# A New Measure of Quantum Starlike Functions Connected with Julia Functions

Authors: Samir B. Hadid; Rabha W. Ibrahim; Shaher Momani
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In a complex domain, the investigation of the quantum differential subordinations for starlike functions is newly considered by few research studies. In this note, we arrange a set of necessary conditions utilizing the concept of the quantum differential subordinations for starlike functions related to the set of parametric Julia functions. Our method is based on the usage of quantum Jack lemma, where this lemma is generalized recently by the quantum derivative (Jackson calculus). We illustrate a starlike formula dominated by different types of Julia functions. The sufficient conditions are computed in the quantum and the Julia fractional parameters. We indicate a relationship between these two parameters.


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## Body

## 1. Introduction

The notion of differential subordination and superordination (DSS) shows a dynamic model in the investigation of geometric possessions of holomorphic functions in the open unit disk. Lindelof first presented it, while Littlewood [1] did the extraordinary exertion in this area of study. Numerous investigators added information in the application of DSS. Antiquity and the improvement of mechanisms in the area connected with DSS are concisely designated and incorporated in the hardcover by Miller and Mocanu [2]. The main growth in the area of derivative of DSS began by Miller et al. [3]. Generally, the concept is defined for univalent function ω by(1)φ≺ψ⇔φ0=ψ0,and φω⊂ψω. In general, if there is a function with the properties ω0=0,ωξ<ξ, satisfying φξ=ψωξ, then(2)φξ≺ψξ,where ξ∈⊔≔ξ∈ℂ:ξ<1.Ismail et al. [4] presented a class of complex functions for each fractional number q, 0<q<1 as the class of analytic functions φ on the open unit disk ⊔, φ0=0,φ′0=1, and φqξ≤φξ on ⊔. This class is investigated, as well as the links between it and other analytic function classes. Agrawal and Sahoo [5] extended this notion by suggesting the q-starlike functions family in a logical order. Srivastava et al. [6] explored the link between the Janowski functions and several known types of q-starlike functions. The Janowski functions are a novel subclass of q-starlike functions that they introduced and presented. Recent investigations can be located in works by Mahmood et al. [7] and Ul-Haq et al. [8].Parametric Julia functions are usually utilized to determine the upper bound solutions of different types of differential equations of a complex variable [4–11]. In the recent study, we shall extend this concept applying the quantum calculus (Jackson calculus) and employ it to define special classes of analytic function types normalized analytically in the open unit disk φ0=0,φ′0=1 and dominated by different kinds of the parametric Julia functions. Our method is based on the quantum Jack lemma.


## 2. Quantum Starlike Formula

The effort of Ma and Minda [12] in this area of studies is not minor as they considered the normalized analytic function p0=1 and the condition of a positive real part ℜp′ξ>0. They have formulated the famous subclasses for starlike and convex functions, as follows, respectively:(3)S∗p=φ∈Δ:ξφ′ξφξ≺pξ,ξ∈⊔,Cp=φ∈Δ:ξφ″ξφ′ξ+1≺pξ,ξ∈⊔,where Δ indicates the class of normalized function φ0=0=φ′0−1.Quantum calculus (QC) is the novel part of mathematical analysis and its applications and is correspondingly significant for its appearances, both in physics and in mathematics as well. Jackson [13,14] formulated the functions of q-differentiation and q-integration and decorated their meanings for the first stage. Later, Ismail et al. [4] contributed the indication of q-calculus in geometric function theory.Nowadays, different classes of Ma and Minda are suggested and developed, using QC by researchers. For instant, Seoudy and Aouf [15] introduced subclass of quantum starlike functions involving q-derivative. Recently, Zainab et al. [16] presented a sufficient condition for q-starlikeness using a special curve. In addition, different differential and integral operators are generalized utilizing QC [17–20].Definition 1.
Jackson derivative is indicated in the following difference operator:(4)∂qhξ=hξ−hqξξ1−q,q∈0,1,such that(5)∂qξv=1−qv1−qξv−1.
Moreover, Maclaurin’s series representation takes the sum(6)∂qhξ=∑ℓ=0∞hℓℓqξℓ−1,where(7)ℓq≔1−qℓ1−q.Note that(8)limq⟶1−∂qhξ=h′ξ.The multiplication rule takes the following formula:(9)∂qfξgξ=gξ∂qfξ+fqξ∂qgξ=gqξ∂qfξ+fξ∂qgξ.We proceed to define our q-starlike class using the q-parametric Julia functions and connecting with the subclass of normalized functions in ⊔ (Figure 1):(10)J1βξ=1+ξ−βξ3(11)J2βξ=1+ξ−βξ22,(12)J3βξ=1+ξ−βξ2,β∈ℂ,ξ∈⊔Figure 1
Plot ofJiβ,i=1,2,3 for β=1/2,1/3,1/4 and β=1, respectively. The plot is connected when β∈0,1; otherwise, it is disconnected.Definition 2.
For a normalized functionφξ∈Δ of the formula(13)φξ=ξ+∑n=2∞φnξn,ξ∈⊔,the q-starlike is defined by the subordination formula:(14)ξ∂qφξφξ≺Jiβξi=1,2,3,q∈0,1,β∈ℂ.
We denote the subclass of these functions byΔqβ, where(15)∂qφ=1+∑n=2∞φn1−qn1−qξn.
Moreover, a functionφ∈Δ is called q-bounded turning if it satisfies the inequality(16)∂qφξ≺Jiβξ.
We denote this class byBqβ.We aim to find the range ofβ in terms of q satisfying the inequality (14). For this purpose, we need the following result.Lemma 1.
(see [21]). Let ϖ be analytic in ⊔, such that ϖ0=0. Then, the upper value of ϖ on the circle ξ=1 at the point ξ0=reiθ,θ∈−π,π,q∈0,1, is(17)ξ0∂qϖξ0=μϖξ0,μ≥1.


