξξDkϑυξ′=p1q,ξp1q,ξ+p0q,ξξ+∑n=2∞E∧nkvnξn+p0q,ξp1q,ξ+p0ν,ξξ+∑n=2∞E∧nknvnξn=ξ+∑n=2∞p1q,ξ+np0q,ξp1q,ξ+p0q,ξE∧nkvnξn≔ξ+∑n=2∞θnq,ξE∧nkvnξn∧1Dkϑυξ=ξDkϑυξ⋮∧℘Dkϑυξ=∧1∧℘−1Dkϑυξ,where
(18)θnq,ξ≔p1q,ξ+np0q,ξp1ν,ξ+p0q,ξ,and the functions p1,p0:0,1×Ω⟶Ω are analytic in Ω with
(19)p1q,ξ≠−p0q,ξ,limq⟶0p1q,ξ=1,limq⟶1p1q,ξ=0,p1q,ξ≠0,∀ξ∈Ω,q∈0,1,limq⟶0p0q,ξ=0,limq⟶1p0q,ξ=1,p0q,ξ≠0,∀ξ∈Ω,q∈0,1.Remark 3.
(i)
For constant coefficients, the operator∧qDkϑυξ is normalized in Ω. Moreover, if ℘−℘=0,ϑ=1, then we realize the Sàlàgean differential operator [9]. If k=0, we attain the conformable differential operator in [10], which is based on the same assumptions. Similarly, we can replace the local fractional integral operator using the k−LFJOJkϑυξ(ii)
The authors in [11] presented a conformable fractional differential operator by using a combination of fractional integral and differential operators, as follows:(20)Dqυξ=ξ+∑n=2∞θnq,ξΓ3−qΓn+1Γn+2−qkvnξn,ξ∈Ω,where
(21)θnq,ξ=p1q,ξ+np0q,ξp1ν,ξ+p0q,ξWe proceed to discuss the most important geometric properties ofDkϑυξ and ∧℘Dkϑυξ, in the next section.


### 1.3. Lemmas

We request the next preliminaries.Lemma 4 (see [8]/p135).
Letυ be analytic and ω be univalent in Ω such that υ0=ω0. Furthermore, let ϕ be analytic in a domain involving ωΩ and ωΩ. If ξω′ξϕωξ is starlike, then the subordination
(22)ξυ′ξϕυξ≺ξω′ξϕωξyields
(23)υξ≺ωξ,and ω is the best dominant.Lemma 5 (see [12]).
For two analytic functionsυ1 and υ2 in a complex domain such that υ10=υ20 and
(24)υ1ξ≺υ2ξ,then
(25)∫02πυ1ξtdμ≤∫02πυ2ξtdμ,where
(26)ξ=νexpiμ,ν∈0,1,t∈ℝ+.The next section is about our results of the operator∧qDkϑυξ, and as a special consequence, we connect the operator Dkϑυξ.


## 1.1. Convoluted Operators

The local derivative suggests the subsequent functional operator: forn≥1.
(9)ΔνξϑnΓ1+ϑξnϑ−1−ϑ=Γ1+nϑ/Γ1+n−1ϑξn−1ϑΓ1+νξnϑ−1−ϑ=Γ1+nϑΓ1+ϑΓ1+n−1ϑξn.As a result, we define the following function for alln≥1,
(10)E∧ϑξ=ξ+∑n=2∞Γ1+nϑΓ1+ϑΓ1+n−1ϑξn.Obviously,E^∈A. By using the convolution operator, we define the the following LFDO: Dϑ:Ω⟶Ω, such that
(11)Dϑυξ≔E∧ϑ∗υξ=ξ+∑n=2∞Γ1+nϑΓ1+ϑΓ1+n−1ϑvnξn.Correspondingly, the fractional locally integral operator (FLIO) is defined as follows:(12)Jϑξ≔Ëϑ∗υξ=ξ+∑n=2∞Γ1+ϑΓ1+n−1ϑΓ1+nϑvnξn.We proceed to define thek−LFDO and k−LFIO, as follows:
(13)Dkϑυξ≔E∧ϑ∗⋯∗E∧ϑ⏟k−times∗υξ=ξ+∑n=2∞Γ1+nϑΓ1+ϑΓ1+n−1ϑkvnξn≔ξ+∑n=2∞E∧nkvnξn.Jkϑυξ≔Ëϑ∗⋯∗Ëϑ⏟k−times∗υξ=ξ+∑n=2∞Γ1+ϑΓ1+n−1ϑΓ1+nϑkvnξn≔ξ+∑n=2∞Ënkvnξn.It is obvious that we reach the well-known Salagean differential and integral operators, respectively, whenϑ=1.Definition 1.
Define two formulas of analytic functionsυ∈A as follows:
(14)Sυξ≔ξυ′ξυξ,ξ∈Ω,Kυξ≔1+ξυ′′ξυ′ξ,ξ∈Ω.
Consequently, there are two classes for this investigation, the starlike and convex classes with(15)RSυξ>0,RKυξ>0,ξ∈Ω,respectively.


