Coefficient Estimates for Some Classes of Biunivalent Function Associated with Jackson <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>Difference Operator

Ali, Ekram E.; Lashin, Abdel Moneim; Albalahi, Abeer M.

doi:https://doi.org/10.1155/2022/2365918

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Volume 2022 | Article ID 2365918 | https://doi.org/10.1155/2022/2365918

Coefficient Estimates for Some Classes of Biunivalent Function Associated with Jackson -Difference Operator

Ekram E. Ali,^1,2Abdel Moneim Lashin ,^3,4and Abeer M. Albalahi¹

Academic Editor: Baowei Feng

Received24 Jun 2022

Revised09 Aug 2022

Accepted13 Aug 2022

Published09 Sept 2022

Abstract

Using the Jackson -difference operator, we present two new subclasses of biunivalent functions. Furthermore, we estimate the initial Taylor-Maclaurin coefficients and of functions belonging to these new subclasses. Our results generalize some of the previously related works of several authors.

1. Introduction

Let denote the normalized analytical function family of the formula in the open unit disc Further, by we shall denote the class of all functions in which are univalent in . If and are analytic functions in , we say that is subordinate to , written if there exists a Schwarz function and for all , such that Furthermore, if the function is univalent in , then we have the following equivalence (cf., e.g., [1, 2])

-calculus plays an important role in the theory of hypergeometric series and quantum theory, number theory, statistical mechanics, etc. In the early 1900s, studies on -difference equations were intensified by Jackson [3, 4], Carmichael [5], Mason [6], and Trjitzinsky [7]. It was Ismail et al. [8] who introduced geometric function theory and -theory together for the first time. Following the same idea, the -difference operator has been extensively investigated in the field of geometric function theory by many authors; for some recent works related to this operator on the classes of analytic functions, we refer to [9–11]. For any nonnegative integer the -number (or basic number) is defined by

For nonnegative integer , the -factorial is defined by

We note that when reduces to classical definition of factorial. Throughout in this paper, we will assume to be a fixed number between and

For , the -derivative operator or -difference operator is defined as and From (5), we deduce that

Recently, Govindaraj and Sivasubramanian [12] defined Salagean -derivative operator as follows:

A simple calculation implies where

Making use of (8) and (9), the power series of for of the form (1) is given by and , which is the familiar Salagean derivative [13]. Shams et al. [14] introduced and investigated the class of parabolic starlike functions and the class of parabolic convex functions of order as

Since if and only if (see [15]), then the conditions (11) and (12) can be written as

Let be analytic function with positive real part and normalized by the conditions and maps onto a region starlike with respect to 1 and symmetric with respect to the real axis

Definition 1. A function is said to be in the class if it satisfies

Remark 2. Taking , , and in the class , we get the well-known class of -convex functions which was studied by [16].

Definition 3. A function is said to be in the class if it satisfies

In Definition 3, if we set , we obtain a new class given below.

Example 1. A function is said to be in the class if it satisfies

Remark 4. Taking and in the class , we get the class of -starlike functions of order which was introduced by Seoudy and Aouf [17].

The well-known Koebe one-quarter theorem [18] ensures the range of every function of the class contains the disc . Thus, every univalent function has an inverse which is defined by where

If a function and its inverse are both univalent in , then a member of is called biunivalent in . We symbolize by the family of biunivalent functions in given by (1). Lewin [19] examined the family and proved that for elements of the family . Later, Brannan et al. [20] claimed that for . Subsequently, Tan [21] obtained some initial coefficient estimates of functions belonging to the class . Brannan and Taha in [22] proposed biconvex and bistarlike functions, which are similar to well-known subfamilies of . The research trend in the last decade was the study of subfamilies of . Generally, interest was shown to obtain the initial coefficient bounds for certain subfamilies of . In 2010, Srivastava et al. [23] introduced two interesting subfamilies of the function family and found bounds for and of functions belonging to these subfamilies. Subsequently, other writers explored related problems in this direction (see [9, 10, 24–30]).