## 3. Results

In this section, we shall illustrate the sufficient conditions on functionsφ∈Δ to be in Δqβ.Theorem 1.
Let the functionρ∈⊔, such that ρ0=1 and(18)1+ξ∂qρξ≺1+ξ,ξ⊔.
If one of the cases(19)β≠q+1q2−q+1,β<q+1+2q2−q+1,β>q+1−2q2−q+1,q∈0,1,holds, then(20)ρξ≺J1βξ=1+ξ−βξ3.Proof.
Define a functionϱ as follows:(21)ϱξ≔1+ξ∂qρξ.
By the assumption (18) and the definition of the subordination, we have(22)1+ξ∂qρξ=1+ϖξ,ϖ0=0,ϖξ≤ξ<1,which leads to(23)ϖξ=ϱ2ξ−1.
We aim to show thatϱ2ξ−1<1 for some values ξ0∈⊔, such that(24)ρξ=1+ϖξ−βϖ3ξ.
Assume not; then, the above conclusion implies that(25)ϱξ=1+ξ∂q1+ϖξ−βϖ3ξ.
By using the rules of Jackson derivative, we obtain(26)∂qϖ2ξ=∂qϖξϖξ+ϖqξ,∂qϖ3ξ=∂qϖξϖ2ξ+ϖξϖqξ+ϖ2qξ.
Consequently, we get(27)ϱξ=1+Wqξξ∂qϖξ,where(28)Wqξ≔ϖξ+ϖqξ−βϖ2ξ+ϖξϖqξ+ϖ2qξ.
But(29)ϖqξ=ϖξ−1−qξ∂qϖξ.
Hence, this yields(30)Wqξ=ϖξ2−3βϖξ+ξ∂qϖξ−1−q+3−qϖξ−β1−q2ξ∂qϖξ.
Suppose that there exists a pointξ0∈⊔, such that(31)maxξ≤ξ0ϖξ=ϖξ0=1,ξ0∂qϖξ0=μϖξ0,μ≥1.
By Jack Lemma1 and by letting ϖξ0=eiθ, we have(32)Wqξ0=ϖξ2−3βϖξ+ξ∂qϖξ−1−q+β3−qϖξ−β1−q2ξ∂qϖξξ=ξ0=eiθ2−3βeiθ+μeiθ−1−q+β3−qeiθ−β1−q2μeiθ≥ℜeiθ2−3βeiθ+μeiθ−1−q+β3−qeiθ−β1−q2μeiθ=cosθ2−3βcosθ+μcosθ−1−q+β3−qcosθ−β1−q2μcosθ=β−3+μ3−q−μ1−q2cos2θ+2−μ1−qcosθ=βμ3−q−1−q2−3cos2θ+2−μ1−qcosθ.
Accordingly, we conclude that(33)ϱξ2−1ξ=ξ0=1+Wqξξ∂qϖξ2−1ξ=ξ0≥Wqξ0ξ0∂qϖξ02−1=μβμ3−q−1−q2−3cos3θ+μ2−μ1−qcos2θ2−1≥β3−q−1−q2−3+1+q2−1≥1,provided one of the following cases holds(34)Υ≤−2,Υ=0,Υ≥2,where(35)Υ≔β3−q−1−q2−3+1+q.Hence, we obtain one of the following arguments:(36)β=q+1q2−q+1,β≥q+1+2q2−q+1,β≤q+1−2q2−q+1,which are all contradict (19), that is(37)ρξ≺J1βξ=1+ξ−βξ3.As a special case, we have the following result.Corollary 1.
Letφ∈Δ be satisfied the subordination:(38)1+ξ∂qξ∂qφξφξ≺1+ξ,ξ∈⊔.
If one of the cases in (21) is occurred, then φ∈Δqβ.Proof.
Assume(39)ρξ=ξ∂qφξφξ.
Obviously,ρ0=1. Thus, in virtue of Theorem 1, we have φ∈Δqβ.Similarly, by assumingρξ=∂qφξ,φ∈Δ, we have the following result.Corollary 2.
Letφ∈Δ be the satisfied subordination:(40)1+ξ∂qφξ≺1+ξ,ξ∈⊔.
If one of the cases in (21) is occurred, then φ∈Bqβ.Theorem 2.
Let the functionh∈⊔, such that h0=1 and(41)1+ξ∂qhξ≺1+ξ,ξ∈⊔,if one of the cases(42)β≠1q+1,β≠2,where for 0.418341<q<1,(43)0.52q+2q+32−10.3784q+1+3q+1≥β≥0.52q−2q+32−5.62159q+1+3q+1,and for 0<q<0.418341,(44)0.52q+2q+32−10.3784q+1+3q+1≥β≥0.52q−2q+32−10.3784q+1+3q+1,hold; then,(45)hξ≺J2βξ=1+ξ−βξ22.Proof.
Define a functionp as follows:(46)pξ≔1+ξ∂qhξ.