## 1.2. Locally Fraction Conformable Operator

Definition 2.
Let℘ be a nonnegative number, such that ℘ be the integer part of ℘. By using the k−LFDO, we have the following locally fractional conformable differential operator:
(16)∧℘Dkϑυξ=∧℘−℘∧℘Dkϑυξ=p1℘−℘,ξp1℘−℘,ξ+p0℘−℘,ξ∧℘Dkϑυξ+p0℘−℘,ξp1℘−℘,ξ+p0℘−℘,ζξ∧℘Dkϑυξ,where for q≔℘−℘∈0,1,
(17)∧0Dkϑυξ=Dkϑυξ,∧qDkϑυξ=p1q,ξp1q,ξ+p0q,ξDkϑυξ+p0q,ξp1q,ξ+p0q,ξξDkϑυξ′=p1q,ξp1q,ξ+p0q,ξξ+∑n=2∞E∧nkvnξn+p0q,ξp1q,ξ+p0ν,ξξ+∑n=2∞E∧nknvnξn=ξ+∑n=2∞p1q,ξ+np0q,ξp1q,ξ+p0q,ξE∧nkvnξn≔ξ+∑n=2∞θnq,ξE∧nkvnξn∧1Dkϑυξ=ξDkϑυξ⋮∧℘Dkϑυξ=∧1∧℘−1Dkϑυξ,where
(18)θnq,ξ≔p1q,ξ+np0q,ξp1ν,ξ+p0q,ξ,and the functions p1,p0:0,1×Ω⟶Ω are analytic in Ω with
(19)p1q,ξ≠−p0q,ξ,limq⟶0p1q,ξ=1,limq⟶1p1q,ξ=0,p1q,ξ≠0,∀ξ∈Ω,q∈0,1,limq⟶0p0q,ξ=0,limq⟶1p0q,ξ=1,p0q,ξ≠0,∀ξ∈Ω,q∈0,1.Remark 3.
(i)
For constant coefficients, the operator∧qDkϑυξ is normalized in Ω. Moreover, if ℘−℘=0,ϑ=1, then we realize the Sàlàgean differential operator [9]. If k=0, we attain the conformable differential operator in [10], which is based on the same assumptions. Similarly, we can replace the local fractional integral operator using the k−LFJOJkϑυξ(ii)
The authors in [11] presented a conformable fractional differential operator by using a combination of fractional integral and differential operators, as follows:(20)Dqυξ=ξ+∑n=2∞θnq,ξΓ3−qΓn+1Γn+2−qkvnξn,ξ∈Ω,where
(21)θnq,ξ=p1q,ξ+np0q,ξp1ν,ξ+p0q,ξWe proceed to discuss the most important geometric properties ofDkϑυξ and ∧℘Dkϑυξ, in the next section.


## 1.3. Lemmas

We request the next preliminaries.Lemma 4 (see [8]/p135).
Letυ be analytic and ω be univalent in Ω such that υ0=ω0. Furthermore, let ϕ be analytic in a domain involving ωΩ and ωΩ. If ξω′ξϕωξ is starlike, then the subordination
(22)ξυ′ξϕυξ≺ξω′ξϕωξyields
(23)υξ≺ωξ,and ω is the best dominant.Lemma 5 (see [12]).
For two analytic functionsυ1 and υ2 in a complex domain such that υ10=υ20 and
(24)υ1ξ≺υ2ξ,then
(25)∫02πυ1ξtdμ≤∫02πυ2ξtdμ,where
(26)ξ=νexpiμ,ν∈0,1,t∈ℝ+.The next section is about our results of the operator∧qDkϑυξ, and as a special consequence, we connect the operator Dkϑυξ.


