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

Abstract

In this paper, we introduce and investigate new subclasses of bi-univalent functions with respect to the symmetric points in $U=\left\{z\in \mathbb{C}:\left|z\right|<1\right\}$ defined by Bernoulli polynomials. We obtain upper bounds for Taylor–Maclaurin coefficients $\left|a_2\right|,$$\left|a_3\right|$ and Fekete–Szegö inequalities $\left|a_3-\mu a_2^2\right|$ for these new subclasses.

1. Introduction

Let the class of analytic functions in $U=\left\{z\in \mathbb{C}:\left|z\right|<1\right\}$, denoted by $A,$ contain all the functions of the type

$$l\left(z\right)=z+\sum _{k=2}^{\infty }{a}_k{z}^k,(z\in U),$$

(1)

which satisfy the usual normalization condition $l\left(0\right)=l^{\prime }\left(0\right)-1=0.$

Let S be the subclass of A consisting of all functions $l\in A$, which are also univalent in U. The Koebe one quarter theorem [1] ensures that the image of $U$ under every univalent function $l\in A$ contains a disk of radius $\frac{1}{4}$. Thus, every univalent function l has an inverse $l^{-1}$ satisfying

$$l^{-1}\left(l\left(z\right)\right)=z,(z\in U)\mathrm{and}l\left(l^{-1}\left(\omega \right)\right)=\omega ,(\left|\omega \right|<r_0\left(l\right),r_0\left(l\right)\ge \frac{1}{4}).$$

If l and $l^{-1}$ are univalent in$U$, then $l\in A$ is said to be bi-univalent in $U,$ and the class of bi-univalent functions defined in the unit disk U is denoted by $\mathrm{\Sigma }$. Since $l\in \mathrm{\Sigma }$ has the Maclaurin series given by (1), a computation shows that $m=l^{-1}$ has the expansion

$$m\left(\omega \right)=l^{-1}\left(\omega \right)=\omega -a_2{\omega }^2+\left(2a_2^2-a_3\right){\omega }^3+\cdots .$$

(2)

The expression $\mathrm{\Sigma }$ is a non-empty class of functions, as it contains at least the functions

$$l_1\left(z\right)=-\frac{z}{1-z},l_2\left(z\right)=\frac{1}{2}log\frac{1+z}{1-z},$$

with their corresponding inverses

$$l_1^{-1}\left(\omega \right)=\frac{\omega }{1+\omega },l_2^{-1}\left(\omega \right)=\frac{e^{2\omega }-1}{e^{2\omega }+1}.$$

In addition, the Koebe function $l\left(z\right)=\frac{z}{{(1-z)}^2}\notin$$\mathrm{\Sigma }$.

The study of analytical and bi-univalent functions is reintroduced in the publication of [2] and is then followed by work such as [3,4,5,6,7,8]. The initial coefficient constraints have been determined by several authors who have also presented new subclasses of bi-univalent functions (see [2,3,4,6,9,10,11]).

Consider $\alpha$ and $\beta$ to be analytic functions in$U.$ We say that $\alpha$ is subordinate to $\beta ,$ if a Schwarz function w exists that is analytic in U with $w\left(0\right)=0$ and $\left|w\left(z\right)\right|<1,$$\left(z\in U\right)$ such that

$$\alpha \left(z\right)=\beta \left(w\left(z\right)\right),\left(z\in U\right).$$

This subordination is denoted by $\alpha \prec \beta$ or $\alpha \left(z\right)\prec \beta \left(z\right),\left(z\in U\right).$ Given that $\beta$ is a univalent function in U, then

$$\alpha \left(z\right)\prec \beta \left(z\right)\iff \alpha \left(0\right)=\beta \left(0\right)\mathrm{and}\alpha \left(U\right)\subset \beta \left(U\right).$$

Using Loewner’s technique, the Fekete–Szegö problem for the coefficients of $l\in S$ in [6] is

$$\left|a_3-\mu a_2^2\right|\le 1+2exp\left(\frac{-2\mu }{1-\mu }\right)\mathrm{for}0\le \mu <1.$$

The elementary inequality $\left|a_3-a_2^2\right|\le 1$ is obtained as $\mu \to 1.$ The coefficient functional

$$F_{\mu }\left(l\right)=a_3-\mu a_2^2$$

on the normalized analytic functions l in the open unit disk U also has a significant impact on geometric function theory. The Fekete–Szegö problem is known as the maximization problem for functional $\left|F_{\mu }\left(l\right)\right|$.