By the assumption (41) and the meninges of the subordination, we have(47)1+ξ∂qhξ=1+wξ,w0=0,wξ≤ξ<1,which yields(48)wξ=p2ξ−1.
We have to prove that(49)wξ=p2ξ−1<1,for some values ξ0∈⊔, such that(50)hξ=1+wξ−βw2ξ2.
Assume not; then, the above conclusion imposes(51)pξ=1+ξ∂q1+wξ−βw2ξ2.
By using the rules of Jackson derivative and the facts(52)wqξ=wξ−1−qξ∂qwξ,∂qw2ξ=∂qwξ2wξ−1−qξ∂qwξ,we obtain(53)∂q1+wξ−βw2ξ2=1+wξ−βw2ξ∂q1+wξ−βw2ξ+1+wqξ−βw2qξ∂q1+wξ−βw2ξ=21+wξ−βw2ξ∂qwξ−β∂qwξ2wξ−1−qξ∂qwξ=21+wξ−βw2ξ∂qwξ1−β2wξ−1−qξ∂qwξ.
Consequently, we get(54)pξ=1+2ξ1+wξ−βw2ξ∂qwξ1−β2wξ−1−qξ∂qwξ.
Suppose that there exists a pointξ0∈⊔, such that(55)maxξ≤ξ0wξ=wξ0=1,ξ0∂qwξ0=μwξ0,μ≥1.
We aim to show that(56)wξ=p2ξ−1<1.
Our method is based on Jack Lemma1. Assume not.
Then, by consumingwξ0=eiθ, we get(57)p2ξ0−1=1+2ξ1+wξ−βw2ξ∂qwξ1−β2wξ−1−qξ∂qwξ2−1ξ=ξ0≥41+eiθ−βe2iθ2μeiθ1−β2eiθ−1−qμeiθ2−1≥ℜ41+eiθ−βe2iθ2μeiθ1−β2eiθ−1−qμeiθ2−1=41+cosθ−βcos2θ2μcosθ1−β2cosθ−1−qμcosθ2−1≥1.
Then, the solution whencosθ=1 of(58)41+cosθ−βcos2θ2μcosθ1−β2cosθ−1−qμcosθ2−1≥1brings one of the following cases:(59)⋏=0,⋏≥2,⋏≤−2,where(60)⋏:=42−β21−β1+q2.
Hence, we obtain one of the following arguments:(61)β=1q+1,β=2,and for 0.418341<q<1,(62)0.52q+2q+32−10.3784q+1+3q+1≤β≤0.52q−2q+32−5.62159q+1+3q+1.
Moreover, for0<q<0.418341, we have(63)0.52q+2q+32−10.3784q+1+3q+1≤β≤0.52q−2q+32−10.3784q+1+3q+1.
All the above inequalities contradict the assumptions of the theorem, which lead to(64)hξ≺J2βξ=1+ξ−βξ22.Corollary 3.
Letφ∈Δ be the satisfied subordination:(65)1+ξξ∂qφξφξ≺1+ξ.If one of the assumptions of Theorem 2 is occurred, then φ∈Δqβ.Proof.
Assume(66)pξ=ξ∂qφξφξ.
Obviously,p0=1. Thus, according to Theorem 2, we get φ∈Δqβ.In the same manner of the above result, we obtain the next one whenpξ=∂qφξ,φ∈Δ.Corollary 4.
Letφ∈Δ be the satisfied subordination:(67)1+ξ∂qφξ≺1+ξ.If one of the assumptions of Theorem 2 is occurred, then φ∈Bqβ.Theorem 3.
Let the functiong∈⊔, such that g0=1 and(68)1+ξ∂qgξ≺1+ξ,ξ∈⊔.
If one of the cases(69)β≠1q+1;β≥−1.189211/q+12q+1.189211/q+12−1q+1,0<q<1,β≤1.189211/q+12q+1.189211/q+12+1q+1,0<q<1,holds, then(70)gξ≺J3βξ=1+ξ−βξ2.Proof.
Define a functionσ as follows:(71)σξ≔1+ξ∂qgξ.
By the assumption (68) and the meninges of the subordination, we have(72)1+ξ∂qgξ=1+uξ,u0=0,uξ≤ξ<1,which yields(73)uξ=σ2ξ−1.
We have to prove that(74)uξ=σ2ξ−1<1,for some values ξ0∈⊔, such that(75)gξ=1+uξ−βu2ξ.
Assume not; then, the above conclusion imposes(76)σξ=1+ξ∂q1+uξ−βu2ξ.
