Research Article | Open Access

Rabha W. Ibrahim, Rafida M. Elobaid, Suzan J. Obaiys, "Geometric Inequalities via a Symmetric Differential Operator Defined by Quantum Calculus in the Open Unit Disk", *Journal of Function Spaces*, vol. 2020, Article ID 6932739, 8 pages, 2020. https://doi.org/10.1155/2020/6932739

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

**Academic Editor:**Adrian Petrusel

#### 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.

#### References

- R. D. Carmichael, “The general theory of linear q-difference equations,”
*American Journal of Mathematics*, vol. 34, no. 2, pp. 147–168, 1912. View at: Publisher Site | Google Scholar - D. O. Jackson, T. Fukuda, O. Dunn, and English Majors and Core Skill-based, “On q-definite integrals,”
*Quarterly Journal of Pure and Applied Mathematics*, vol. 41, pp. 193–203, 1910. View at: Publisher Site | Google Scholar - T. E. Mason, “On properties of the Solutions of linear q-difference equations with entire function coefficients,”
*American Journal of Mathematics*, vol. 37, no. 4, pp. 439–444, 1915. View at: Publisher Site | Google Scholar - W. J. Trjitzinsky, “Analytic theory of linear q-difference equations,”
*Acta Mathematica*, vol. 61, pp. 1–38, 1933. View at: Publisher Site | Google Scholar - M. E. H. Ismail, E. Merkes, and D. Styer, “A generalization of starlike functions,”
*Complex Variables, Theory and Application: An International Journal*, vol. 14, pp. 77–84, 2007. View at: Publisher Site | Google Scholar - G. S. Salagean,
*Subclasses of univalent functions, Complex Analysis-Fifth Romanian-Finnish Seminar, Part 1 (Bucharest, 1981)*, vol. 1013 of*Lecture Notes in Mathematics*, Springer, Berlin, 1983. - F. M. Al-Oboudi, “On univalent functions defined by a generalized Sălăgean operator,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2004, no. 27, Article ID 172525, pp. 1429–1436, 2004. View at: Publisher Site | Google Scholar - R. W. Ibrahim and M. Darus, “New symmetric differential and integral operators defined in the complex domain,”
*Symmetry*, vol. 11, no. 7, p. 906, 2019. View at: Publisher Site | Google Scholar - M. Naeem, S. Hussain, T. Mahmood, S. Khan, and M. Darus, “A new subclass of analytic functions defined by using Salagean q-differential operator,”
*Mathematics*, vol. 7, no. 5, p. 458, 2019. View at: Publisher Site | Google Scholar - P. Duren,
*Univalent Functions*, vol. 259 of*Grundlehren der mathematischen Wissenschaften*, Springer-Verlag New York Inc, 1983. - S. S. Miller and P. T. Mocanu,
*Differential Subordinations: Theory and Applications*, CRC Press, 2000. - M. Govindaraj and S. Sivasubramanian, “On a class of analytic functions related to conic domains involving
*q*-calculus,”*Analysis Mathematica*, vol. 43, no. 3, pp. 475–487, 2017. View at: Publisher Site | Google Scholar - Z. J. Jakubowski and J. Kaminski, “On some properties of Mocanu-Janowski functions,”
*Revue Roumaine de Mathématiques Pures et Appliquées*, vol. 23, pp. 1523–1532, 1978. View at: Google Scholar - M. Obradović and S. Owa, “On certain properties for some classes of starlike functions,”
*Journal of Mathematical Analysis and Applications*, vol. 145, no. 2, pp. 357–364, 1990. View at: Publisher Site | Google Scholar - W. C. Ma and D. Minda, “A unified treatment of some special classes of univalent functions,” in
*Proceedings of the Conference on Complex Analysis*, pp. 19–23, Tianjin, China, 1992. View at: Google Scholar - P. Goel and S. S. Kumar, “Certain class of starlike functions associated with modified sigmoid function,”
