Initial Coefficient Behavior of Bi-Univalent Functions Defined Through Bernoulli Polynomial Subordination

Abstract

The study of coefficient problems for bi-univalent functions continues to play a central role in geometric function theory due to its analytical depth and wide range of applications. In this paper, we introduce a new subclass of bi-univalent functions defined through subordination to the generating function of Bernoulli polynomials. We derive explicit upper bounds for the initial Taylor–Maclaurin coefficients and establish a corresponding Fekete–Szegö-type inequality for functions in this class. The results obtained provide refined estimates that extend several known findings in the literature and reveal the effectiveness of Bernoulli polynomial subordination as a unifying framework for investigating coefficient problems in the theory of bi-univalent functions. Various special cases are also discussed to demonstrate the scope and applicability of the main results.

1. Introduction

The investigation of analytic and univalent functions has long been a central theme in geometric function theory, with particular emphasis placed on understanding the behavior of their Taylor–Maclaurin coefficients [1,2,3]. Among the many subclasses studied, bi-univalent functions have attracted significant attention due to the additional geometric constraints imposed by requiring both the function and its inverse to be univalent in the unit disk [4,5,6,7,8]. Despite substantial progress in this area, determining sharp bounds for the initial coefficients of bi-univalent functions remains a challenging and active line of research [9].

Over the past decades, various approaches have been developed to analyze coefficient problems for subclasses of bi-univalent functions. These methods often involve imposing geometric or analytic conditions through subordination, integral representations, or special functional relationships [10,11,12]. Such frameworks not only facilitate the derivation of coefficient estimates, but also provide deeper insights into the structural properties of the functions under consideration.

In recent years, special polynomials have emerged as an effective tool for constructing and studying subclasses of analytic and bi-univalent functions. Polynomials such as Chebyshev, Fibonacci, Gegenbauer, and Bernoulli have been used to generate subclasses with rich geometric interpretations and tractable analytical properties [13,14,15,16]. Among these, Bernoulli polynomials occupy a distinguished place due to their well-known generating function and their broad applications in number theory, approximation theory, and mathematical analysis [17].

The use of Bernoulli polynomials is mathematically well-motivated by their fundamental role in summation theory, particularly in the Euler–Maclaurin summation formula. This formula provides a powerful bridge between discrete sums and continuous integrals, with Bernoulli numbers and polynomials appearing as essential coefficients in the associated expansions. The work of Leinartas and Shishkina [18,19] further supports this perspective, demonstrating that Bernoulli polynomials arise naturally in the evaluation of sums over lattice points in simplices. This highlights their intrinsic importance in addressing a wide range of combinatorial and analytic problems.

Several recent contributions have significantly advanced the study of subclasses of bi-univalent functions through the use of special functions and structured subordination conditions [20,21,22,23,24]. These works have demonstrated that polynomial-based subordinations provide a productive avenue for analyzing geometric properties of analytic and bi-univalent functions. The present study continues in this direction by introducing a Bernoulli polynomial-based subclass, which differs from earlier formulations and leads to new coefficient bounds and functional inequalities.

Motivated by these developments, the present paper introduces a new subclass of bi-univalent functions defined through subordination to the generating function of Bernoulli polynomials. This approach provides a flexible framework that allows the derivation of explicit coefficient bounds while preserving a clear connection to classical analytic function theory. In particular, we obtain upper estimates for the initial Taylor–Maclaurin coefficients and establish a Fekete–Szegö-type inequality for functions belonging to this subclass, thereby extending several earlier results in the literature [25,26,27,28].

The structure of the paper is organized as follows. In the next section, we present the necessary definitions and preliminary concepts required for our analysis. We then introduce the new subclass and derive coefficient bounds for the initial terms of the corresponding series expansion. Subsequently, we establish a Fekete–Szegö inequality and discuss several special cases that highlight the generality of our results. Finally, we conclude with remarks on possible directions for future research.