## 2. Results

This section deals with the operator∧qDkϑυξ≔∧qξ.


### 2.1. Starlikeness of∧qξ

Theorem 6.
If the following conditions occur:(i)
Λ is univalent in Ω(ii)
ξΛ′ξ/ΛξΛξ−1 is starlike in Ω(iii)
K∧qξ−1/S∧qξ−1≺1+ξΛ′ξ/ΛξΛξ−1
Then,(27)S∧qξ≺Λξ,ξ∈Ω,and Λ is the best dominant. Moreover,
(28)∫02πS∧qξtdμ≤∫02πΛξtdμ,t∈ℝ+.Proof.
Put the functionH, as follows:
(29)Hξ≔S∧qξ,ξ∈Ω.
A simple calculation yields(30)SHξ=K∧qξ−Hξ.
Consequently, we attain(31)K∧qξ−1S∧qξ−1=SHξ+Hξ−1Hξ−1=1+ξH′ξHξHξ−1.
Thus, we have(32)ξH′ξHξHξ−1≺ξΛ′ξΛξΛξ−1,ξ∈Ω.
Based on Lemma4, we attain the requested result. The second part is a direct application of Lemma 5.Theorem 7.
If the following conditions hold:(i)
Λ is univalent in Ω(ii)
ξΛ′ξ/Λξ−1 is starlike in Ω(iii)
S∧qξK∧qξ−1/S∧qξ−1S∧qξ−1−1≺ξΛ′ξ/Λξ−1
Then,(33)S∧qξ≺Λξ,ξ∈Ω,and Λ is the best dominant. Furthermore,
(34)∫02πS∧qξtdμ≤∫02πΛξtdμ,t∈ℝ+.Proof.
Formulate the functionH as follows:
(35)Hξ≔S∧qξ,ξ∈Ω.
Then, we get the equality(36)SHξ+Hξ=K∧qξ.
Replacing produces the following results:(37)S∧qξK∧qξ−1S∧qξ−1−1=ξH′ξHξ−1.
Hence,(38)ξH′ξHξ−1≺ξΛ′ξΛξ−1,ξ∈Ω.
In view of Lemma4, we have the outcome. Lemma 5 implies the integral inequality.Theorem 8.
If the following conditions are fulfilled:(i)
Λ is univalent in Ω(ii)
SΛ is starlike in Ω(iii)
K∧qξ−S∧qξ≺SΛξ
Then,(39)S∧qξ≺Λξ,ξ∈Ω,and Λ is the best dominant. In addition,
(40)∫02πS∧qξtdμ≤∫02πΛξtdμ,t∈ℝ+.Proof.
Present the functionH as follows:
(41)Hξ≔S∧qξ,ξ∈Ω.
Therefore, we get(42)SHξ+Hξ=K∧qξ.
Replacing brings that(43)K∧qξ−S∧qξ=SHξ.
Hence,(44)SHξ≺SΛξ,ξ∈Ω.
Hence, Lemma4 implies Hξ≺Λξ. And Lemma 5 yields the last integral inequality
(45)∫02πHξtdμ≤∫02πΛξtdμ,t∈ℝ+.Theorem 9.
If the following conditions are applied:(i)
Λ is univalent in Ω(ii)
ξΛ′ξ is starlike in Ω(iii)
S∧qξK∧qξ−S∧qξ≺ξΛ′ξ
Then,(46)S∧qξ≺Λξ,ξ∈Ω,and Λ is the best dominant. Also,
(47)∫02πS∧qξtdμ≤∫02πΛξtdμ,t∈ℝ+.Proof.
