# New Subclasses of Bi-Univalent Functions with Respect to the Symmetric Points Defined by Bernoulli Polynomials

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

**Definition 1.**

**Definition 2.**

**Lemma 1**

**.**Suppose that $c\left(z\right)={\sum}_{n=1}^{\infty}{c}_{n}$z${}^{n},$$\left(\right)open="|"\; close="|">c\left(z\right)$$z\in U,$ is an analytic function in $U.$ Then,

## 2. Coefficients Estimates for the Class ${\mathit{S}}_{\mathit{s}}^{\mathsf{\Sigma}}\left(\mathit{x}\right)$

**Theorem 1.**

**Proof.**

## 3. The Fekete–Szegö Problem for the Function Class ${\mathit{S}}_{\mathit{s}}^{\mathsf{\Sigma}}\left(\mathit{x}\right)$

**Theorem 2.**

**Proof.**

## 4. Coefficients Estimates for the Class ${\mathit{C}}_{\mathit{s}}^{\mathsf{\Sigma}}\left(\mathit{x}\right)$

**Theorem 3.**

**Proof.**

## 5. The Fekete–Szegö Problem for the Function Class ${\mathit{C}}_{\mathit{s}}^{\mathsf{\Sigma}}\left(\mathit{x}\right)$

**Theorem 4.**

**Proof.**

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Buyankara, M.; Çağlar, M.; Cotîrlă, L.-I.
New Subclasses of Bi-Univalent Functions with Respect to the Symmetric Points Defined by Bernoulli Polynomials. *Axioms* **2022**, *11*, 652.
https://doi.org/10.3390/axioms11110652

**AMA Style**

Buyankara M, Çağlar M, Cotîrlă L-I.
New Subclasses of Bi-Univalent Functions with Respect to the Symmetric Points Defined by Bernoulli Polynomials. *Axioms*. 2022; 11(11):652.
https://doi.org/10.3390/axioms11110652

**Chicago/Turabian Style**

Buyankara, Mucahit, Murat Çağlar, and Luminiţa-Ioana Cotîrlă.
2022. "New Subclasses of Bi-Univalent Functions with Respect to the Symmetric Points Defined by Bernoulli Polynomials" *Axioms* 11, no. 11: 652.
https://doi.org/10.3390/axioms11110652