2. Preliminaries

Let $\mathbb{U}=\{\xi \in \mathbb{C}:|\xi |<1\}$ denote the open unit disk in the complex plane, and let $\mathcal{A}$ represent the class of functions that are analytic in $\mathbb{U}$ and satisfy the normalization conditions

$$f\left(0\right)=0\mathrm{and}{f}^{\prime }\left(0\right)=1.$$

Each function $f\in \mathcal{A}$ can be expressed through a Taylor–Maclaurin series expansion of the form

$$f\left(\xi \right)=\xi +{\displaystyle \sum _{n=2}^{\infty }}{a}_n{\xi }^n.$$

(1)

A function $f\in \mathcal{A}$ is said to be bi-univalent in $\mathbb{U}$ if both f and its inverse function $f^{-1}$ are univalent in $\mathbb{U}$. We denote by $\Sigma$ the class of all such functions. If $f\in \Sigma$ is represented as above, then the inverse function $g=f^{-1}$ admits the expansion

$$g\left(w\right)=w-a_2{w}^2+(2a_2^2-a_3)w^3+\dots ,$$

(2)

where the coefficients follow from the classical inverse function relations [29,30].

A function $\omega$ analytic in $\mathbb{U}$ with $\omega \left(0\right)=0$ and $\left|\omega \right(\xi \left)\right|<1$ is commonly referred to as a Schwarz function. If

$$\omega \left(\xi \right)={\displaystyle \sum _{n=1}^{\infty }}{c}_n{\xi }^n,$$

then it is known that, as in [31], the best possible estimates of the coefficients are

$$|c_n|\le 1\mathrm{for}\mathrm{all}n\in \mathbb{N}.$$

The Bernoulli polynomials $\left\{B_n\left(x\right)\right\}$ are defined through the generating function

$${\displaystyle \frac{\xi e^{x\xi }}{e^{\xi }-1}}=1+B_1\left(x\right)\xi +{\displaystyle \frac{B_2\left(x\right)}{2!}}{\xi }^2+\dots ,\left|\xi \right|<2\pi ,$$

where the first two Bernoulli polynomials are defined in [32] as

$$B_1\left(x\right)=x-{\displaystyle \frac{1}{2}}\mathrm{and}{B}_2\left(x\right)=x^2-x+{\displaystyle \frac{1}{6}}.$$

These polynomials provide the foundation for defining the subclass of bi-univalent functions studied in the present work.

In [33], Yousef et al. introduced the following subclass, which serves as the foundation for defining our new subclass.

Definition 1.

Let $\lambda \ge 1$, $\mu \ge 0$, $\delta \ge 0$, and $0\le \alpha <1$. A function $f\in \Sigma$ given by (1) is said to belong to the class ${\mathcal{L}}_{\Sigma }^{\mu }(\alpha ,\lambda ,\delta )$ if the following conditions hold for all $z,w\in \mathbb{U}$:

$$\Re \left((1-\lambda ){\left({\displaystyle \frac{f\left(z\right)}{z}}\right)}^{\mu }+\lambda f^{\prime }\left(z\right){\left({\displaystyle \frac{f\left(z\right)}{z}}\right)}^{\mu -1}+\varsigma \delta z{f}^{″}\left(z\right)\right)>\alpha ,$$

(3)

and

$$\Re \left((1-\lambda ){\left({\displaystyle \frac{g\left(w\right)}{w}}\right)}^{\mu }+\lambda g^{\prime }\left(w\right){\left({\displaystyle \frac{g\left(w\right)}{w}}\right)}^{\mu -1}+\varsigma \delta w{g}^{″}\left(w\right)\right)>\alpha ,$$

(4)

where $g\left(w\right)=f^{-1}\left(w\right)$ and

$$\varsigma ={\displaystyle \frac{2\lambda +\mu }{2\lambda +1}}.$$

3. Definition of the New Subclass

In this section, we introduce a new subclass of bi-univalent functions, ${\mathcal{B}}_{\Sigma }^{\mu }(\gamma ,\lambda ,\delta ;x)$, defined through subordination to the generating function of Bernoulli polynomials.

Definition 2.