Define the functionH as follows:
(48)Hξ≔S∧qξ,ξ∈Ω.
Consequently, we have(49)SHξ+Hξ=K∧qξ.
Substituting implies(50)S∧qξK∧qξ−S∧qξ=ξH′ξ.
Hence,(51)ξH′ξ≺ξΛ′ξ,ξ∈Ω.
According to Lemma4, we have result Hξ≺Λξ, while Lemma 5 gives
(52)∫02πHξtdμ≤∫02πΛξtdμ,t∈ℝ+.Remark 10.
Note that, whenq=0 in Theorems 6–9, we have the starlikeness of the operator Dkϑυξ, as follows:
(53)SDkϑυξ≺Λξ,ξ∈Ω.
Moreover, whenk=0,q=0, we obtain the Ma-Minda class of starlikeness [13], as follows:
(54)Sυξ≺Λξ,ξ∈Ω.
The last class is studied widely by many researchers for different types of functions [8] Λξ. For example, the inequality Λξ≔1+iξ/1−jξ,−1≤j<i≤1, which indicates that the image of Ω under the description Sυξ is centered on the x-axis owning diameter of end points 1−i/1−j and 1+i/1+j (see [14]). Another recent example is that Λξ=cosξ [15].
We proceed to discover more properties of the locally fractional conformable operator.Theorem 11.
If the following conditions hold:(i)
Λ is convex univalent in Ω(ii)
S∧qξ≺Λξ,Λ0=1
Then,(55)∧qξ≺ξexp∫0ξΛΞzzdz,where Ξ fulfills Ξ0=0 and Ξξ<1. In addition, the inequality ξ≔τ<1 yields
(56)exp∫01Λ−ττdτ≤∧qξξ≤exp∫01Λττdτ.Proof.
The conditions (i)-(ii) yield(57)∧qξ∧qξ−1ξ=ΛΞξ−1ξ.
Integration implies that(58)∧qξ≺ξexp∫0ξΛΞzzdz,which is equivalent to
(59)∧qξξ≺exp∫0ξΛΞzzdz.
But, we have(60)Λ−τξ≤RΛΞξτ≤Λτξ.
Then, we obtain(61)∫01Λ−τξρdτ≤∫01RΛΞξττdτ≤∫01Λτξτdτ.
The above conclusion leads to(62)∫01Λ−τξρdτ≤log∧qξξ≤∫01Λτξτdτ.
This leads to(63)exp∫01Λ−ττdτ≤∧qξξ≤exp∫01Λττdτ.Corollary 12.
Let the conditions of Theorem11 hold for q=0. Then,
(64)Dkϑυξ≺ξexp∫0ξΛΞzzdz,where Ξ achieves Ξ0=0 and Ξξ<1. Moreover, the inequality ξ≔τ<1 yields
(65)exp∫01Λ−ττdτ≤Dkϑυξξ≤exp∫01Λττdτ.Corollary 13.
Let the conditions of Theorem11 hold for q=0 and k=0. Then,
(66)υξ≺ξexp∫0ξΛΞzzdz,where Ξ satisfies Ξ0=0 and Ξξ<1. Moreover, the inequality ξ≔τ<1 yields
(67)exp∫01Λ−ττdτ≤υξξ≤exp∫01Λττdτ.