By employing the rules of Jackson derivative and the facts(77)uqξ=uξ−1−qξ∂quξ,∂qu2ξ=∂quξ2uξ−1−qξ∂quξ,we obtain(78)∂q1+uξ−βu2ξ=∂quξ−β∂quξ2uξ−1−qξ∂quξ=∂quξ1−β2uξ−1−qξ∂quξ.
Following the above structure, we get(79)σξ=1+ξ∂quξ1−β2uξ−1−qξ∂quξ.
Suppose that there exists a pointξ0∈⊔, such that(80)maxξ≤ξ0uξ=uξ0=1,(81)ξ0∂quξ0=μuξ0,μ≥1.
We aim to show that(82)uξ=σ2ξ−1<1.
Our method is based on Jack Lemma1. Assume not.
Then, by consuminguξ0=eiθ, we get(83)σ2ξ0−1=1+ξ∂quξ1−β2uξ−1−qξ∂quξ2−1ξ=ξ0≥ξ∂quξ1−β2uξ−1−qξ∂quξ2−1ξ=ξ0=ξ0∂quξ01−β2uξ0−1−qξ0∂quξ02−1=μeiθ1−β2eiθ−1−qμeiθ2−1≥ℜμeiθ1−β2eiθ−1−qμeiθ2−1=μcosθ1−β2cosθ−1−qμcosθ2−1≥1.
Thus, forcosθ=1,μ=1, the solution of(84)μcosθ1−β2cosθ−1−qμcosθ2−1≥1,provided one of the following cases:(85)⋎=0,⋎≥2,⋎≤−2,where(86)⋎≔1−β1+q2.
Hence, we obtain one of the following arguments:(87)β=1q+1;β≤−1.189211/q+12q+1.189211/q+12−1q+1,0<q<1,β≥1.189211/q+12q+1.189211/q+12+1q+1,0<q<1.
All the above inequalities contradict the assumptions of the theorem, which mean that(88)gξ≺J3βξ=1+ξ−βξ2.Corollary 5.
Letφ∈Δ be the satisfied subordination:(89)1+ξξ∂qφξφξ≺1+ξ.
If one of the assumptions of Theorem3 occurred, then φ∈Δqβ.Proof.
Assume(90)σξ=ξ∂qφξφξ.
Obviously,σ0=1. Thus, according to Theorem 3, we obtain φ∈Δqβ.Similarly, forσξ=∂qφξ, we have the following consequence.Corollary 6.
Letφ∈Δ be the satisfied subordination:(91)1+ξ∂qφξ≺1+ξ.
If one of the assumptions of Theorem3 is occurred, then φ∈Bqβ.


## 4. Conclusion

From above, we investigate the sufficient conditions to obtain theq-subordination of the q-starlike class(92)ξ∂qφξφξ≺Jξ,ξ∈⊔,where Jξ=Jiβ,i=1,2,3. Differential inequalities are illustrated, involving the q-differential subordination. Nice geometric presentation is included describing the connected Julia functions of different orders. Our class is called 2D parametric subclass of analytic function, and β is given in terms of q. Note that the case of 1D parametric subclass is given by(93)Jξ=1+ξ1−qξ.It is studied in [21], while null parametric subclass is formulated by(94)Jξ=1+43ξ+23ξ2,and it is investigated in [16].For future works, one can suggest any types of parametric analytic functions (geometric functions) in the open unit disk. The aboveq-differential subordination formula can be suggested to study the solution of many classes of generalized differential equations such as the class of Briot-Bouquet differential equation (2).



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