Let $\gamma >0$, $\lambda \ge 1$, $\mu \ge 0$, $\delta \ge 0$, and $x\in \left({\displaystyle \frac{1}{2}},1\right)$. A function $f\in \Sigma$ of the form (1), is said to belong to the class ${\mathcal{B}}_{\Sigma }^{\mu }(\gamma ,\lambda ,\delta ;x)$ if the following subordination conditions hold:

$$1+{\displaystyle \frac{1}{\gamma }}\left[(1-\lambda ){\left({\displaystyle \frac{f\left(\xi \right)}{\xi }}\right)}^{\mu }+\lambda f^{\prime }\left(\xi \right){\left({\displaystyle \frac{f\left(\xi \right)}{\xi }}\right)}^{\mu -1}+\varsigma \delta \xi f^{″}\left(\xi \right)-1\right]\prec {\displaystyle \frac{\xi e^{x\xi }}{e^{\xi }-1}}$$

and

$$1+{\displaystyle \frac{1}{\gamma }}\left[(1-\lambda ){\left({\displaystyle \frac{g\left(w\right)}{w}}\right)}^{\mu }+\lambda g^{\prime }\left(w\right){\left({\displaystyle \frac{g\left(w\right)}{w}}\right)}^{\mu -1}+\varsigma \delta w{g}^{″}\left(w\right)-1\right]\prec {\displaystyle \frac{w{e}^{xw}}{e^w-1}},$$

where $g=f^{-1}$ and ${\displaystyle \varsigma =\frac{2\lambda +\mu }{2\lambda +1}.}$

The above definition introduces a Bernoulli polynomial-based subclass of bi-univalent functions that depends on four adjustable parameters. Different choices of these parameters lead to several meaningful reductions, as illustrated in the following five examples.

Example 1.

Let $\lambda \ge 1$, $\mu \ge 0$, $\delta \ge 0$, and $x\in \left({\displaystyle \frac{1}{2}},1\right)$. A function $f\in \Sigma$ of the form (1), is said to belong to the class ${\mathcal{B}}_{\Sigma }^{\mu }(1,\lambda ,\delta ;x):={}^1{\mathfrak{B}}_{\Sigma }^{\mu }(\lambda ,\delta ;x)$ if the following subordination conditions hold:

$$(1-\lambda ){\left({\displaystyle \frac{f\left(\xi \right)}{\xi }}\right)}^{\mu }+\lambda f^{\prime }\left(\xi \right){\left({\displaystyle \frac{f\left(\xi \right)}{\xi }}\right)}^{\mu -1}+\varsigma \delta \xi f^{″}\left(\xi \right)\prec {\displaystyle \frac{\xi e^{x\xi }}{e^{\xi }-1}}$$

and

$$(1-\lambda ){\left({\displaystyle \frac{g\left(w\right)}{w}}\right)}^{\mu }+\lambda g^{\prime }\left(w\right){\left({\displaystyle \frac{g\left(w\right)}{w}}\right)}^{\mu -1}+\varsigma \delta w{g}^{″}\left(w\right)\prec {\displaystyle \frac{w{e}^{xw}}{e^w-1}},$$

where $g=f^{-1}$ and ${\displaystyle \varsigma =\frac{2\lambda +\mu }{2\lambda +1}.}$

Example 2.

Let $\lambda \ge 1$, $\mu \ge 0$, and $x\in \left({\displaystyle \frac{1}{2}},1\right)$. A function $f\in \Sigma$ of the form (1), is said to belong to the class ${\mathcal{B}}_{\Sigma }^{\mu }(1,\lambda ,0;x):={}^2{\mathfrak{B}}_{\Sigma }^{\mu }(\lambda ;x)$ if the following subordination conditions hold:

$$(1-\lambda ){\left({\displaystyle \frac{f\left(\xi \right)}{\xi }}\right)}^{\mu }+\lambda f^{\prime }\left(\xi \right){\left({\displaystyle \frac{f\left(\xi \right)}{\xi }}\right)}^{\mu -1}\prec {\displaystyle \frac{\xi e^{x\xi }}{e^{\xi }-1}}$$

and

$$(1-\lambda ){\left({\displaystyle \frac{g\left(w\right)}{w}}\right)}^{\mu }+\lambda g^{\prime }\left(w\right){\left({\displaystyle \frac{g\left(w\right)}{w}}\right)}^{\mu -1}\prec {\displaystyle \frac{w{e}^{xw}}{e^w-1}},$$

where $g=f^{-1}$.

Example 3.