### 2.2. Positive Real Part of∧qξ

In this part, we aim to present the sufficient conditions for the operator∧q to be in the class of the real positive part (P.).Theorem 14.
Letℓ∈0,1 and ℘∈0,∞. If
(68)p0℘−℘,ξ=ℓ1−ℓp1℘−℘,ξ;q=℘−℘,then
(69)∧℘+2ξ∧℘+1ξ∈P⟹∧℘+1ξ∧℘ξ∈P.Proof.
By the conditionp0℘−℘,ξ=ℓ/1−ℓp1℘−℘,ξ and by Definition 2, we have
(70)∧℘ξ=1−ℓ∧℘ξ+ℓξ∧℘ξ,∧℘+1ξ=ξ∧℘ξ+ℓξ2∧℘ξ,∧℘+2ξ=ξ∧℘ξ+1+2ℓ℘2∧℘ξ+ℓξ3∧℘ξ.
We realize that(71)R∧℘+2ξ∧℘+1ξ>0,whenever the following real is positive:
(72)R1+1+ℓξ∧℘ξ+ℓξ2∧℘ξ∧℘ξ+ℓξ∧℘ξ>0.
Hence, from the inequality (74) with the idea of convex functional
(73)1−ℓ∧℘ξ+ℓξ∧℘ξ,we have
(74)R1+ξ1−ℓ∧℘ξ+ℓξ∧℘ξ1−ℓ∧℘ξ+ℓξ∧℘ξ>0.
Since convexity implies starlikeness, then we realize(75)Rξ1−ℓ∧℘ξ+ℓξ∧℘ξ1−ℓ∧℘ξ+ℓξ∧℘ξ>0.
The inequality (75) holds whenever
(76)R∧℘+1ξ∧℘ξ>0.Note that whenk=0 in Theorem 14, we have [10]—Theorem 3.1.Next, we deal with the functional(77)∧℘+1ξξ∧℘ξ′to be in the class of positive real part P.Theorem 15.
Letℓ∈0,1, ℘∈0,∞, and
(78)p1℘−℘,ξ=ℓ1−ℓp0℘−℘,ξ.
If∧℘ξ∈C (the class of convex univalent functions in Ω), then
(79)∧℘+1ξξ∧℘ξ∈Pℓ.Proof.
Applying the differential operator rule to(80)∧℘+1ξ=∧1∧℘ξyields
(81)∧℘+1ξ=∧℘−℘∧℘+1ξ=∧℘−℘∧∧℘ξ=∧℘−℘ξ∧℘ξ=p1℘−℘,ξp1℘−℘,ξ+p0℘−℘,ξξ∧℘ξ+p0℘−℘,ξp1℘−℘,ξ+p0℘−℘,ξ×ξ∧℘ξ+ξ∧℘ξ=p1℘−℘,ξp1℘−℘,ξ+p0℘−℘,ξξ∧℘ξ+p0℘−℘,ξp1℘−℘,ξ+p0℘−℘,ξξ∧℘ξ+p0℘−℘,ξp1℘−℘,ξ+p0℘−℘,ξξ2∧℘ξ=ξ∧℘ξ+p0℘−℘,ξp1℘−℘,ξ+p0℘−℘,ξξ2∧℘ξ.
Dividing equation (80) by the formula ξ∧℘ξ and employing the condition
(82)p1℘−℘,ξ=ℓ1−ℓp0℘−℘,ξ,we attain
(83)∧℘+1ξξ∧℘ξ=1+1−ℓξ∧℘ξ∧℘ξ.
The convexity of∧℘ξ,ξ∈Ω, brings
(84)R1+ξ∧℘ξ∧℘ξ>0.
Hence, it yields that(85)R∧℘+1ξξ∧℘ξ>ℓ.Note that whenk=0 in Theorem 15, we have [10]—Theorem 3.2.


