Inclusion and Neighborhood on a Multivalent <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.5268pt" id="M1" height="12.3952pt" version="1.1" viewBox="-0.0657574 -7.8684 8.58866 12.3952" width="8.58866pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-114" d="M474 429L457 433C435 440 389 448 367 448C348 448 323 446 309 443C266 434 195 406 148 366C78 307 23 210 23 101C23 35 55 -12 92 -12C118 -12 146 1 196 35C247 70 311 130 346 173H348L281 -148C268 -211 256 -221 208 -229L191 -232L187 -257L433 -245L437 -219L411 -216C357 -210 350 -205 362 -140L427 207C447 315 461 381 474 429ZM387 387C379 337 363 262 355 236C318 180 201 57 142 57C126 57 112 81 112 128C112 205 150 321 220 376C244 395 280 403 312 403C345 403 370 396 387 387Z"/></g></svg>-</span>Symmetric Function with Poisson Distribution Operators

Amini, Ebrahim; Al-Omari, Shrideh; Suthar, Dayalal

doi:https://doi.org/10.1155/2024/3697215

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Research Article | Open Access

Volume 2024 | Article ID 3697215 | https://doi.org/10.1155/2024/3697215

Inclusion and Neighborhood on a Multivalent -Symmetric Function with Poisson Distribution Operators

Ebrahim Amini,¹Shrideh Al-Omari,^2,3and Dayalal Suthar⁴

Academic Editor: Ding-Xuan Zhou

Received18 Nov 2023

Revised05 May 2024

Accepted06 May 2024

Published21 May 2024

Abstract

In this paper, by using Poisson distribution probability, some characteristics of analytic multivalent -symmetric starlike and -symmetric convex functions of order are examined. Then, by utilizing the Poisson distribution and the concept of the -analogue Salagean integral operator, the -valent convergence polynomial was introduced. Furthermore, a number of subclasses of analytic symmetric -valent functions linked to novel polynomials are also deduced. After that, specific coefficient constraints are determined and symmetric -neighborhoods for -valent functions are defined. In relation to symmetric -neighborhoods of -symmetric -valent functions formed by Poisson distributions, this paper presents new inclusion results. In addition, a detailed discussion of certain -symmetric inequalities of analytic functions with negative coefficients is also provided.

1. Introduction

Recently, the concept of -calculus has attracted many researchers due to its prominent use in the development of different classes of univalent functions. Although an extensive review of the -calculus theory was given by Jackson [1, 2], Srivastava [3] establishes a connection between the geometric nature of the univalent function and the -derivative operator. In [4], the authors studied a subclass of biunivalent functions by using -difference operators. Kanas et al. [5] described a symmetric operator by employing a -derivative on a conic region, while Arif et al. [6] studied a symmetric operator to popularize the multivalent analytic functions. Moreover, the authors in [7] investigated a subclass of -valent functions by using probability Borel distribution operators and established some properties of several normalized analytic functions. On the other hand, Khan et al. [8] discussed the -Ruscheweyh-type derivative operator and its application to multivalent functions. Alpay et al. in [9] introduced polyanalytic functions. They studied integral representations on the quaternionic unit ball. In addition, the authors in [10] investigated some subclasses of multivalent functions associated with the -calculus theory. However, various difference and -difference operators are investigated for some subclasses of univalent functions; see, e.g., [11–13] and references cited therein. For , the -analogue of the derivative of a function is introduced by

Then, the relation between the -derivative operator and the ordinary derivative operator is given by

The symmetric -calculus has been successfully applied in many areas of science associated with the geometric function theory and the quantum mechanics. Brito and Martins [14] studied the symmetric -derivative and derived better convergence properties than the classical -derivative. For fixed , the symmetric -derivative or the -symmetric derivative of a function in is defined bywhereas the symmetric -number of a natural number is defined by [15]

Let be the class of -valent analytic functions defined on the open unite disk such that the series form holds. Then, for , the new subclass of univalent functions was investigated by Graham and Kohr [16]. By simple computations, we obtain that(i),(ii),(iii).