Let $\lambda \ge 1$ and $x\in \left({\displaystyle \frac{1}{2}},1\right)$. A function $f\in \Sigma$ of the form (1), is said to belong to the class ${\mathcal{B}}_{\Sigma }^1(1,\lambda ,0;x):={}^3{\mathfrak{B}}_{\Sigma }(\lambda ;x)$ if the following subordination conditions hold:

$$(1-\lambda )\left({\displaystyle \frac{f\left(\xi \right)}{\xi }}\right)+\lambda f^{\prime }\left(\xi \right)\prec {\displaystyle \frac{\xi e^{x\xi }}{e^{\xi }-1}}$$

and

$$(1-\lambda )\left({\displaystyle \frac{g\left(w\right)}{w}}\right)+\lambda g^{\prime }\left(w\right)\prec {\displaystyle \frac{w{e}^{xw}}{e^w-1}},$$

where $g=f^{-1}$.

Example 4.

Let $x\in \left({\displaystyle \frac{1}{2}},1\right)$. A function $f\in \Sigma$ of the form (1), is said to belong to the class ${\mathcal{B}}_{\Sigma }^0(1,1,0;x):={}^4{\mathfrak{B}}_{\Sigma }\left(x\right)$ if the following subordination conditions hold:

$${\displaystyle \frac{\xi f^{\prime }\left(\xi \right)}{f\left(\xi \right)}}\prec {\displaystyle \frac{\xi e^{x\xi }}{e^{\xi }-1}}$$

and

$${\displaystyle \frac{w{g}^{\prime }\left(w\right)}{g\left(w\right)}}\prec {\displaystyle \frac{w{e}^{xw}}{e^w-1}},$$

where $g=f^{-1}$.

Example 5.

Let $x\in \left({\displaystyle \frac{1}{2}},1\right)$. A function $f\in \Sigma$ of the form (1) is said to belong to the class ${\mathcal{B}}_{\Sigma }^1(1,1,0;x):={}^5{\mathfrak{B}}_{\Sigma }\left(x\right)$ if the following subordination conditions hold:

$$f^{\prime }\left(\xi \right)\prec {\displaystyle \frac{\xi e^{x\xi }}{e^{\xi }-1}}$$

and

$$g^{\prime }\left(w\right)\prec {\displaystyle \frac{w{e}^{xw}}{e^w-1}},$$

where $g=f^{-1}$.

Remark 1.

It is worth mentioning that:

• The subclass in Example 2 was introduced by Saravanan et al. [34] under the assumption $k_t=1$.
• The subclass in Example 3 was introduced by Saravanan et al. [35] under the assumption ${\Gamma }_n[{\mu }_1;v_1]$.
• The subclass in Example 4 was introduced by Wanas and Khachi [36] under the assumption $q\to 1^{-}$.
• The subclass in Example 5 was introduced by Saravanan et al. [34].

4. Coefficient Bounds

We now derive estimates for the initial coefficients of the functions belonging to the class ${\mathcal{B}}_{\Sigma }^{\mu }(\gamma ,\lambda ,\delta ;x)$.

Theorem 1.

Let $f\in {\mathcal{B}}_{\Sigma }^{\mu }(\gamma ,\lambda ,\delta ;x)$ be given by (1), then the following bounds hold:

$$|a_2|\le {\displaystyle \frac{\gamma |B_1\left(x\right)|\sqrt{2|B_1\left(x\right)|}}{\sqrt{\left|\gamma B_1^2\left(x\right)\left[(2\lambda +\mu )(1+\mu )+12\varsigma \delta \right]-B_2\left(x\right){(\lambda +\mu +2\varsigma \delta )}^2\right|}}}$$

and

$$|a_3|\le {\displaystyle \frac{{\gamma }^2{B}_1^2\left(x\right)}{{(\lambda +\mu +2\varsigma \delta )}^2}}+{\displaystyle \frac{|B_1\left(x\right)|}{\varsigma (2\lambda +6\delta +1)}}.$$

Proof. 

From the definition of the class, there exist analytic Schwarz functions

$$\omega \left(\xi \right)=c_1\xi +c_2{\xi }^2+\dots \mathrm{and}\nu \left(w\right)=d_{1}w+d_2{w}^2+\dots ,$$

with $|c_n|\le 1$ and $|d_n|\le 1$ satisfying the subordination conditions.

Equating the coefficients of $\xi$ from the earlier expansions gives

$${\displaystyle \frac{\lambda +\mu +2\varsigma \delta }{\gamma }}{a}_2=B_1\left(x\right)c_1\mathrm{and}-{\displaystyle \frac{\lambda +\mu +2\varsigma \delta }{\gamma }}{a}_2=B_1\left(x\right)d_1,$$

(5)

which implies $c_1=-d_1$.

