Faber Polynomial Coefficient Bounds for -Fold Symmetric Analytic and Bi-univalent Functions Involving -Calculus

Abstract

In our present investigation, by applying -calculus operator theory, we define some new subclasses of -fold symmetric analytic and bi-univalent functions in the open unit disk and use the Faber polynomial expansion to find upper bounds of and initial coefficient bounds for and as well as Fekete-Szego inequalities for the functions belonging to newly defined subclasses. Also, we highlight some new and known corollaries of our main results.

1. Introduction, Definitions, and Motivation

Let denote the class of all analytic functions in the open unit disk and have the series expansion of the form

By , we mean the subclass of consisting of univalent functions. The inverse of univalent function can be defined as where

According to the Koebe one-quarter theorem [1], an analytic function is called bi-univalent in if both and are univalent in . Let denote the class all bi-univalent functions in . For , Lewin [2] showed that and Brannan and Cluni [3] proved that . Netanyahu [4] showed that Brannan and Taha [5] introduced a certain subclass of bi-univalent functions for class . In recent years, Srivastava et al. [6], Frasin and Aouf [7], Altinkaya and Yalcin [8, 9], and Hayami and Owa [10] studied the various subclasses of analytic and bi-univalent function. For a brief history, see [11].

In [12], Faber introduced Faber polynomials, and after that, Gong [13] studied Faber polynomials in geometric function theory. In their published works, some contributions have been made to finding the general coefficient bounds by applying Faber polynomial expansions. By using Faber polynomial expansions, very little work has been done for the coefficient bounds for of Maclaurin’s series. For more studies, see [14–17].

A domain is said to be -fold symmetric if

The univalent function maps the unit disk into a region with -fold symmetry and can be defined as

A function is said to be -fold symmetric [18] if it has the series expansion of the form

The class of all -fold symmetric univalent functions is denoted by , and for , then .

In [19], Srivastava et al. proved the inverse series expansion for , which is given as follows:

Here, we will denote -fold symmetric bi-univalent functions by . For , equation (7) coincides with equation (3) of the class The coefficient problem for is one of the favorite subjects of geometric function theory in these days (see [20–23]).

The quantum (or -) calculus has great importance because of its applications in several fields of mathematics, physics, and some related areas. The importance of -derivative operator is pretty recognizable by its applications in the study of numerous subclasses of analytic functions. Initially, in 1908, Jackson [24] introduced a -derivative operator and studied its applications. Further, in [25], Ismail et al. defined a class of -starlike functions; after that, Srivastava [26] studied -calculus in the context of univalent function theory; also, numerous mathematicians studied -calculus in the context of univalent function theory Further, the -analogue of the Ruscheweyh differential operator was defined by Kanas and Raducanu [27] and Arif et al. [28] discussed some of its applications for multivalent functions while Zhang et al. in [29] studied -starlike functions related with the generalized conic domain. Srivastava et al. published the articles (see [30, 31]) in which they studied the class of -starlike functions. For some more recent investigations about -calculus, we may refer to [32–34].

For a better understanding of the article, we recall some concept details and definitions of the -difference calculus. Throughout the article, we presume that

Definition 1. The -factorial is defined as and the -generalized Pochhammer symbol , , is defined as

Remark 2. For , then , and .

Definition 3. The -number for is defined as

Definition 4 (see [24]). The -derivative (or -difference) operator of a function is defined, in a given subset of , by provided that exists.

From Definition 4, we can observe that for a differentiable function in a given subset of . It is also known from (1) and (12) that

Here, in this paper, we use the -difference operator to define new subclasses of -fold symmetric analytic and bi-univalent functions and then apply the Faber polynomial expansion technique to determine the general coefficient bounds and initial coefficient bounds and as well as Fekete-Szego inequalities.

Definition 5. A function is said to be in the class if and only if where , , and, and is defined by (7).

Remark 6. For and , then the class reduces into the class introduced by Hamidi and Jahangiri in [35].