Researchers were concerned about several classes of univalent functions (see [12,13,14,15]) due to the Fekete–Szegö problem, proposed in 1933 ([16]); therefore, it stands to reason that similar inequalities were also discovered for bi-univalent functions, and fairly recent publications can be cited to back up the claim that the subject still yields intriguing findings [17,18,19].

Because of their importance in probability theory, mathematical statistics, mathematical physics, and engineering, orthogonal polynomials have been the subject of substantial research in recent years from a variety of angles. The classical orthogonal polynomials are the orthogonal polynomials that are most commonly used in applications (Hermite polynomials, Laguerre polynomials, Jacobi polynomials, and Bernoulli). We point out [17,18,20,21,22,23,24] as more recent examples of the relationship between geometric function theory and classical orthogonal polynomials.

Fractional calculus, a classical branch of mathematical analysis whose foundations were laid by Liouville in an 1832 paper and is currently a very active research field [25], is one of many special functions that are studied. This branch of mathematics is known as the Bernoulli polynomials, named after Jacob Bernoulli (1654–1705). A novel approximation method based on orthonormal Bernoulli’s polynomials has been developed to solve fractional order differential equations of the Lane–Emden type [26], whereas in [27,28,29], Bernoulli polynomials are utilized to numerically resolve Fredholm fractional integro-differential equations with right-sided Caputo derivatives.

The Bernoulli polynomials $B_n\left(x\right)$ are often defined (see, e.g., [30]) using the generating function:

$$F\left(x,t\right)=\frac{t{e}^{xt}}{e^t-1}=\sum _{n=0}^{\infty }\frac{B_n\left(x\right)}{n!}{t}^n,\left|t\right|<2\pi ,$$

(3)

where $B_n\left(x\right)$ are polynomials in x, for each nonnegative integer n.

The Bernoulli polynomials are easily computed by recursion since

$$\sum _{j=0}^{n-1}\left(\underset{j}{\overset{n}{}}\right)B_j\left(x\right)=n{x}^{n-1},n=2,3,\cdots .$$

(4)

The initial few polynomials of Bernoulli are

$$B_0\left(x\right)=1,B_1\left(x\right)=x-\frac{1}{2},B_2\left(x\right)=x^2-x+\frac{1}{6},B_3\left(x\right)=x^3-\frac{3}{2}{x}^2+\frac{1}{2}x,\cdots .$$

(5)

Sakaguchi [31] introduced the class $S_s^{*}$ of functions starlike with respect to symmetric points, which consists of functions $l\in S$ satisfying the condition

$$Re\left\{\frac{z{l}^{\prime }\left(z\right)}{l\left(z\right)-l(-z)}\right\}>0,\left(z\in U\right).$$

In addition, Wang et al. [32] introduced the class $C_s$ of functions convex with respect to symmetric points, which consists of functions $l\in S$ satisfying the condition

$$Re\left\{\frac{{\left[z{l}^{\prime }\left(z\right)\right]}^{\prime }}{{\left[l\left(z\right)-l(-z)\right]}^{\prime }}\right\}>0,\left(z\in U\right).$$

In this paper, we consider two subclasses of $\mathrm{\Sigma }$: the class $S_s^{\mathrm{\Sigma }}\left(x\right)$ of functions bi-starlike with respect to the symmetric points and the relative class $C_s^{\mathrm{\Sigma }}\left(x\right)$ of functions bi-convex with respect to the symmetric points associated with Bernoulli polynomials. The definitions are as follows:

Definition 1. 

$l\in S_s^{\mathrm{\Sigma }}\left(x\right)$, if the next subordinations hold:

$$\frac{2z{l}^{\prime }\left(z\right)}{l(-z)-l\left(z\right)}\prec F\left(x,z\right),$$

(6)

and

$$\frac{2\omega m^{\prime }\left(\omega \right)}{m\left(\omega \right)-m(-\omega )}\prec F(x,\omega ),$$

(7)

where $z,\omega \in U,$$F\left(x,z\right)$ is given by (3), and $m=l^{-1}$ is given by (2).

Definition 2. 

$l\in C_s^{\mathrm{\Sigma }}\left(x\right)$, if the following subordinations hold:

$$\frac{2{\left[z{l}^{\prime }\left(z\right)\right]}^{\prime }}{{\left[l\left(z\right)-l(-z)\right]}^{\prime }}\prec F\left(x,z\right),$$

(8)

and

$$\frac{2{\left[\omega m^{\prime }\left(\omega \right)\right]}^{\prime }}{{\left[m\left(\omega \right)-m(-\omega )\right]}^{\prime }}\prec F(x,\omega ),$$

(9)

where $z,\omega \in U,$$F\left(x,z\right)$ is given by (3), and $m=l^{-1}$ is given by (2).