Comparing the coefficients of ${\xi }^2$ yields

$${\displaystyle \frac{2\lambda +\mu }{\gamma }}\left[{\displaystyle \frac{\mu -1}{2}}{a}_2^2+\left(1+{\displaystyle \frac{6\delta }{2\lambda +1}}\right)a_3\right]=B_1\left(x\right)c_2+{\displaystyle \frac{B_2\left(x\right)}{2}}{c}_1^2$$

(6)

and

$${\displaystyle \frac{2\lambda +\mu }{\gamma }}\left[\left({\displaystyle \frac{\mu +3}{2}}+{\displaystyle \frac{12\delta }{2\lambda +1}}\right)a_2^2-\left(1+{\displaystyle \frac{6\delta }{2\lambda +1}}\right)a_3\right]=B_1\left(x\right)d_2+{\displaystyle \frac{B_2\left(x\right)}{2}}{d}_1^2.$$

(7)

Adding (6) and (7) and using $c_1=-d_1$, we obtain

$${\displaystyle \frac{2\lambda +\mu }{\gamma }}\left[(1+\mu )+{\displaystyle \frac{12\delta }{2\lambda +1}}\right]a_2^2=B_1\left(x\right)(c_2+d_2)+{\displaystyle \frac{B_2\left(x\right)}{2}}(c_1^2+d_1^2).$$

(8)

Squaring the relations in (5) and adding gives

$${\displaystyle \frac{2{(\lambda +\mu +2\varsigma \delta )}^2}{{\gamma }^2}}{a}_2^2=B_1^2\left(x\right)(c_1^2+d_1^2).$$

(9)

Substituting (9) into (8) yields

$${\displaystyle \frac{1}{\gamma }}\left[(2\lambda +\mu )(1+\mu )+12\varsigma \delta -{\displaystyle \frac{{(\lambda +\mu +2\varsigma \delta )}^2{B}_2\left(x\right)}{\gamma B_1^2\left(x\right)}}\right]a_2^2=B_1\left(x\right)(c_2+d_2).$$

(10)

Using $|c_2|\le 1$ and $|d_2|\le 1$, we have

$$|c_2+d_2|\le 2,$$

and hence

$$|a_2|\le {\displaystyle \frac{\gamma |B_1\left(x\right)|\sqrt{2|B_1\left(x\right)|}}{\sqrt{\left|\gamma B_1^2\left(x\right)\left[(2\lambda +\mu )(1+\mu )+12\varsigma \delta \right]-B_2\left(x\right){(\lambda +\mu +2\varsigma \delta )}^2\right|}}}.$$

Subtracting (6) and (7) gives

$$a_3=a_2^2+{\displaystyle \frac{\gamma B_1\left(x\right)}{2\varsigma (2\lambda +6\delta +1)}}(c_2-d_2).$$

(11)

Taking absolute values and using $|c_2-d_2|\le 2$, we obtain

$$|a_3|\le |a_2{|}^2+{\displaystyle \frac{\gamma |B_1\left(x\right)|}{\varsigma (2\lambda +6\delta +1)}}.$$

Finally, from (5),

$$|a_2|\le {\displaystyle \frac{\gamma |B_1\left(x\right)|}{\lambda +\mu +2\varsigma \delta }},$$

which leads to

$$|a_3|\le {\displaystyle \frac{{\gamma }^2{B}_1^2\left(x\right)}{{(\lambda +\mu +2\varsigma \delta )}^2}}+{\displaystyle \frac{|B_1\left(x\right)|}{\varsigma (2\lambda +6\delta +1)}}.$$

□

5. Fekete–Szegö Inequality

This section is devoted to establishing a Fekete–Szegö-type inequality for functions belonging to the class ${\mathcal{B}}_{\Sigma }^{\mu }(\gamma ,\lambda ,\delta ;x)$. For a function f of the form $f\left(\xi \right)=\xi +a_2{\xi }^2+a_3{\xi }^3+\dots$, we obtain an upper bound for the functional $|a_3-\eta a_2^2|$, where $\eta \in \mathbb{R}$. The derived estimate depends explicitly on the parameters $\gamma$, $\lambda$, $\mu$, $\delta$, and the coefficients $B_1\left(x\right)$ and $B_2\left(x\right)$ of the associated generating function. The result provides a piecewise bound that generalizes several known Fekete–Szegö inequalities as special cases under suitable choices of the parameters.

