A Class of Quantum Briot–Bouquet Differential Equations with Complex Coefficients

Quantum inequalities (QI) are local restraints on the magnitude and range of formulas. Quantum inequalities have been established to have a different range of applications. In this paper, we aim to introduce a study of QI in a complex domain. The idea basically, comes from employing the notion of subordination. We shall formulate a new q-differential operator (generalized of Dunkl operator of the first type) and employ it to define the classes of QI. Moreover, we employ the q-Dunkl operator to extend the class of Briot–Bouquet differential equations. We investigate the upper solution and exam the oscillation solution under some analytic functions.


Introduction
Quantum calculus exchanges the traditional derivative by a difference operator, which permits dealing with sets of non-differentiable curves and admits several formulas. The most common formula of quantum calculus is constructed by the q-operator (q-indicates for the quantum), which is created by the Jackson q-difference operator [1] as follows: let δ q be the q-calculus which is formulated by δ q ( (ξ)) = (qξ) − (ξ) , then the derivatives of functions are presented as fractions by the q-derivative D q ( (ξ)) = δ q ( (ξ)) δ q (ξ) = (qξ) − (ξ) (q − 1)ξ , ξ = 0.
Recently, quantum inequalities (differential and integral) have extensive applications not only in mathematical physics but also in other sciences. In variation problems, Cruz et al. [2] presented a new variational calculus created by the general quantum difference operator of D q . Rouze and Datta

Related Works
The quantum calculus receives the attention of many investigators. This calculus, for the first time appeared in complex analysis by Ismail et al. [9]. They defined a class of complex analytic functions dealing with the inequality condition | (qξ)| < | (ξ)| on the open unit disk. Grinshpan [10] presented some interesting outcomes filled with geometric observations are of very significant in the univalent function theory. Newly, q-calculus becomes very attractive in the field of special functions. Srivastava and Bansal [11] presented a generalization of the well-known Mittag-Leffler functions and they studied the sufficient conditions under which it is close-to-convex in the open unit disk. Srivastava et al. [12] established a new subclass of normalized smooth and starlike functions in ∪. Mahmood et al. [13] introduced a family of q-starlike functions which are based on the Ruscheweyh-type q-derivative operator. Shi et al. [14] examined some recent problems concerning the concept of q-starlike functions. Ibrahim and Darus [15] employed the notion of quantum calculus and the Hadamard product to amend an extended Sàlàgean q-differential operator. Srivastava [16] developed many functions and classes of smooth functions based on the q-calculus. The q-Subordination inequality presented by Ul-Haq et al. [17]. Govindaraj and Sivasubramanian [18] as well as Ibrahim et al. [7] used the quantum calculus and the Hadamard product to deliver some subclasses of analytic functions involving the modified Sàlàgean q-differential operator and the generalized symmetric Sàlàgean q-differential operator respectively.

q-Differential Operator
Assume that is the set of the smooth functions formulating by the followed power series For a function ∈ , the Sàlàgean operator expansion is formulated by the expansion For ∈ , we get D q (ξ) = ∞ ∑ n=1 n [n] q ξ n−1 , ξ ∈ ∪, 1 = 1. Now, let ∈ , the Sàlàgean q-differential operator [18] is formulated by where m is a positive integer. A calculation associated by the formula of D q , yields ς m [n] m q n ξ n .
Next, we present the q-differential operator as follows: where λ ∈ C. For λ = 0, q → 1 − , the operator subjects to the Sàlàgean operator [19]. In addition, the operator q Λ m λ represents to the q-Dunkl operator of first rank [20], such that the value of λ is the Dunkl parameter. The term [n] q + λ(1 + (−1) n+1 ) m indicates a major law in oscillation study (see [21]). Furthermore, the term e 2iπ is denoting the quantum number q when = 1. That is there is a connection between the definition of the q Λ m λ and its coefficients. Two functions and in are subordinated ( ≺ ), if there occurs a Schwarz function ζ ∈ ∪ with ζ(0) = 0 and |ζ(ξ)| < 1, whenever (ξ) = (ζ(ξ)) for all ξ ∈ ∪ (see [22]). Literally, the subordination inequality is indicated the equality at the origin and inclusion regarding ∪.
The next result can be found in [22].

q-Subordination Relations
In this section, we deal with the set q S * m (λ, ς) for some ς.
Theorem 2. Assume that q S * m (λ, ς) fulfills the next relation: where γ is non-negative real number (depending on β) and ς is one of the form in Theorem 1 and According to Theorem 1, one can find which leads to the requested result. Assume that ς(ξ) = (1 − β) e ξ + β, then we conclude the next minimization and maximization inequality A calculation yields
Next outcome shows the inclusion relation between the class q S * m (λ, σ) and other geometric class.
Theorem 3. The set q S * m (λ, ς) satisfies the inclusion: where ς is given in Theorem 1 .
The following theorem confirms the belonging of a normalized function in the class q S * m (λ, ς), where ς indicates the Janowski formula of order ℘ > 0.

Theorem 4. If ∈ satisfies the subordination
Proof. In virtue of Lemma 1, a computation yields Now, according to Lemma 1, we attain The next result shows the iteration inequality including the q-differential operator q Λ m λ and q Λ m+1 λ .

Theorem 5.
Suppose that ϕ is convex with ϕ(0) = 0 and g is defined as follows: Proof. For all ξ ∈ ∪, we define Please note that the term Consequently, we indicate that In virtue of Lemma 2, we conclude that (ξ) ≺ g(ξ), which leads to

Q-Differential Equations
This section deals with a class of differential equations type complex Briot-Bouquet (see [30,31] for recent works) and its analytic solutions. The main formula of BBE is ξ( (ξ)) (ξ) = Υ(ξ). The operator (1) can be used to extend q-BBE as follows: where Υ(ξ) ∈ C (the set of univalent and convex in ∪). The aim is to discuss the maximum outcome of (2) by using q−inequalities.
Proof. By the condition of the theorem, we get the following assertions: Moreover, we have ( q Λ m λ )(0) = 0, which leads to Hence the proof.
We illustrate an example to find the upper and oscillation solution of q-BBE when q = 1/2, see Tables 1 and 2.

-BBE (2) Oscillation Solution Graph
Where Ci is the cos integral function and Si is the sin integral function. The first example of 1 2 -BBE of (2) is Y(ξ) = cos(ξ) which has an oscillation solution with one branch point at the origin and has a local maximum at ξ = π 2 + 2nπ and local minimum at ξ = π 2 − 2nπ. While, for Y(ξ) = 1 − sin(ξ), the oscillation solution has no branch point in the disk. Moreover, for Y(ξ) = 1/(1 − ξ) the oscillation solution has no branch point. Finally, when Y(ξ) = 1 − ξ, the oscillation solution has a global maximum equal to 1/e at ξ = 1.

Conclusions
From above, we conclude that in view of the quantum calculus, some generalized differential operators in the open unit disk can have connections (coefficients) convergence of quantum numbers. These numbers might change the behavior of the operator and its classes of analytic functions. We investigated the oscillation solution and asymptotic solutions of different differential equations of the Briot-Bouquet type. For future work, one can employ the q-operator (1) in different classes of analytic functions such as the meromorphic and multivalent functions (see [32][33][34]). Funding: This work is financially supported by the Prince Sultan University.

Acknowledgments:
The authors would like to thank both anonymous reviewers and editor for their helpful advice.

Conflicts of Interest:
The authors declare no conflict of interest.