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# The study of coefficient estimates and Fekete–Szegö inequalities for the new classes of m-fold symmetric bi-univalent functions defined using an operator

*Journal of Inequalities and Applications*
**volume 2023**, Article number: 15 (2023)

## Abstract

The objective of this paper is to introduce new classes of m-fold symmetric bi-univalent functions. We discuss estimates on the Taylor–Maclaurin coefficients \(|a_{m+1}|\) and \(|a_{2m+1}|\), and the Fekete–Szegő problem is also considered for the new classes of functions introduced. We denote these classes by \(MF-S_{\Sigma ,m}^{p,q}(h)\), \(MF-S_{\Sigma , m}^{p,q}(s)\), and \(MF-S_{\Sigma , m}^{b,d}\). Quantum calculus aspects are also considered in this study to enhance its novelty and to obtain more interesting results.

## 1 Introduction and preliminary results

Let \(\mathcal{A}\) denote the family of functions of the form

which are analytic in the open unit disk \(U=\{z\in \mathbb{C}:|z|<1\}\) and normalized by the conditions \(f(0)=0\), \(f'(0)=1\).

The subclass \(S\subset \mathcal{A}\) is formed of all functions in the class \(\mathcal {A}\) that are univalent in *U* (see[14]).

The Koebe one-quarter theorem ensures that the image of the unit disk under every \(f\in S\) function contains a disk of radius \(1/4\), see [14].

If the function \(f\in S\), then it has an inverse \(f^{-1}\), which is defined by

and

where

We say that a function \(f\in \mathcal {A}\) is bi-univalent in *U* if both *f* and \(f^{-1}\) are univalent in *U*.

We denote by Σ the class of all bi-univalent functions in *U* given by (1).

The study on bi-univalent functions has its origins in the article published by Lewin in [25], where it was shown that \(|a_{2}|<1.51\).

The domain *D* is *m*-fold symmetric if a rotation of *D* about the origin through an angle \(2\pi /m\) carries *D* on itself.

The holomorphic function *f* in the domain *D* is *m*-fold symmetric if the following condition is true: \(f(e^{\frac{2\pi i}{m}}z)=e^{\frac{2\pi i}{m}}f(z)\).

### Definition 1

([36])

A function *f* is said to be *m*-fold symmetric if it has the following normalized form:

The normalized form of *f* is given as in (3), and the series expansion for \(f^{-1}(z)\) is given below (see[4]):

Examples of *m*-fold symmetric bi-univalent functions are:

Srivastava et al. in the paper [36] defined m-fold symmetric bi-univalent functions following the concept of m-fold symmetric univalent functions.

The interest in bi-univalent functions resurfaced in 2010 when a paper authored by Srivastava et al. in [35] was published. It opened the door for many interesting developments on the topic. Soon other new subclasses of bi-univalent functions were introduced [19–21] and special classes of bi-univalent functions were investigated such as Ma–Minda starlike and convex functions [3], analytic bi-Bazilevic functions [23], and recently a family of bi-univalent functions associated with Bazilevic functions and the *λ*-pseudo-starlike functions [38]. Brannan and Clunie’s conjecture [8] was further investigated [32] and subordination properties were also obtained for certain subclasses of bi-univalent functions [11]. New results continued to emerge in the recent years such as coefficient estimates for some general subclasses of analytic and bi-univalent functions [13, 27, 34]. Horadam polynomials were used for applications on Bazilevic bi-univalent functions satisfying subordination conditions [40] and for introducing certain classes of bi-univalent functions [1]. Operators were also included in the study as it can be seen in earlier publications [9] and in very recent ones [28]. Interesting results regarding m-fold symmetric bi-univalent functions were published in the same year when this notion was introduced [21]. This continued to appear in the following years [4, 16, 31, 33] and is still researched today [10, 37], proving that the topic remains in development.

The Fekete–Szegö problem is the problem of maximizing the absolute value of the functional \(|a_{3}-\mu a_{2}^{2}|\).

The Fekete–Szegö inequalities introduced in 1933, see [18], preoccupied researchers regarding different classes of univalent functions [15, 24]. Hence it is obvious that such inequalities were obtained regarding bi-univalent functions too and very recently published papers can be cited to support the assertion that the topic still provides interesting results [2, 6, 41]. Inspiring new results emerged when quantum calculus was involved in the studies, as can be seen in many papers [30] and in studies published very recently [5, 12, 17, 39]. Some elements of the \((p,q)\)-calculus must be used for obtaining the original results contained in this paper. Further information can be found in [22, 30]. The tremendous impact quantum calculus has had when associated with univalent functions theory is nicely highlighted in the recent review paper [22, 30].

