Journal of Function Spaces
Volume 2022, Issue 1 4865785
Research Article
Open Access

A New Measure of Quantum Starlike Functions Connected with Julia Functions

Samir B. Hadid, 

Samir B. Hadid

Department of Mathematics and Sciences, College of Humanities and Sciences, Ajman University, Ajman, UAE ajman.ac.ae

Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, UAE ajman.ac.ae

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Rabha W. Ibrahim, 

Corresponding Author

Rabha W. Ibrahim

IEEE: 94086547, Perrysburg, Ohio, USA

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Shaher Momani, 

Shaher Momani

Department of Mathematics and Sciences, College of Humanities and Sciences, Ajman University, Ajman, UAE ajman.ac.ae

Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, UAE ajman.ac.ae

Department of Mathematics, Faculty of Science, University of Jordan, Amman 11942, Jordan ju.edu.jo

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First published: 30 January 2022
https://doi.org/10.1155/2022/4865785
Citations: 5
Academic Editor: Sarfraz Nawaz Malik

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.

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)
and φ(ω) ⊂ ψ(ω). In general, if there is a function with the properties ω(0) = 0, |ω(ξ)| < |ξ|, satisfying φ(ξ) = ψ(ω(ξ)), then
(2)
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 (⊔), φ(0) = 0, φ′(0) = 1, and |φ(qξ)| ≤ |φ(ξ)| 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 (φ(0) = 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 p(0) = 1 and the condition of a positive real part ℜ(p′(ξ)) > 0. They have formulated the famous subclasses for starlike and convex functions, as follows, respectively:
(3)
where Δ indicates the class of normalized function φ(0) = 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)
such that
(5)

Moreover, Maclaurin’s series representation takes the sum

(6)
where
(7)

Note that
(8)
The multiplication rule takes the following formula:
(9)
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)
(11)
(12)
Details are in the caption following the image
Figure 1
Open in figure viewerPowerPoint
Plot of 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 function φ(ξ) ∈ Δ of the formula

(13)
the q-starlike is defined by the subordination formula:
(14)

We denote the subclass of these functions by , where

(15)

Moreover, a function φ ∈ Δ is called q-bounded turning if it satisfies the inequality

(16)

We denote this class by .

We aim to find the range of β 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 ϖ(0) = 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)

3. Results

In this section, we shall illustrate the sufficient conditions on functions φ ∈ Δ to be in .

Theorem 1. Let the function ρ ∈ ⊔, such that ρ(0) = 1 and

(18)

If one of the cases

(19)
holds, then
(20)

Proof. Define a function ϱ as follows:

(21)

By the assumption (18) and the definition of the subordination, we have

(22)
which leads to
(23)

We aim to show that |ϱ2(ξ) − 1| < 1 for some values ξ0 ∈ ⊔, such that

(24)

Assume not; then, the above conclusion implies that

(25)

By using the rules of Jackson derivative, we obtain

(26)

Consequently, we get

(27)
where
(28)

But

(29)

Hence, this yields

(30)

Suppose that there exists a point ξ0 ∈ ⊔, such that

(31)

By Jack Lemma 1 and by letting ϖ(ξ0) = eiθ, we have

(32)

Accordingly, we conclude that

(33)
provided one of the following cases holds
(34)
where
(35)
Hence, we obtain one of the following arguments:
(36)
which are all contradict (19), that is
(37)

As a special case, we have the following result.

Corollary 1. Let φ ∈ Δ be satisfied the subordination:

(38)

If one of the cases in (21) is occurred, then .

Proof. Assume

(39)

Obviously, ρ(0) = 1. Thus, in virtue of Theorem 1, we have .

Similarly, by assuming ρ(ξ) = ∂qφ(ξ), φ ∈ Δ, we have the following result.

Corollary 2. Let φ ∈ Δ be the satisfied subordination:

(40)

If one of the cases in (21) is occurred, then .

Theorem 2. Let the function h ∈ ⊔, such that h(0) = 1 and

(41)
if one of the cases
(42)
where for 0.418341 < q < 1,
(43)
and for 0 < q < 0.418341,
(44)
hold; then,
(45)

Proof. Define a function p as follows:

(46)

By the assumption (41) and the meninges of the subordination, we have

(47)
which yields
(48)

We have to prove that

(49)
for some values ξ0 ∈ ⊔, such that
(50)

Assume not; then, the above conclusion imposes

(51)

By using the rules of Jackson derivative and the facts

(52)
we obtain
(53)

Consequently, we get

(54)

Suppose that there exists a point ξ0 ∈ ⊔, such that

(55)

We aim to show that

(56)

Our method is based on Jack Lemma 1. Assume not.

Then, by consuming w(ξ0) = eiθ, we get

(57)

Then, the solution when cos(θ) = 1 of

(58)
brings one of the following cases:
(59)
where
(60)

Hence, we obtain one of the following arguments:

(61)
and for 0.418341 < q < 1,
(62)

Moreover, for 0 < q < 0.418341, we have

(63)

All the above inequalities contradict the assumptions of the theorem, which lead to

(64)

Corollary 3. Let φ ∈ Δ be the satisfied subordination:

(65)
If one of the assumptions of Theorem 2 is occurred, then .

Proof. Assume

(66)

Obviously, p(0) = 1. Thus, according to Theorem 2, we get .

In the same manner of the above result, we obtain the next one when p(ξ) = ∂qφ(ξ), φ ∈ Δ.

Corollary 4. Let φ ∈ Δ be the satisfied subordination:

(67)
If one of the assumptions of Theorem 2 is occurred, then .

Theorem 3. Let the function g ∈ ⊔, such that g(0) = 1 and

(68)

If one of the cases

(69)
holds, then
(70)

Proof. Define a function σ as follows:

(71)

By the assumption (68) and the meninges of the subordination, we have

(72)
which yields
(73)

We have to prove that

(74)
for some values ξ0 ∈ ⊔, such that
(75)

Assume not; then, the above conclusion imposes

(76)

By employing the rules of Jackson derivative and the facts

(77)
we obtain
(78)

Following the above structure, we get

(79)

Suppose that there exists a point ξ0 ∈ ⊔, such that

(80)
(81)

We aim to show that

(82)

Our method is based on Jack Lemma 1. Assume not.

Then, by consuming u(ξ0) = eiθ, we get

(83)

Thus, for cos(θ) = 1, μ = 1, the solution of

(84)
provided one of the following cases:
(85)
where
(86)

Hence, we obtain one of the following arguments:

(87)

All the above inequalities contradict the assumptions of the theorem, which mean that

(88)

Corollary 5. Let φ ∈ Δ be the satisfied subordination:

(89)

If one of the assumptions of Theorem 3 occurred, then .

Proof. Assume

(90)

Obviously, σ(0) = 1. Thus, according to Theorem 3, we obtain .

Similarly, for σ(ξ) = ∂qφ(ξ), we have the following consequence.

Corollary 6. Let φ ∈ Δ be the satisfied subordination:

(91)

If one of the assumptions of Theorem 3 is occurred, then .

4. Conclusion

From above, we investigate the sufficient conditions to obtain the q-subordination of the q-starlike class
(92)
where . 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)
It is studied in [21], while null parametric subclass is formulated by
(94)
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 above q-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).

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally and significantly to writing this article and read and agreed to the published version of the manuscript.

Acknowledgments

This research was supported by Ajman University(2021-IRG-HBS-24).

    Data Availability

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