Abstract

In this paper, using the basic concepts of symmetric -calculus operator theory, we define a symmetric -difference operator for -fold symmetric functions. By considering this operator, we define a new subclass of -fold symmetric bi-univalent functions in open unit disk . As in applications of Faber polynomial expansions for , we find general coefficient for , Fekete–Szegő problems, and initial coefficients and . Also, we construct -Bernardi integral operator for -fold symmetric functions, and with the help of this newly defined operator, we discuss some applications of our main results. For validity of our result, we have chosen to give some known special cases of our main results in the form of corollaries and remarks.

1. Introduction and Definitions

Let denote the set of all analytic functions in the open unit disk , and thus each analytic function can be written in terms of power series:

A function in open unit disk is considered to be normalized function if it fulfills the condition of normalization, that is,

A function is said to be univalent in the open unit disk at opints if and here represent the class of univalent function. We know that every has an inverse , which is given aswhere

An analytic function and its inverse are univalent in ; then, is called bi-univalent. An analytic function is called bi-Bazilevic in if and are Bazilevic in (see [1]). The behavior of these types of functions is unpredictable and not much is known about their coefficients. Let denote the class of analytic and bi-univalent functions in . For , in [2], Levin showed that , and after that, Branan and Clunie [3] investigated . Furthermore, in paper [4], Netanyahu showed that maximum of is .

In 1986, Branan and Taha [5] defined the subclass of the bi-univalent functions of class , and also, Srivastava et al. in [6] investigated various subclasses of class . After that, Frasin and Aouf [7], Xu et al. [8, 9], and Hayami and Owa [10] followed their work by introducing new subclasses of class . For more studies, we refer the readers to [11, 12]. The performance of the coefficients of the functions and is unpredictable due to the bi-univalency requirements; therefore, we cannot much investigate about general coefficient for .

In [13], Faber introduced Faber polynomials and used it to determine the general coefficient bounds for . Furthermore, Gong [14] explained importance of Faber polynomials in mathematical sciences, especially in geometric function theory. Recently, Airault [15] gave some remarks on Faber polynomials, and in [16], he discussed more about Faber polynomials on differential calculus.

By using the Faber polynomial expansion technique, Hamidi and Jahangiri [17, 18] investigated some new coefficient estimates for analytic bi-close-to-convex functions. In the literature, there are only a few works determining the general coefficient bounds for given by (1) by using Faber polynomial expansions. By using Faber polynomial expansion, we have known very little about the bounds of Maclaurin’s series coefficient for . Recently, many authors have conducted some studies about Faber polynomial expansion and determined general coefficient bounds for (for details, see [1922]).

A domain is recognized -fold symmetric ifwhere is a positive integer. The function , of the formis univalent and maps the unit disk into a region with -fold symmetry.

A function is called -fold symmetric if it has the following normalized form:

Let denote the set of all -fold symmetric univalent functions and note that (for details, see [14, 23]).

The univalent function of form (8) is said to be -fold symmetric bi-univalent function in if its inverse is univalent and the series expansion for is given as follows:whereand series for was proved by Srivastava et al. in [24] and denoted by . For , the series in (9) corresponds with (5) of the class . Srivastava et al. [24] defined a subclass of class and investigated initial coefficient bounds while Hamidi and Jahangiri [25] also defined -fold symmetric bi-starlike functions.

Many authors investigated several subclasses of by using the basic concepts of -calculus and fractional -calculus. Ismail et al. [26] were the first ones to introduce a -difference operator for the class of normalized starlike functions in . A number of researchers have got inspired by the -calculus because of its divers applications in mathematics and physics [24]. Historically, Srivastava [27] was the first who made used of -calculus in the context of geometric function theory. In 1909, Jackson [28, 29] defined the -analogue of derivative and integral operator and discussed some of its applications. Aral and Gupta [30, 31] introduced the -Baskakov–Durrmeyer operator while the author in [32] studied the -Picard and -Gauss–Weierstrass singular integral operators. Recently, Kanas and Raducanu [33] introduced the -analogue of Ruscheweyh differential operator while Aldweby and Darus [34] and Mahmood and Sokoł [35] discussed some applications of this differential operator. In [36], using the symmetric -derivative operator, a subclass of analytic and bi-univalent functions has been introduced and discussed. Some properties of -close-to-convex functions have been obtained in [37], while Hu et al. [38] considered some subclasses of starlike functions and have obtained some sufficiency criteria for their defined functions class. For some recent investigations, we refer the readers to [39, 40].

The theory of -symmetric calculus has been used in different fields of mathematics and physics, for example, Lavagno studied quantum mechanics in [41] while Da Cruz et al. [42] discussed -symmetric variation calculus. After that, several authors used basic concepts of -symmetric calculus in geometric function theory from different aspects and defined some new subclasses of analytic functions and investigated some new results. Kanas et al. [43] implemented some basics concepts of -symmetric calculus and defined -symmetric derivative. They discussed some applications of this operator on new subclasses of analytic functions. Furthermore, Khan et al. [44] defined symmetric conic domain by using the concepts of -symmetric calculus and used this domain to investigate some new subclasses of analytic functions. But in geometric function theory, using -symmetric calculus very little work has been done. Especially, very few articles have been published so far on this topic.