Theorem 2.

Let $f\in {\mathcal{B}}_{\Sigma }^{\mu }(\gamma ,\lambda ,\delta ;x)$ be of the form (1), and let $\eta \in \mathbb{R}$. Set

$$\Delta ={\displaystyle \frac{1}{\gamma }}\left[(2\lambda +\mu )(1+\mu )+12\varsigma \delta -{\displaystyle \frac{{(\lambda +\mu +2\varsigma \delta )}^2{B}_2\left(x\right)}{\gamma B_1{\left(x\right)}^2}}\right]\mathit{and}A={\displaystyle \frac{\gamma }{2\varsigma (2\lambda +6\delta +1)}}.$$

Then

$$|a_3-\eta a_2^2|\le \left\{\begin{array}{cc}2A|B_1\left(x\right)|,\hfill & {\displaystyle |\eta -1|\le |\Delta |A,}\hfill \\ {\displaystyle \frac{2|\eta -1|}{|\Delta |}\left|B_1\left(x\right)\right|,}\hfill & {\displaystyle |\eta -1|\ge |\Delta |A.}\hfill \end{array}\right.$$

Proof. 

As in the proof of the coefficient bounds, there exist Schwarz functions

$$\omega \left(\xi \right)=c_1\xi +c_2{\xi }^2+\dots \mathrm{and}\nu \left(w\right)=d_{1}w+d_2{w}^2+\dots ,$$

with $|c_n|\le 1$ and $|d_n|\le 1$, such that

$${\displaystyle \frac{\lambda +\mu +2\varsigma \delta }{\gamma }}{a}_2=B_1\left(x\right)c_1\mathrm{and}-{\displaystyle \frac{\lambda +\mu +2\varsigma \delta }{\gamma }}{a}_2=B_1\left(x\right)d_1,$$

(12)

hence $c_1=-d_1$.

From the summed second-order relations, we obtain

$$\Delta a_2^2=B_1\left(x\right)(c_2+d_2),$$

(13)

and from the difference of the second-order relations,

$$a_3=a_2^2+B_1\left(x\right)A(c_2-d_2).$$

(14)

Using (13) and (14), we write

$$\begin{array}{ccc}\hfill a_3-\eta a_2^2& \hfill =\hfill & (1-\eta )a_2^2+B_1\left(x\right)A(c_2-d_2)\hfill \\ \hfill & \hfill =\hfill & B_1\left(x\right)\left[{\displaystyle \frac{1-\eta }{\Delta }}(c_2+d_2)+A(c_2-d_2)\right]\hfill \\ \hfill & \hfill =\hfill & B_1\left(x\right)\left[\left({\displaystyle \frac{1-\eta }{\Delta }}+A\right)c_2+\left({\displaystyle \frac{1-\eta }{\Delta }}-A\right)d_2\right].\hfill \end{array}$$

(15)

Taking the absolute values in (15) and using $|c_2|\le 1$ and $|d_2|\le 1$, we obtain

$$|a_3-\eta a_2^2|\le |B_1\left(x\right)|\left(\left|{\displaystyle \frac{1-\eta }{\Delta }}+A\right|+\left|{\displaystyle \frac{1-\eta }{\Delta }}-A\right|\right).$$

Consequently,

$$|a_3-\eta a_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{\gamma |B_1\left(x\right)|}{\varsigma (2\lambda +6\delta +1)},}\hfill & {\displaystyle |\eta -1|\le \frac{\gamma |\Delta |}{2\varsigma (2\lambda +6\delta +1)},}\hfill \\ {\displaystyle \frac{2|B_1\left(x\right)||\eta -1|}{|\Delta |},}\hfill & {\displaystyle |\eta -1|\ge \frac{\gamma |\Delta |}{2\varsigma (2\lambda +6\delta +1)}.}\hfill \end{array}\right.$$

This completes the proof. □

Remark 2.

The coefficient bounds obtained in Theorems 1 and 2 are not claimed to be sharp. Determining the sharp estimates for the initial coefficients, as well as for the Fekete–Szegö functional associated with the introduced subclasses, remains an open problem for future research.

By choosing particular values of the parameters $\gamma$, $\lambda$, $\mu$, and $\delta$, we obtain the following corollaries.

Corollary 1.