## 2.1. Starlikeness of∧qξ

Theorem 6.
If the following conditions occur:(i)
Λ is univalent in Ω(ii)
ξΛ′ξ/ΛξΛξ−1 is starlike in Ω(iii)
K∧qξ−1/S∧qξ−1≺1+ξΛ′ξ/ΛξΛξ−1
Then,(27)S∧qξ≺Λξ,ξ∈Ω,and Λ is the best dominant. Moreover,
(28)∫02πS∧qξtdμ≤∫02πΛξtdμ,t∈ℝ+.Proof.
Put the functionH, as follows:
(29)Hξ≔S∧qξ,ξ∈Ω.
A simple calculation yields(30)SHξ=K∧qξ−Hξ.
Consequently, we attain(31)K∧qξ−1S∧qξ−1=SHξ+Hξ−1Hξ−1=1+ξH′ξHξHξ−1.
Thus, we have(32)ξH′ξHξHξ−1≺ξΛ′ξΛξΛξ−1,ξ∈Ω.
Based on Lemma4, we attain the requested result. The second part is a direct application of Lemma 5.Theorem 7.
If the following conditions hold:(i)
Λ is univalent in Ω(ii)
ξΛ′ξ/Λξ−1 is starlike in Ω(iii)
S∧qξK∧qξ−1/S∧qξ−1S∧qξ−1−1≺ξΛ′ξ/Λξ−1
Then,(33)S∧qξ≺Λξ,ξ∈Ω,and Λ is the best dominant. Furthermore,
(34)∫02πS∧qξtdμ≤∫02πΛξtdμ,t∈ℝ+.Proof.
Formulate the functionH as follows:
(35)Hξ≔S∧qξ,ξ∈Ω.
Then, we get the equality(36)SHξ+Hξ=K∧qξ.
Replacing produces the following results:(37)S∧qξK∧qξ−1S∧qξ−1−1=ξH′ξHξ−1.
Hence,(38)ξH′ξHξ−1≺ξΛ′ξΛξ−1,ξ∈Ω.
In view of Lemma4, we have the outcome. Lemma 5 implies the integral inequality.Theorem 8.
If the following conditions are fulfilled:(i)
Λ is univalent in Ω(ii)
SΛ is starlike in Ω(iii)
K∧qξ−S∧qξ≺SΛξ
Then,(39)S∧qξ≺Λξ,ξ∈Ω,and Λ is the best dominant. In addition,
(40)∫02πS∧qξtdμ≤∫02πΛξtdμ,t∈ℝ+.Proof.
Present the functionH as follows:
(41)Hξ≔S∧qξ,ξ∈Ω.
Therefore, we get(42)SHξ+Hξ=K∧qξ.
Replacing brings that(43)K∧qξ−S∧qξ=SHξ.
Hence,(44)SHξ≺SΛξ,ξ∈Ω.
Hence, Lemma4 implies Hξ≺Λξ. And Lemma 5 yields the last integral inequality
(45)∫02πHξtdμ≤∫02πΛξtdμ,t∈ℝ+.Theorem 9.
If the following conditions are applied:(i)
Λ is univalent in Ω(ii)
ξΛ′ξ is starlike in Ω(iii)
S∧qξK∧qξ−S∧qξ≺ξΛ′ξ
Then,(46)S∧qξ≺Λξ,ξ∈Ω,and Λ is the best dominant. Also,
(47)∫02πS∧qξtdμ≤∫02πΛξtdμ,t∈ℝ+.Proof.
Define the functionH as follows:
(48)Hξ≔S∧qξ,ξ∈Ω.
Consequently, we have(49)SHξ+Hξ=K∧qξ.
Substituting implies(50)S∧qξK∧qξ−S∧qξ=ξH′ξ.
Hence,(51)ξH′ξ≺ξΛ′ξ,ξ∈Ω.
According to Lemma4, we have result Hξ≺Λξ, while Lemma 5 gives
(52)∫02πHξtdμ≤∫02πΛξtdμ,t∈ℝ+.Remark 10.
Note that, whenq=0 in Theorems 6–9, we have the starlikeness of the operator Dkϑυξ, as follows:
(53)SDkϑυξ≺Λξ,ξ∈Ω.
Moreover, whenk=0,q=0, we obtain the Ma-Minda class of starlikeness [13], as follows:
(54)Sυξ≺Λξ,ξ∈Ω.
The last class is studied widely by many researchers for different types of functions [8] Λξ. For example, the inequality Λξ≔1+iξ/1−jξ,−1≤j<i≤1, which indicates that the image of Ω under the description Sυξ is centered on the x-axis owning diameter of end points 1−i/1−j and 1+i/1+j (see [14]). Another recent example is that Λξ=cosξ [15].
We proceed to discover more properties of the locally fractional conformable operator.Theorem 11.
If the following conditions hold:(i)
Λ is convex univalent in Ω(ii)
S∧qξ≺Λξ,Λ0=1
Then,(55)∧qξ≺ξexp∫0ξΛΞzzdz,where Ξ fulfills Ξ0=0 and Ξξ<1. In addition, the inequality ξ≔τ<1 yields
(56)exp∫01Λ−ττdτ≤∧qξξ≤exp∫01Λττdτ.Proof.
The conditions (i)-(ii) yield(57)∧qξ∧qξ−1ξ=ΛΞξ−1ξ.
Integration implies that(58)∧qξ≺ξexp∫0ξΛΞzzdz,which is equivalent to
(59)∧qξξ≺exp∫0ξΛΞzzdz.
But, we have(60)Λ−τξ≤RΛΞξτ≤Λτξ.
Then, we obtain(61)∫01Λ−τξρdτ≤∫01RΛΞξττdτ≤∫01Λτξτdτ.
The above conclusion leads to(62)∫01Λ−τξρdτ≤log∧qξξ≤∫01Λτξτdτ.
This leads to(63)exp∫01Λ−ττdτ≤∧qξξ≤exp∫01Λττdτ.Corollary 12.
Let the conditions of Theorem11 hold for q=0. Then,
(64)Dkϑυξ≺ξexp∫0ξΛΞzzdz,where Ξ achieves Ξ0=0 and Ξξ<1. Moreover, the inequality ξ≔τ<1 yields
(65)exp∫01Λ−ττdτ≤Dkϑυξξ≤exp∫01Λττdτ.Corollary 13.
Let the conditions of Theorem11 hold for q=0 and k=0. Then,
(66)υξ≺ξexp∫0ξΛΞzzdz,where Ξ satisfies Ξ0=0 and Ξξ<1. Moreover, the inequality ξ≔τ<1 yields
(67)exp∫01Λ−ττdτ≤υξξ≤exp∫01Λττdτ.