Definition 7. A function is said to be in the class if and only if where , , and, and is defined by (7).

Remark 8. For , , and , then the class reduces into the class , introduced by Hamidi and Jahangiri in [36].

2. Main Results

Using the Faber polynomial expansion of functions of the form (1), the coefficients of its inverse map may be expressed as [15] given by for an expansion of (see [37]). In particular, the first three terms of are

In general, for any and , an expansion of is as (see [15]) where , and by [37], while , and the sum is taken over all nonnegative integers satisfying

Evidently, (see [14]), or equivalently, while , and the sum is taken over all nonnegative integers satisfying

It is clear that , and the first and last polynomials are and

Similarly, using the Faber polynomial expansion of functions of the form (6), that is,

The coefficients of its inverse map may be expressed as

Theorem 9. For , let be given by (6), and if, , then

Proof. By definition, for the function of the form (6), we have and for its inverse map , we have where On the other hand, since and by definition, we have where Comparing the coefficients of (27) and (31), we have Similarly, comparing coefficients of (28) and (32), we have Note that for , , we have and so Now taking the absolute of (36) and (37) and using the fact that , , and , we have which completes the proof of Theorem 9.

For and , in Theorem 9, we obtain the following corollary.

Corollary 10. For , let , and if, , then

For , , and , in Theorem 9, we obtain the following known corollary.

Corollary 11 (see [35]). For , let , and if, , then

Theorem 12. For , let be given by (6), and then

Proof. Replacing by and in (33) and (34), respectively, we have From (42) and (44), we have Adding (43) and (45), we have Taking the absolute value (47), we have Now, the bounds given for can be justified since From (43), we have Next, we subtract (45) from (43), and we have or After some simple calculation and by taking the absolute, we have Using the assertion (46), we have From (50) and (54), we note that Now, we rewrite (45) as Taking the absolute value, we have Finally, from (51), we have Taking the absolute value, we have

For and , in Theorem 12, we obtain the following corollary.

Corollary 13. For , let be given by (1), and then

For , , and , in Theorem 12, we obtain the following corollary.

Corollary 14 (see [35]). For , let be given by (1), and then

Theorem 15. Let be given by (6), and if, , then

Proof. By definition, for the function of the form (6), we have where the first few coefficients of are In general, where For the inverse map , we obtain where On the other hand, since and by definition, we have where Comparing the coefficients of (63) and (70), we have Similarly, comparing the coefficients of (67) and (71), we have Note that for , , we have and so Taking the absolute values of (75) and (76) and using the fact that , , and , we have Hence, Theorem 15 is complete.

For , , and , in Theorem 15, we obtain the following corollary.

Corollary 16. , and if, , then

Theorem 17. Let be given by (6), and then

Proof. Replacing by and in (72) and (73), respectively, we have From (80) and (82), we have Adding (81) and (83), we have Taking the absolute value (85), we have Next, we subtract (83) from (81), and we have or After some simple calculation of (88) and by taking the absolute, we have Using the assertion (86), we have For the third part, we rewrite (83) as Taking the absolute value, we have Finally, from (87), we have Taking the absolute value, we have

For , , and , in Theorem 17, we get the following corollary.

Corollary 18. Let be given by (1), and then

3. Conclusion

In this paper, we have applied -calculus operator theory to define some new subclasses of -fold symmetric analytic and bi-univalent functions in open unit disk and used the Faber polynomial expansion to find upper bounds and initial coefficient bounds and as well as Fekete-Szego inequalities for the functions belonging to newly defined subclasses of -fold symmetric analytic and bi-univalent function. Also, we highlighted some new and known consequences of our main results.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All authors jointly worked on the results, and they read and approved the final manuscript.

Acknowledgments

This work was supported by the Key Scientific Research Project of Colleges and Universities in Henan Province (No. 19A110024) and the Natural Science Foundation of Henan Province (CN) (No. 212300410204).