We denote by the set of all -valent functions which are holomorphic or analytic on a subset in the complex plan with the series form

Let be given by (5) and . We use the following notation to denote the convolution of two functions and :

In recent decades, inequalities and inclusions of various types of analytic functions are studied in the geometric function theory. Liu and Xu in [17] considered the concept of -convex functions and discovered new Hermite–Hadamard type inequalities on intervals of Riemann integrable functions. The authors in [18] discussed the concept of -derivative and studied several inequalities of analytic functions by using the generalized Sǎlǎgean differential operator. On the other hand, Lashin et al. in [19] discovered some inclusion relations of subclasses of analytic functions associated with the Pascal distribution probability.A random variable with Poisson distribution takes the exponential probability that is given bywhere and is a parameter (see [20]).

The main properties of the Poisson distribution probability are in fact a limiting case of the binomial distribution. The Poisson distribution, like the binomial distribution, is a counted number of times of something happens. The observed difference is that there is no specified number of possible tries. Here is one way that it can arise. If an event happens independently and randomly over a time and the mean rate of occurrence is constant over the time, then the number of occurrences in a fixed amount of time lead to a Poisson distribution. The Poisson model is used extensively for modeling a count of data in a range of different scientific fields (see, e.g., [21]).

Al-Shaqsi [22] defined the convergence polynomial as follows:

We note that this polynomial has radius of convergence at infinity. Now, we introduce the following -valent functions.

We define a linear operator as follows:

Then, benefited from the definition of the -symmetric derivative of a function , we establish that

Let . By using the idea of Salagean in [23], we will define the -symmetric derivative of the linear Poisson distribution operator as follows:

By simple notation, we writewhere

We now state the following necessary definitions.

Definition 1. Let , , , and . Then, the class of -valent symmetric -starlike functions of order with Poisson distribution is defined byNote that if , then the calls is denoted by . If and , we have .

Example 1. In this example, we show that the set is nonempty.
We set . Since , we can find a function such thatBy using the assertion (13), we can writeor equivalentlyThis, indeed, implies thatTherefore, we can obtain the coefficient of function as follows:Hence, the set is nonempty.

Definition 2. Let , , , and . Then, the class of -valent symmetric -convex functions of order with Poisson distribution is defined byNote that if , then the calls is denoted by . If and , we have .

Example 2. In this example, we show that the set is nonempty.
We set . Since , we can find a function such thatBy using the assertion (13), we can writeor equivalentlyThis, indeed, implies thatTherefore, we can obtain the coefficient of function as follows.Hence, the set is nonempty.

For , the subclasses , resp., , of symmetric -starlike functions of order and the class of symmetric -convex functions of order were thoroughly discussed in [24].

In [25], the authors defined the neighborhood of function . Then, Altinta et al. in [26] introduced the -neighborhood for a starlike function with negative coefficient. Moreover, the authors of [27–29] investigated some neighborhood properties for a subclass of complex analytic functions. Thus, similar to the previous result, we can define the symmetric -neighborhood of analytic function as follows.

Definition 3. Let andThe symmetric -neighborhood of a function , is defined asThen, it follows from (28) that if , then

Our object, in this present paper, is to investigate various properties and characteristics of symmetric analytic -valent functions belonging to the class of all functions such that the following inequality holds.

Certainly, we, in terms of the simpler classes and , respectively, have

Also, we denote by the set of all functions satisfying the inequality

Then, we establish several inclusion relationships involving symmetric -neighborhoods of analytic -valent functions belonging to such subclasses in details.

2. A Set of Coefficient Inequalities

In this section, we aim to obtain coefficient inequalities for functions in the subclasses and .

Theorem 4. Let ,,,, and be given by (13). Then, if and only ifwhere and is given by (14). The result is also sharp.

Proof. Suppose that . Then, we haveBy choosing to be real and , we obtainOr, equivalently, we getHence, we reach to assertion (33) of Theorem 4. Conversely, suppose that inequality (33) holds true and . Then, from definition (5), we find thatwhere . Thus, inequality (33) satisfies. Hence, by the maximum modulus theorem, we conclude .
Finally, we note that inequality (33) of Theorem 4 is sharp. Therefore, the extremal function is given byBy using the abovementioned theorem, we can easily obtain the following result.

Corollary 5. Let ,,,, and be given by (13). Then, if and only ifwhere is given by (14). The result is sharp for the function

Corollary 6. Let , ,, and be given by 1.3. Then, if and only if

The result is sharp for the function.

Corollary 7. Let , , ,, and be given by (13). Then, if and only if

The result is sharp for the function.