If $f\in {}^1{\mathfrak{B}}_{\Sigma }^{\mu }(\lambda ,\delta ;x)$ is given by (1), then

$$|a_2|\le {\displaystyle \frac{|B_1\left(x\right)|\sqrt{2|B_1\left(x\right)|}}{\sqrt{\left|B_1^2\left(x\right)\left[(2\lambda +\mu )(1+\mu )+12\varsigma \delta \right]-B_2\left(x\right){(\lambda +\mu +2\varsigma \delta )}^2\right|}}},$$

$$|a_3|\le {\displaystyle \frac{B_1^2\left(x\right)}{{(\lambda +\mu +2\varsigma \delta )}^2}}+{\displaystyle \frac{|B_1\left(x\right)|}{\varsigma (2\lambda +6\delta +1)}},$$

and

$$|a_3-\eta a_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{|B_1\left(x\right)|}{\varsigma (2\lambda +6\delta +1)},}\hfill & {\displaystyle |\eta -1|\le \frac{|{\Delta }_1|}{2\varsigma (2\lambda +6\delta +1)},}\hfill \\ {\displaystyle \frac{2|B_1\left(x\right)\left|\right|\eta -1|}{|{\Delta }_1|},}\hfill & {\displaystyle |\eta -1|\ge \frac{|{\Delta }_1|}{2\varsigma (2\lambda +6\delta +1)},}\hfill \end{array}\right.$$

where

$${\Delta }_1=(2\lambda +\mu )(1+\mu )+12\varsigma \delta -{\displaystyle \frac{{(\lambda +\mu +2\varsigma \delta )}^2{B}_2\left(x\right)}{B_1^2\left(x\right)}}.$$

Setting $\gamma =1$ and $\delta =0$, we obtain the following consequence (see Theorems 3.1 and 4.1 in [34], with $k_2=k_3=1$).

Corollary 2.

If $f\in {}^2{\mathfrak{B}}_{\Sigma }^{\mu }(\lambda ;x)$ is given by (1), then

$$|a_2|\le {\displaystyle \frac{|B_1\left(x\right)|\sqrt{2|B_1\left(x\right)|}}{\sqrt{\left|B_1^2\left(x\right)\left[(2\lambda +\mu )(1+\mu )\right]-B_2\left(x\right){(\lambda +\mu )}^2\right|}}},$$

$$|a_3|\le {\displaystyle \frac{B_1^2\left(x\right)}{{(\lambda +\mu )}^2}}+{\displaystyle \frac{|B_1\left(x\right)|}{(2\lambda +\mu )}},$$

and

$$|a_3-\eta a_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{|B_1\left(x\right)|}{(2\lambda +\mu )},}\hfill & {\displaystyle |\eta -1|\le \frac{|{\Delta }_2|}{2(2\lambda +\mu )},}\hfill \\ {\displaystyle \frac{2|B_1\left(x\right)\left|\right|\eta -1|}{|{\Delta }_2|},}\hfill & {\displaystyle |\eta -1|\ge \frac{|{\Delta }_2|}{2(2\lambda +\mu )},}\hfill \end{array}\right.$$

where

$${\Delta }_2=(2\lambda +\mu )(1+\mu )-{\displaystyle \frac{{(\lambda +\mu )}^2{B}_2\left(x\right)}{B_1^2\left(x\right)}}.$$

Setting $\gamma =1, \delta =0$, and $\mu =1$, we obtain the following consequence (see Theorems 1 and 2 in [35], with ${\Gamma }_2[{\mu }_1;v_1]={\Gamma }_3[{\mu }_1;v_1]=1$).

Corollary 3.

If $f\in {}^3{\mathfrak{B}}_{\Sigma }(\lambda ;x)$ is given by (1), then

$$|a_2|\le {\displaystyle \frac{|B_1\left(x\right)|\sqrt{2|B_1\left(x\right)|}}{\sqrt{\left|2B_1^2\left(x\right)(2\lambda +1)-B_2\left(x\right){(\lambda +1)}^2\right|}}},$$

$$|a_3|\le {\displaystyle \frac{B_1^2\left(x\right)}{{(\lambda +1)}^2}}+{\displaystyle \frac{|B_1\left(x\right)|}{(2\lambda +1)}},$$