*Bulletin of the Malaysian Mathematical Sciences Society*, vol. 43, no. 1, pp. 957–991, 2020. View at: Publisher Site | Google Scholar - J. Dziok, R. K. Raina, and J. Sokół, “Certain results for a class of convex functions related to a shell-like curve connected with Fibonacci numbers,”
*Computers & Mathematcs with Applications*, vol. 61, no. 9, pp. 2605–2613, 2011. View at: Publisher Site | Google Scholar - J. Dziok, R. K. Raina, and J. Sokół, “On a class of starlike functions related to a shell-like curve connected with Fibonacci numbers,”
*Mathematical and Computer Modelling*, vol. 57, no. 5-6, pp. 1203–1211, 2013. View at: Publisher Site | Google Scholar - R. Kargar, A. Ebadian, and J. Sokol, “On subordination of some analytic functions,”
*Siberian Mathematical Journal*, vol. 57, no. 4, pp. 599–605, 2016. View at: Publisher Site | Google Scholar - R. Kargar, A. Ebadian, and J. Sokół, “On booth lemniscate and starlike functions,”
*Analysis and Mathematical Physics*, vol. 9, no. 1, pp. 143–154, 2019. View at: Publisher Site | Google Scholar - J. Sokól, “On some subclass of strongly starlike functions,”
*Demonstratio Mathematica*, vol. 31, no. 1, pp. 81–86, 1998. View at: Publisher Site | Google Scholar - N. E. Cho, V. Kumar, S. S. Kumar, and V. Ravichandran, “Radius problems for starlike functions associated with the sine function,”
*Bulletin of the Iranian Mathematical Society*, vol. 45, no. 1, pp. 213–232, 2019. View at: Publisher Site | Google Scholar - H. Tang, H. M. Srivastava, S.-H. Li, and G.-T. Deng, “Majorization results for subclasses of starlike functions based on the sine and cosine functions,”
*Bulletin of the Iranian Mathematical Society*, vol. 46, no. 2, pp. 381–388, 2020. View at: Publisher Site | Google Scholar - S. Sivasubramanian, M. Govindaraj, G. Murugusundaramoorthy, and N. E. Cho, “Differential subordination for analytic functions associated with left-like domains,”
*Journal of Computational Analysis & Applications*, vol. 26, no. 1, 2019. View at: Google Scholar - M. Arif, K. Ahmad, J. L. Liu, and J. Sokół, “A new class of analytic functions associated with Sălăgean operator,”
*Journal of Function Spaces*, vol. 2019, Article ID 6157394, 8 pages, 2019. View at: Publisher Site | Google Scholar - K. Sakaguchi, “On a certain univalent mapping,”
*Journal of the Mathematical Society of Japan*, vol. 11, no. 1, pp. 72–75, 1959. View at: Publisher Site | Google Scholar - R. N. Das and P. Singh, “On subclasses of schlicht mapping,”
*Indian Journal of Pure and Applied Mathematics*, vol. 8, pp. 864–872, 1977. View at: Google Scholar - R. W. Ibrahim and M. Darus, “Subordination and superordination for univalent solutions for fractional differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 345, no. 2, pp. 871–879, 2008. View at: Publisher Site | Google Scholar - R. W. Ibrahim, “On holomorphic solutions for nonlinear singular fractional differential equations,”
*Computers & Mathematics with Applications*, vol. 62, no. 3, pp. 1084–1090, 2011. View at: Publisher Site | Google Scholar - R. W. Ibrahim, “Existence and uniqueness of holomorphic solutions for fractional Cauchy problem,”
*Journal of Mathematical Analysis and Applications*, vol. 380, no. 1, pp. 232–240, 2011. View at: Publisher Site | Google Scholar - R. W. Ibrahim, “Fractional complex transforms for fractional differential equations,”
*Advances in Difference Equations*, vol. 2012, no. 1, 2012. View at: Publisher Site | Google Scholar

#### Copyright

Copyright © 2020 Rabha W. Ibrahim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.