Here we recall few basic concepts and definitions of the -symmetric difference calculus. Throughout this paper, we suppose that and

The -symmetric number frequently occurs in the study of -deformed quantum mechanical simple harmonic oscillator (see [45]) and can be defined as follows.

Definition 1. For , the -symmetric number is defined by

Remark 1. We note that the -symmetric number does not reduce to the -number.

Definition 2. For any , the -symmetric number shift factorial is defined byNote that

Definition 3. (see [46]). The -symmetric derivative (-difference) operator for is defined byWe can observe thatHere we define -symmetric derivative (-difference) operator for -fold symmetric analytic functions.

Definition 4. Let of form (8); then, -symmetric derivative (-difference) operator for can be defined asNote that for and ,

Recently, Bulut [47] used a Faber polynomial technique on and investigated some useful results. Here, in this article, we define a new subclass of -fold symmetric analytic bi-univalent functions associated with -symmetric derivative (-difference) operator. We shall implement a Faber polynomial expansions technique to determine the estimates for the general coefficient bounds , as well as initial coefficients , and Fekete–Szegő problem for .

Definition 5. A function is said to be in the class , as if and only ifwhere , , , , and is defined by (9).

Remark 2. For , , introduced by Hamidi and Jahangiri in [18].

Here in this paper, the symmetric -difference operator for -fold symmetric functions is defined. Then, by using this newly defined operator, we defined a new subclass of -fold symmetric bi-univalent functions in open unit disk . As in applications of Faber polynomial expansions for , we find general coefficient for , Fekete–Szegő problems, initial and coefficients and . Also, -Bernardi integral operator for -fold symmetric functions is constructed, and with the help of this operator, we discuss some applications of our main results.

2. Main Results

Using the Faber polynomial expansion of functions of form (1), the coefficients of its inverse map may be expressed as [16]wheresuch that with is a homogeneous polynomial in the variables [48]. In particular, the first three terms of are

For more details, see [15, 16, 48].

Similarly, Bulut [47] used the Faber polynomial expansion on (8) and obtained the series of the form

The coefficients of its inverse map can be expressed as

Theorem 1. For , let be given by (8); if , , then

Proof. Let of form (8); then,and for its inverse map , we havewhere , .
On the other hand, since and by definitions and , we havewhereEquating the coefficients of (26) and (29), we haveSimilarly, from (27) and (30), we haveSince , , we haveTaking the absolute values of (34) and (35), we haveNow using the fact that , , and , we haveHence, Theorem 1 is complete.

For , , and , in Theorem 1, we obtain the following known corollary.

Corollary 1. (see [18]). For , let ; if , , then

Theorem 2. For , let be given by (8); then,

Proof. Taking in (31) and in (32), then we haveFrom (42) and (44) and using the fact that , , and , we haveAdding (43) and (45), we haveTaking modulus on (47), we haveNow the bounds can be justified sinceFrom (43), we haveNext we subtract (45) from (43), and we haveorAfter some simple calculation for (52) and taking the absolute values, we getUsing assertion (46) on (53), we haveIt follows from (50) and (54) thatAgain, we rewrite (45) for the result of (40) as follows:Taking the absolute value and using the fact that , , and , we haveFinally, from (51), we haveTaking the absolute value and using the fact that , , and , we have

Putting , , and in Theorem (29), we obtain the following known corollary.

Corollary 2. (see [18]). For , let be given by (1); then,

3. Applications of the Main Results

In this section, firstly we define the -Bernardi integral operator for -fold symmetric analytic functions and then use it to discuss some applications of our main results.

Let of form (8); then, is called the -Bernardi integral operator for functions defined by with , and is given bywhere

Remark 3. If we take and , in (61), then we obtain Bernardi integral operator introduced by Bernardi in [49].

Theorem 3. For , let be given by (8); if , , andwhere is the integral operator given by (61), then

Proof. The proof of Theorem 3 follows by using (62) and Theorem 1.

Theorem 4. For , let be given by (8); in addition, is defined by (61) and of form (62); then,

Proof. The proof of Theorem 4 follows by using (62) and Theorem 2.

4. Conclusion

The applications of operators in geometric function theory are quite significant. Many new subclasses of analytic functions have been defined with the help of operators. In our present investigations, we were motivated by the recent research on operator theory and have defined a new -symmetric derivative (difference) operator for -fold symmetric functions. With the help of this newly defined operator, we have systematically defined a new subclass of -fold symmetric bi-univalent functions. We have then successfully used the Faber polynomial expansion technique to find general coefficient for , Fekete–Szegő problems, and initial coefficients and for the function in the open unit disk . Also, we have defined a -symmetric Bernardi integral operator for -fold symmetric functions and have used it to discuss some applications of our main results. In future, researchers can define certain new subclasses related to -fold symmetric functions associated with (17).

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.