Corollary 8. Let ,,, and be given by (13). Then, if and only if

The result is sharp for the function.

Theorem 9. Let ,,, and be given by (13). Then, if and only if

The result is sharp for a function given by

Proof. Suppose that . Then, we haveBy choosing to be real and , we obtainHence, we reach to assertion (47). Conversely, suppose that (47) holds true and . Then, from definition (5), we find thatwhere , , . Thus, inequality (47) satisfies. Hence, by the maximum modulus theorem, we conclude .

Theorem 10. Let ,,,, and expressed in (13) be in the class . Then,andwhere is defined by

Proof. From inequality (33) of Theorem 4, we can easily see thatwhich immediately yields the first inequality (52) of Theorem 10.
By using inequality (33), we also haveNext, we obtainThis yieldsHence, we obtain the desired inequality (53).
By using the abovementioned theorem, we can easily obtain the following result.

Corollary 11. Let , , , , and given by (13) be in the class . Then, we have

Corollary 12. Let ,,, , and given by (13) be in the class . Then, we have

Similarly, we can obtain a new theorem as follows.

Theorem 13. Let , , , , and given by 1.3 be in the class . Then, we have

Theorem 14. Let ,, , , and be a function given by (13). If , then we havewhere is given by (54).

Proof. Since , we have the following inequality:Moreover, since the function is given by (13), we obtainConsequently, we haveHence, from the assertion (63), we derive the desired inequality (62).
By using the abovementioned theorem, we can easily obtain the following result.

Corollary 15. Let , , ,, and be a function given by (13). If , then we have

Corollary 16. Let ,,, , and be a function given by (13). If , then we have

Alike to the proof applied to Theorem 14, we can establish the following result.

Theorem 17. Let ,,,, and be a function given by (13). If , then we havewhere is given by (54).

By using the abovementioned theorem, we can easily obtain the following result.

Corollary 18. Let ,,,, and be a function given by (13). If , then we have

Corollary 19. Let , , ,, and be a function given by (13). If , then we have

3. Inclusion Results Involving the Symmetric -Neighborhoods

In this section, we determine the neighborhood properties for each of the following function classes:

Theorem 20. Let ,,, and . If is given by (13) belongs to the class , then we havewhere andwhere is given by (54).

Proof. Assertion (72) would follow easily from the definition of , which is given by (29) when is replaced by and the second assertion (52) of Theorem 10.

Theorem 21. Let ,,, and . If given by (13) belongs to the class , then we havewhere and

Putting in Theorem 20 leads to the following corollaries.

Corollary 22. Let ,, and . If given by (13) belongs to the class , then we havewhere and

Putting in Theorem 20 leads to the following corollary.

Corollary 23. Let ,, and . If given by (13) belongs to the class , thenwhere and

We now state the following necessary definitions.

Definition 24. A function is said to be in the class if there exists such thatwhere is given by (27).

Theorem 25. If andthen we havewhere is given by (54) and is given by (27).

Proof. Since , then by using assertion (27), we deriveSince is a nondecreasing sequence, we infer thatThis implies thatBy applying inequality (83), we getHence, under the assumption that the function belongs to the class , given by (27), we, by using the above inequality, haveIn view of the (5) and (27), we writeNow, by settingwe, in view of Definition 24 and using the inequality (88), obtain that .
This completes the proof of Theorem 25.

4. Concluding Remarks

In this paper, a new generalization of the linear operator which is a continuous bridge between the Poisson distribution probability and the geometric function theory has been considered. The new discussed linear operator is defined by the generalized power series, -Sălăgean integral operator, and properties of the symmetric -derivative. The suggestion of this operator was applied in extending the geometric function theory. Then, several subclasses of -valent functions with negative coefficients were introduced by using symmetric -derivatives. Furthermore, symmetric -neighborhoods of -valent functions are defined. Moreover, coefficient bounds for such given functions are derived. In addition, several inclusion relations for each function in the new subclasses are also obtained. Therefore, the results obtained in this research could be further employed for discussing fractional symmetric -derivatives to generalize certain results involving univalent functions.

Data Availability

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

Conflicts of Interest

The authors declare they have no conflicts of interest.

Authors’ Contributions

The authors contributed equally and significantly in writing this paper. All the authors have read and approved the final manuscript.

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Copyright © 2024 Ebrahim Amini 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.

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