Geometric Inequalities via a Symmetric Differential Operator Defined by Quantum Calculus in the Open Unit Disk

Abstract

The present investigation covenants with the concept of quantum calculus besides the convolution operation to impose a comprehensive symmetric -differential operator defining new classes of analytic functions. We study the geometric representations with applications. The applications deliberated to indicate the certainty of resolutions of a category of symmetric differential equations type Briot-Bouquet.

1. Introduction

In this effort, we deal with the structure of -calculus, which develops an interesting technique for calculations and organizes different classes of operators and specific transformations. The significance of -calculus appeared in a huge number of applications including physical problems. The symmetric -activation normally achieves -difference equations (may involve derivative). A close connection between these operators and symmetries of -symmetric operator is accordingly to be estimated (see [1–9]). In recent investigation, we deliver a process for deriving and interpreting from a symmetry possessions and infirm analogy with the traditional cases. By combining the -calculus and the symmetric Salagean differential operator, we introduce a novel modified symmetric Salagean -differential operator. Via employing this operator, we deliver new classes of analytic functions.

2. Preliminaries

This section gives out the mathematical processing to deliver the suggested SDOs and complex conformable operator for some classes of analytic functions in the open unit disk . Let be the category of smooth function elicited as pursue

A function is known as a starlike with respect to if the straight line segment combining the origin to all else point of embedding completely in . The aim is that each point of must be manifested via (0,0). A univalent function () is indicated to be convex in if the linear slice combining two ends of stays completely in . We denote these classes by and for starlike and convex, respectively. In addition, suppose that the category involves all functions analytic in with a positive real part in achieving . Mathematically, if and only if , and if and only if ; equivalently, for starlikeness and for convexity.

For two functions, and belong to the category and are said to be subordinate, noting by , if we can find a Schwarz function with and achieving (the detail can be located in [10]). Obviously, implies that and We employ next facts, one can find it in [11].

For every nonnegative integer the -integer number symbolized by is structured by where , , and . Consequently, the analytic function is written by the formula

Clearly, we have Consequently, for we attain

For , it realized that the Sàlàgean -differential operator [12] has the formula such that represents as a positive integer. A computation based on the definition of implies that , where is the convolution product. and

Clearly, the well-known Salagean differential operator [6].

Consider the role and a constant , we introduce the -symmetric Salagean differential operator using the definition of as follows:

Obviously, we indicate that when , we get the Salagean -differential operator. We can call (11) as the symmetric Sàlàgean -differential operator in . Also, we have the following two limits. which are represented to the well-known Salagean differential operator [6] and the symmetric differential operator [8], respectively.

Depending on the definition of (11), we impose the recognizing classes:

Definition 1. A member is called in the category when We obtain the following special cases: (i) ([8])(ii) ([13–15])(iii) ([16])(iv) ([17, 18])(v) ([19])(vi) ([20])(vii) ([21])(viii) ([22])(ix) ([23])(x) ([24])

Definition 2. If then if and only if (i) [25](ii) [26](iii) [27](iv) [8]In our investigation, we focus on the geometric presentation of the special classes and via utilizing the basis information given in [11].

Lemma 3. Let , let be a positive integer and let (i)Suppose that such that In addition, let and ; thereafter, there are two fixed positive numbers and with so that (ii)Assume that and , subsequently, there occurs a fixed number with such that (iii)Consider with posterity or for such that thus

Remark 4. Concerning Lemma 3 (i), more information about and can be found in [11] (Theorem 3.1c, p. 73). About (ii), more information about and can be found in [11] (Theorem 3.1e or Corollary 3.1e.2, p. 77 and p. 79, respectively). And for (iii), one can see Theorem 4.1g in [11] (p. 198) and, for the inequality contains a Briot-Bouquet formula.

3. Main Results

Here, we focus in the geometric representations of the classes and and the outcomes of these classes.

Theorem 5. For if one of the recognizing determinations are indicated by (i) is of bounded turning(ii) satisfies the subordination formula (iii) achieves the formula (iv) satisfies the relation (v) admits the inequality then where .

Proof. Sort out a function in the following construction: Via the main information, is of constrained limit turning; it infers that Accordingly, via Lemma 3 (i), we acquire which incomes the first requested statement of the theorem. In opinion of the additional information, we obligate the subsequent subordination relative Now, based on Lemma 3 (i), there exists a fixed positive number satisfying and the subordination inequality This leads to the conclusion Lastly, we assume the third fact, a direct reckoning reaches to According to the virtue of Lemma 3 (ii), there occurs a fixed positive number, say achieving Consequently, we obtain Hence, via Equation (29), it indicates that So by Noshiro-Warschawski and Kaplan Propositions, and of bounded turning in .
Via the derivative (25) and operating the real, one can attain the real relation Hence, according to Lemma 3 (ii), one get
Via considering the logarithmic derivative on (25) and acting as areal part, we obtain the consequence conversation: Thus, according to Lemma 3 (iii), where we attain that

Theorem 6. Suppose that with . Then where (analytic) satisfies that and . In addition, for , reckoning fulfills the formula statement

Proof. We note that , then one can gain this confirms that there occurs analytic function type Schwarz achieving the relations and A calculation gives us Via integrating left and right parts, one can achieve Thus, we have Via the definition and the properties of subordination, one can have Furthermore, we deliver that maps the disk onto a convex symmetric domain corresponding to the real axis, that is which implies that Via applying Equation (40), one can indicate that which leads to Hence, we have

Corollary 7 [8]. If in Theorem 6, then

Theorem 8. Suppose that , then the odd construction formula fulfills the consequently subordination

Proof. Let Subsequently, we get that there occurs a function with the formula This yields In addition, since achieves the inequality taking account that the fractional functional express is univalent and hence, consequently, we attain the relation Furthermore, the expression , which leads to the inequality that is, there occurs a Schwarz function with the property which implies that there is such that An operation, one can indicate that Thus, one can inform the recognizing inequality This implies the consequence result Next consequence outcomes of the above result can be located in [8, 25, 26] accordingly.

Corollary 9. If in Theorem 8, then

Corollary 10. If , and in Theorem 8, then

Corollary 11. If in Theorem 8, then

4. Applications

We introduce an application of our outcomes based on finding the outcome of Briot-Bouquet equation (BBE) (see [11] for more information). This category of ODE is an association of ODE whose outcomes are formulas in the complex plane. Existence and uniqueness theorems include the utility of majors and minors (or subordination and superordination concepts) (see [28–31]). Investigation of rational first ODEs in the complex region implies the finding of new transcendental special functions which are now known as symmetric BBE

By employing the -differential operator (11), we have the -formula of BBE

A simple result of (64) can be recognized at Thus, we investigate the status, and The initial condition will be

Theorem 12. Suppose that Equation (64) with and with nonnegative coefficients. If is convex in , then there occurs a major solution achieving the inequality such that indicates analytic function in satisfying the relations and .

Proof. In virtue of every conditions indicated by (64), and we realize the information where Moreover, by the definition of we indicate that . Consequently, yields that Thus, via Theorem 6, we reach the desired outcome in (65).

Data Availability

No data were used to support the findings of the study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.