For obtaining the original results contained in this paper, some elements of the \((p,q)\)-calculus must be used.

### Definition 2

([22], p. 2)

Let \(f\in \mathcal{A}\) given by (1) and \(0< q< p\leq 1\). Then the \((p,q)\)-derivative operator for the function *f* of the form (1) is defined by

and

it follows that the function *f* is differentiable at 0.

We can deduce from relation (2) that

where the \((p,q)\)-bracket number is given by

which is a natural generalization of the *q*-number.

We can see that \(\lim_{p\rightarrow 1^{-}}[k]_{p,q}=[k]_{q}=\frac{1-q^{k}}{1-q}\), see the papers [17, 22].

### Definition 3

([7], p.137)

Let the function \(f\in \mathcal{A}\), where \(0\leq d<1\), \(s\geq 1\) is real. The function \(f\in L_{s}(d)\) of an s-pseudo-starlike function of order *d* in the unit disk *U* if and only if

### Lemma 4

[14, 29] *Let the function* \(w\in \mathcal {P}\) *be given by the following series* \(w(z)=1+w_{1}z+w_{2} z^{2}+\cdots\) , \(z\in U\), *where we denote by* \(\mathcal{P}\) *the class of Carathéodory functions analytic in the open disk* *U*,

*The sharp estimate given by* \(|w_{n}|\leq 2\), \(n\in \mathbb{N}^{*}\) *holds true*.

In the next section of the paper, the original results obtained are presented in three definitions of new subclasses of m-fold symmetric bi-univalent functions and theorems concerning coefficient estimates and Fekete–Szegő problem for the newly defined classes.

## 2 Main results

### Definition 5

The function class \(M-FS_{\Sigma ,m}^{p,q}(h), (m\in \mathbb{N}, 0< q< p\leq 1, 0< h\leq 1, (z,w) \in U)\), contains all the functions *f* given by relation (3) that satisfy the following conditions:

and

where *g* is given by relation (4).

The coefficient bounds for the functions class \(MF-S_{\Sigma ,m}^{p,q}(h)\) are obtained in the next theorem.

### Theorem 6

*If the function* *f*, *given by relation* (3), *is in the function class* \(MF-S_{\Sigma ,m}^{p,q}(h), (m\in \mathbb{N}, 0< q< p\leq 1, 0< h\leq 1, (z,w) \in U)\), *then the following inequalities are true*:

*and*

### Proof

If we use (8) and (9), we obtain

and

where \(\alpha (z)\) and \(\beta (w)\) in \(\mathcal {P}\) are given by

and

Comparing the coefficients in (12) and (13), we obtain

and

Now, from (17), (19), and (21) we obtain that

Therefore, we obtain that

Now, for the coefficients \(\alpha _{2m}\) and \(\beta _{2m}\), if we apply Lemma 4, we obtain

If we use (17) and (19), then we obtain

From (20), (21), and (22), we obtain

If we apply Lemma 4 for the coefficients \(\alpha _{m}\), \(\alpha _{2m}\), \(\beta _{m}\), \(\beta _{2m}\), we obtain

□

The Fekete–Szegö functional for the class \(MF-S_{\Sigma , m}^{p,q}(h)\) is given in the next theorem.

### Theorem 7

*Let* *f* *be a function of the form* (3) *in the class* \(MF-S_{\Sigma , m}^{p,q}(h)\). *Then*

*where we denote*

### Proof

The values of the coefficients \(a_{m+1}^{2}\) and \(a_{2m+1}\) are given in the proof of Theorem 6 as follows:

We start to compute \(a_{2m+1}-\rho a_{m+1}^{2}\).

It follows that

After some computations and according to Lemma 4, we obtain

□

### Definition 8

The function class \(MF-S_{\Sigma , m}^{p,q}(s), (0< q< p\leq 1, 0\leq s<1, m\in \mathbb{N}, (z, w)\in U)\), contains all the functions *f* given by relation (3) that satisfy the following conditions:

where the function *g* is of the form (4).