Definition 5. A function given by (1) is said to be in the class if both and its inverse map are in

Remark 6. Note the following: (1)If and then the class is equivalent to the class introduced by [10](2)If then the class is equivalent to the class the class obtained by Attiya et al. [32](3) (see Darwish et al. [33])(4) (see Hamidi and Jahangiri [34])(5) (see Goyal and Kumar [35] and also Zireh et al. [36])

Definition 7. A function given by (1) is said to be in the class if both and its inverse map are in .

In Definition 7, if we set and , we obtain a new class given below.

Example 2. A function given by (1) is said to be in the class if

Remark 8. Note the following: (1)If , then the class is equivalent to the class introduced by Murugusundaramoorthy et al. [31](2)If , then the class is equivalent to the class introduced by Attiya et al. [32](3) (see Darwish et al. [33])(4) (see Hamidi and Jahangiri [34])(5) (see Goyal and Kumar [35] and also Zireh et al. [36])

In order to prove our main results, we will need the following result.

Lemma 9 (see [37]). If , then for each , where is the family of all functions analytic in for which for .

The aim of this paper is to find bounds on first two coefficients in the Taylor-Maclaurin expansion functional problem for functions belonging to the classes and . We also indicate interesting cases of the main results.

2. Coefficient Estimates

Unless otherwise mentioned, we shall assume throughout the remainder of this paper that , and

Theorem 10. Let the function given by (1) be in the class Then

Proof. Let ; then and its inverse map are in the class ; there exist two analytic functions with and , such that Define the functions and by or equivalently, We observe that , and in view of Lemma 9, we have that and , for . Further, using (26) and (27) together with (14), it is evident that Therefore, in view of (23), (24), (28), and (29), we have Since has the Taylor series expansion (1) and the series (19), we have Comparing the corresponding coefficients of (30) and (32) yields Similarly, from (31) and (33), we have From (34) and (36), it follows that Adding (35) and (37) yields From (39) and (40), we get Applying Lemma 9 for the coefficients and , we immediately have This gives the bound on as asserted in (21). Next, in order to find the bound on , by subtracting (37) from (35), we get Upon substituting the value of from (39), we obtain Applying Lemma 9 for the coefficients , and , we get which yield the estimate given by (22), and so the proof of Theorem 10 is completed.

If we set in Definition 5, of the biunivalent function class , we obtain a new class given by Definition 11.

Definition 11. For , a function is said to be in the class if it satisfies the following conditions: where

Using the parameter setting of Definition 11, in Theorem 10, we get the following corollary.

Corollary 12. For , let the function be in the class . Then Let in Definition 5, of the biunivalent function class , we obtain a new class given by Definition 13.

Definition 13. For , a function is said to be in the class if it satisfies the following conditions: where

Using the parameter setting of Definition 13 in Theorem 10, we get the following corollary.

Corollary 14. For , let the function be in the class . Then

Theorem 15. Let the function given by (1) be in the class Then

Proof. Let ; there are two Schwarz functions and defined by (26) and (27), respectively, such that Since Now, upon equating the coefficients in (30) and (55) and in (31) and (56), we get From (57) and (59), it follows that and after a few additional measurements using (58)–(60), we find Applying Lemma 9 is followed by the estimates in (52) and (53).

If we set in Definition 7, of the biunivalent function class , we obtain a new class given by Definition 16.

Definition 16. For , a function is said to be in the class if it satisfies the following conditions: where

Using the parameter setting of Definition 16, in Theorem 15, we get the following corollary.

Corollary 17. For , let the function be in the class . Then

If we set in Definition 7, of the biunivalent function class , we obtain a new class given by Definition 18.

Definition 18. For , a function is said to be in the class if it satisfies the following conditions: where

Using the parameter setting of Definition 18 in Theorem 15, we get the following corollary.

Corollary 19. For , let the function be in the class . Then

3. Conclusions

This study introduces two new subclasses of biunivalent functions associated with the Jackson -difference operator in the open unit disc. We have determined upper bounds for the Taylor-Maclaurin coefficients and of functions belonging to these new subclasses. Our results generalize some of the earlier work of several authors.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare no conflict of interest.

Authors’ Contributions

The authors contributed equally to the writing of this paper. All authors have read and agreed to the published version of the manuscript.

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Copyright © 2022 Ekram E. Ali 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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