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A New Measure of Quantum Starlike Functions Connected with Julia Functions
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.
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  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 . The main growth in the area of derivative of DSS began by Miller et al. . Generally, the concept is defined for univalent function byand . In general, if there is a function with the properties satisfying , thenwhere .
Ismail et al.  presented a class of complex functions for each fractional number , as the class of analytic functions on the open unit disk , , and on . This class is investigated, as well as the links between it and other analytic function classes. Agrawal and Sahoo  extended this notion by suggesting the -starlike functions family in a logical order. Srivastava et al.  explored the link between the Janowski functions and several known types of -starlike functions. The Janowski functions are a novel subclass of -starlike functions that they introduced and presented. Recent investigations can be located in works by Mahmood et al.  and Ul-Haq et al. .
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 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  in this area of studies is not minor as they considered the normalized analytic function and the condition of a positive real part . They have formulated the famous subclasses for starlike and convex functions, as follows, respectively:where indicates the class of normalized function .
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 -integration and decorated their meanings for the first stage. Later, Ismail et al.  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  introduced subclass of quantum starlike functions involving q-derivative. Recently, Zainab et al.  presented a sufficient condition for -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:such thatMoreover, Maclaurin’s series representation takes the sumwhere
The multiplication rule takes the following formula:We proceed to define our -starlike class using the -parametric Julia functions and connecting with the subclass of normalized functions in (Figure 1):
Definition 2. For a normalized function of the formulathe -starlike is defined by the subordination formula:We denote the subclass of these functions by , whereMoreover, a function is called -bounded turning if it satisfies the inequalityWe denote this class by .
We aim to find the range of in terms of satisfying the inequality (14). For this purpose, we need the following result.
Lemma 1. (see ). Let be analytic in , such that . Then, the upper value of on the circle at the point , is
In this section, we shall illustrate the sufficient conditions on functions to be in .
Theorem 1. Let the function , such that andIf one of the casesholds, then
Proof. Define a function as follows:By the assumption (18) and the definition of the subordination, we havewhich leads toWe aim to show that for some values , such thatAssume not; then, the above conclusion implies thatBy using the rules of Jackson derivative, we obtainConsequently, we getwhereButHence, this yieldsSuppose that there exists a point , such thatBy Jack Lemma 1 and by letting , we haveAccordingly, we conclude thatprovided one of the following cases holdswhereHence, we obtain one of the following arguments:which are all contradict (19), that is
As a special case, we have the following result.
Corollary 1. Let be satisfied the subordination:If one of the cases in (21) is occurred, then .
Proof. AssumeObviously, . Thus, in virtue of Theorem 1, we have .
Similarly, by assuming , we have the following result.
Corollary 2. Let be the satisfied subordination:If one of the cases in (21) is occurred, then .
Theorem 2. Let the function , such that andif one of the caseswhere for ,and for ,hold; then,
Proof. Define a function as follows:By the assumption (41) and the meninges of the subordination, we havewhich yieldsWe have to prove thatfor some values , such thatAssume not; then, the above conclusion imposesBy using the rules of Jackson derivative and the factswe obtainConsequently, we getSuppose that there exists a point , such thatWe aim to show thatOur method is based on Jack Lemma 1. Assume not.
Then, by consuming , we getThen, the solution when ofbrings one of the following cases:whereHence, we obtain one of the following arguments:and for ,Moreover, for , we haveAll the above inequalities contradict the assumptions of the theorem, which lead to
Corollary 3. Let be the satisfied subordination:If one of the assumptions of Theorem 2 is occurred, then .
Proof. AssumeObviously, . Thus, according to Theorem 2, we get .
In the same manner of the above result, we obtain the next one when .
Corollary 4. Let be the satisfied subordination:If one of the assumptions of Theorem 2 is occurred, then .
Theorem 3. Let the function , such that andIf one of the casesholds, then
Proof. Define a function as follows:By the assumption (68) and the meninges of the subordination, we havewhich yieldsWe have to prove thatfor some values , such thatAssume not; then, the above conclusion imposesBy employing the rules of Jackson derivative and the factswe obtainFollowing the above structure, we getSuppose that there exists a point , such thatWe aim to show thatOur method is based on Jack Lemma 1. Assume not.
Then, by consuming , we getThus, for , the solution ofprovided one of the following cases:whereHence, we obtain one of the following arguments:All the above inequalities contradict the assumptions of the theorem, which mean that
Corollary 5. Let be the satisfied subordination:If one of the assumptions of Theorem 3 occurred, then .
Proof. AssumeObviously, . Thus, according to Theorem 3, we obtain .
Similarly, for , we have the following consequence.
Corollary 6. Let be the satisfied subordination:If one of the assumptions of Theorem 3 is occurred, then .
From above, we investigate the sufficient conditions to obtain the -subordination of the -starlike classwhere . 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 is given in terms of . Note that the case of 1D parametric subclass is given by
It is studied in , while null parametric subclass is formulated byand it is investigated in .
For future works, one can suggest any types of parametric analytic functions (geometric functions) in the open unit disk. The above -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).
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
All authors contributed equally and significantly to writing this article and read and agreed to the published version of the manuscript.
This research was supported by Ajman University(2021-IRG-HBS-24).
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