Coefficient bounds for the functions class \(MF-S_{\Sigma , m}^{p,q}(s)\) are obtained in the next theorem.

### Theorem 9

*Let* *f* *be a function in the class* \(MF-S_{\Sigma , m}^{p,q}(s), (m\in \mathbb{N}, 0< q< p\leq 1, 0\leq s<1, (z, w)\in U)\), *which has the form* (3). *Then*

*and*

### Proof

We can see that from (24) and (25) we obtain

and

where \(\alpha (z)\) and \(\beta (w)\) in \(\mathcal {P}\) are given by (14) and (15).

Now we compare the coefficients from (28) and (29), and we obtain

We obtain from (30) and (32) that

and

If we apply Lemma 4 for the coefficients \(\alpha _{m}\), \(\alpha _{2m}\), \(\beta _{m}\), \(\beta _{2m}\), then we obtain

Using (33) and (31) to find the bound on \(|a_{2m+1}|\), we obtain

or equivalently

From (35) we substitute the value of \(a_{m+1}^{2}\) and obtain

Now, we will apply Lemma 4 for the coefficients \(\alpha _{m}\), \(\alpha _{2m}\), \(\beta _{m}\), \(\beta _{2m}\), and we obtain

From (36) and (38), if we apply again Lemma 4, then we obtain

□

In the next theorem we compute the Fekete–Szegö functional for the class \(MF-S_{\Sigma , m}^{p,q}(s)\).

### Theorem 10

*Let* *f* *be a function of the form* (3) *in the class* \(MF-S_{\Sigma , m}^{p,q}(s)\). *Then*

*where* \(l(\rho )\) *is given by*

### Proof

Using the values of \(a_{m+1}^{2}\) and \(a_{2m+1}\) from the proof of Theorem 9, we can compute \(a_{2m+1}-\rho a_{m+1}^{2}\).

We obtain

The next inequality is obtained after some computations and according to Lemma 4:

□

### Definition 11

Let \(b, d:U\rightarrow \mathbb{C}\) be analytic functions with the property \(\min \{\operatorname{Re}(b(z)), \operatorname{Re}(d(z))\}>0\), where \(z\in U\), \(b(0)=d(0)=1\).

The class \(MF-S_{\Sigma , m}^{b,d}\) contains all the functions *f* given by (3) if the following conditions are satisfied:

and

where the function *g* is given by (4).

In the next theorem we obtain the coefficient bounds for the function class \(MF-S_{\Sigma , m}^{b,d}\).

### Theorem 12

*If the function* *f* *of the form* (3) *is in the class* \(MF-S_{\Sigma , m}^{b,d}\), *then the following inequalities are satisfied*:

### Proof

We can write relations (42) and (43) as follows:

and

where the functions *b* and *d* have the following forms and satisfy the conditions from Definition 11:

Substituting relations (49) and (50) into (47) and (48), respectively, and equating the coefficients, we obtain

We obtain from (51) and (53) that

and

Adding relations (52) and (54), we obtain

and

We find from (58) and (59) that

and

We get in this way the desired estimate on the coefficient \(|a_{m+1}|\) as asserted in (44).

By subtracting (54) from (52), we obtain

It follows that

using the value of \(a_{m+1}^{2}\) from (58) into (60).

Hence,

Using in (60) \(a_{m+1}^{2}\) given by (59), we have

It follows that

□

## 3 Conclusion

These classes of functions introduced in this paper can be extended and similar properties to those presented can be studied. Using the same research method as in the paper [36], we introduce in Definitions 5, 8, and 11 three new classes of m-fold symmetric bi-univalent functions. As future research, using other operators or the \((p,q)\)-derivative operator, properties of starlikeness, convexity, and close-to-convexity of the new classes of functions could be investigated, and we can study the properties of symmetry of \((p,q)\)-derivative operator. We believe that this study will motivate a number of researchers to extend this idea for other functions and classes of functions.

## Availability of data and materials

Data sharing not applicable to this paper as no data sets were generated or analyzed during the current study.

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Breaz, D., Cotîrlă, LI. The study of coefficient estimates and Fekete–Szegö inequalities for the new classes of m-fold symmetric bi-univalent functions defined using an operator.
*J Inequal Appl* **2023**, 15 (2023). https://doi.org/10.1186/s13660-023-02920-6

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DOI: https://doi.org/10.1186/s13660-023-02920-6