and

$$|a_3-\eta a_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{|B_1\left(x\right)|}{(2\lambda +1)},}\hfill & {\displaystyle |\eta -1|\le \frac{|{\Delta }_3|}{2(2\lambda +1)},}\hfill \\ {\displaystyle \frac{2|B_1\left(x\right)\left|\right|\eta -1|}{|{\Delta }_3|},}\hfill & {\displaystyle |\eta -1|\ge \frac{|{\Delta }_3|}{2(2\lambda +1)},}\hfill \end{array}\right.$$

where

$${\Delta }_3=2(2\lambda +1)-{\displaystyle \frac{{(\lambda +1)}^2{B}_2\left(x\right)}{B_1^2\left(x\right)}}.$$

Setting $\gamma =1, \delta =0$, $\mu =0$, and $\lambda =1$, we obtain the following consequence (see Theorems 1 and 3 in [36], with $q\to 1^{-}$).

Corollary 4.

If $f\in {}^4{\mathfrak{B}}_{\Sigma }\left(x\right)$ is given by (1), then

$$|a_2|\le {\displaystyle \frac{|B_1\left(x\right)|\sqrt{2|B_1\left(x\right)|}}{\sqrt{\left|2B_1^2\left(x\right)-B_2\left(x\right)\right|}}},$$

$$|a_3|\le B_1^2\left(x\right)+{\displaystyle \frac{|B_1\left(x\right)|}{2}},$$

and

$$|a_3-\eta a_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{|B_1\left(x\right)|}{2},}\hfill & {\displaystyle |\eta -1|\le \frac{|{\Delta }_4|}{4},}\hfill \\ {\displaystyle \frac{2|B_1\left(x\right)\left|\right|\eta -1|}{|{\Delta }_4|},}\hfill & {\displaystyle |\eta -1|\ge \frac{|{\Delta }_4|}{4},}\hfill \end{array}\right.$$

where

$${\Delta }_4=2-{\displaystyle \frac{B_2\left(x\right)}{B_1^2\left(x\right)}}.$$

Setting $\gamma =1, \delta =0$, $\mu =1$, and $\lambda =1$, we obtain the following consequence (see Corollaries 3.7 and 4.7 in [34]).

Corollary 5.

If $f\in {}^5{\mathfrak{B}}_{\Sigma }\left(x\right)$ is given by (1), then

$$|a_2|\le {\displaystyle \frac{|B_1\left(x\right)|\sqrt{|B_1\left(x\right)|}}{\sqrt{\left|3B_1^2\left(x\right)-2B_2\left(x\right)\right|}}},$$

$$|a_3|\le {\displaystyle \frac{B_1^2\left(x\right)}{4}}+{\displaystyle \frac{|B_1\left(x\right)|}{3}},$$

and

$$|a_3-\eta a_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{|B_1\left(x\right)|}{3},}\hfill & {\displaystyle |\eta -1|\le \frac{|{\Delta }_5|}{6},}\hfill \\ {\displaystyle \frac{2|B_1\left(x\right)\left|\right|\eta -1|}{|{\Delta }_5|},}\hfill & {\displaystyle |\eta -1|\ge \frac{|{\Delta }_5|}{6},}\hfill \end{array}\right.$$

where

$${\Delta }_5=6-{\displaystyle \frac{4B_2\left(x\right)}{B_1^2\left(x\right)}}.$$

6. Conclusions

In this paper, we introduced a new subclass of bi-univalent functions defined through subordination to the generating function of Bernoulli polynomials. By employing coefficient comparison techniques and properties of analytic Schwarz functions, we established explicit bounds for the initial Taylor–Maclaurin coefficients. In particular, estimates for $|a_2|$ and $|a_3|$ were obtained, along with a corresponding Fekete–Szegö-type inequality that further characterizes the analytic behavior of functions within the proposed class.

The presence of the parameters $\gamma$, $\lambda$, $\mu$, and $\delta$ allows for a flexible framework that unifies several earlier subclasses of bi-univalent functions as special cases. This demonstrates that Bernoulli polynomial subordination provides a natural and effective tool for investigating coefficient-related problems in geometric function theory.

The approach presented here may be extended in several directions. For example, future research could consider higher-order coefficient estimates, Hankel-determinant problems, or subclasses defined by alternative families of special polynomials. Such investigations would further clarify the role of polynomial-based subordinations in the broader study of analytic and bi-univalent functions.