## 2.2. Positive Real Part of∧qξ

In this part, we aim to present the sufficient conditions for the operator∧q to be in the class of the real positive part (P.).Theorem 14.
Letℓ∈0,1 and ℘∈0,∞. If
(68)p0℘−℘,ξ=ℓ1−ℓp1℘−℘,ξ;q=℘−℘,then
(69)∧℘+2ξ∧℘+1ξ∈P⟹∧℘+1ξ∧℘ξ∈P.Proof.
By the conditionp0℘−℘,ξ=ℓ/1−ℓp1℘−℘,ξ and by Definition 2, we have
(70)∧℘ξ=1−ℓ∧℘ξ+ℓξ∧℘ξ,∧℘+1ξ=ξ∧℘ξ+ℓξ2∧℘ξ,∧℘+2ξ=ξ∧℘ξ+1+2ℓ℘2∧℘ξ+ℓξ3∧℘ξ.
We realize that(71)R∧℘+2ξ∧℘+1ξ>0,whenever the following real is positive:
(72)R1+1+ℓξ∧℘ξ+ℓξ2∧℘ξ∧℘ξ+ℓξ∧℘ξ>0.
Hence, from the inequality (74) with the idea of convex functional
(73)1−ℓ∧℘ξ+ℓξ∧℘ξ,we have
(74)R1+ξ1−ℓ∧℘ξ+ℓξ∧℘ξ1−ℓ∧℘ξ+ℓξ∧℘ξ>0.
Since convexity implies starlikeness, then we realize(75)Rξ1−ℓ∧℘ξ+ℓξ∧℘ξ1−ℓ∧℘ξ+ℓξ∧℘ξ>0.
The inequality (75) holds whenever
(76)R∧℘+1ξ∧℘ξ>0.Note that whenk=0 in Theorem 14, we have [10]—Theorem 3.1.Next, we deal with the functional(77)∧℘+1ξξ∧℘ξ′to be in the class of positive real part P.Theorem 15.
Letℓ∈0,1, ℘∈0,∞, and
(78)p1℘−℘,ξ=ℓ1−ℓp0℘−℘,ξ.
If∧℘ξ∈C (the class of convex univalent functions in Ω), then
(79)∧℘+1ξξ∧℘ξ∈Pℓ.Proof.
Applying the differential operator rule to(80)∧℘+1ξ=∧1∧℘ξyields
(81)∧℘+1ξ=∧℘−℘∧℘+1ξ=∧℘−℘∧∧℘ξ=∧℘−℘ξ∧℘ξ=p1℘−℘,ξp1℘−℘,ξ+p0℘−℘,ξξ∧℘ξ+p0℘−℘,ξp1℘−℘,ξ+p0℘−℘,ξ×ξ∧℘ξ+ξ∧℘ξ=p1℘−℘,ξp1℘−℘,ξ+p0℘−℘,ξξ∧℘ξ+p0℘−℘,ξp1℘−℘,ξ+p0℘−℘,ξξ∧℘ξ+p0℘−℘,ξp1℘−℘,ξ+p0℘−℘,ξξ2∧℘ξ=ξ∧℘ξ+p0℘−℘,ξp1℘−℘,ξ+p0℘−℘,ξξ2∧℘ξ.
Dividing equation (80) by the formula ξ∧℘ξ and employing the condition
(82)p1℘−℘,ξ=ℓ1−ℓp0℘−℘,ξ,we attain
(83)∧℘+1ξξ∧℘ξ=1+1−ℓξ∧℘ξ∧℘ξ.
The convexity of∧℘ξ,ξ∈Ω, brings
(84)R1+ξ∧℘ξ∧℘ξ>0.
Hence, it yields that(85)R∧℘+1ξξ∧℘ξ>ℓ.Note that whenk=0 in Theorem 15, we have [10]—Theorem 3.2.


## 3. Conclusion

By using the concept of locally fractional differential and integral of a complex variable, we illustrated a set of differential and integral operators acting on a class of normalized analytic functions. Moreover, we presented a conformable differential operator linking with the suggested locally fractional differential operator (similarly if we take the locally fractional integral). Different investigations are introduced, including the stralikeness and convexity properties.



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