Lemma 1 

([33], p. 172). Suppose that $c\left(z\right)={\sum }_{n=1}^{\infty }{c}_n$z${}^n,$$\left|c\left(z\right)\right|<1,$$z\in U,$ is an analytic function in $U.$ Then,

$$\left|c_1\right|\le 1,\left|c_n\right|\le 1-{\left|c_1\right|}^2,n=2,3,\cdots .$$

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

We obtain upper bounds of $\left|a_2\right|$and$\left|a_3\right|$ for the functions belonging to the class$S_s^{\mathrm{\Sigma }}\left(x\right).$

Theorem 1. 

If $l\in S_s^{\mathrm{\Sigma }}\left(x\right)$, then

$$\left|a_2\right|\le \left|B_1\left(x\right)\right|\sqrt{6\left|B_1\left(x\right)\right|},$$

(10)

and

$$\left|a_3\right|\le \frac{B_1\left(x\right)}{2}+\frac{{\left[B_1\left(x\right)\right]}^2}{4}.$$

(11)

Proof. 

Let $l$∈$S_s^{\mathrm{\Sigma }}\left(x\right)$ and $m=l^{-1}$. From definition in (6) and (7), we have

$$\frac{2l^{\prime }\left(z\right)z}{l\left(z\right)-l(-z)}=F(x,\phi \left(z\right)),$$

(12)

and

$$\frac{2\omega m^{\prime }\left(\omega \right)}{m\left(\omega \right)-m(-\omega )}=F(x,\chi \left(\omega \right)),$$

(13)

where $\phi$and$\chi$ are analytic functions in U given by

$$\phi \left(z\right)=r_{1}z+r_2{z}^2+\cdots ,$$

(14)

$$\chi \left(\omega \right)=s_1\omega +s_2{\omega }^2+\cdots ,$$

(15)

and $\phi \left(0\right)=\chi \left(0\right)=0$, and $\left|\phi \left(z\right)\right|<1,$$\left|\chi \left(\omega \right)\right|$$<1,$$z,\omega \in U.$

As a result of Lemma 1,

$$\left|r_k\right|\le 1\mathrm{and}\left|s_k\right|\le 1,k\in \mathbb{N}.$$

(16)

If we replace (14) and (15) in (12) and (13), respectively, we obtain

$$\frac{2z{l}^{\prime }\left(z\right)}{l\left(z\right)-l(-z)}=B_0\left(x\right)+B_1\left(x\right)\phi \left(z\right)+\frac{B_2\left(x\right)}{2!}{\phi }^2\left(z\right)+\cdots ,$$

(17)

and

$$\frac{2\omega m^{\prime }\left(\omega \right)}{m\left(\omega \right)-m(-\omega )}=B_0\left(x\right)+B_1\left(x\right)\chi \left(\omega \right)+\frac{B_2\left(x\right)}{2!}{\chi }^2\left(\omega \right)+\cdots .$$

(18)

In view of (1) and (2), from (17) and (18), we obtain

$$1+2a_{2}z+2a_3{z}^2+\cdots =1+B_1\left(x\right)r_{1}z+\left[B_1\left(x\right)r_2+\frac{B_2\left(x\right)}{2!}{r}_1^2\right]z^2+\cdots$$

and

$$1-2a_2\omega +(4a_2^2-2a_3){\omega }^2+\cdots =1+B_1\left(x\right)s_1\omega +\left[B_1\left(x\right)s_2+\frac{B_2\left(x\right)}{2!}{s}_1^2\right]{\omega }^2+\cdots ,$$

which yields the following relations:

$$2a_2=B_1\left(x\right)r_1,$$

(19)

$$2a_3=B_1\left(x\right)r_2+\frac{B_2\left(x\right)}{2!}{r}_1^2,$$

(20)

and

$$-2a_2=B_1\left(x\right)s_1,$$

(21)

$$4a_2^2-2a_3=B_1\left(x\right)s_2+\frac{B_2\left(x\right)}{2!}{s}_1^2.$$

(22)

From (19) and (21), it follows that

$$r_1=-s_1,$$

(23)

and

$$8a_2^2={\left[B_1\left(x\right)\right]}^2\left(r_1^2+s_1^2\right)$$

$$a_2^2=\frac{{\left[B_1\left(x\right)\right]}^2\left(r_1^2+s_1^2\right)}{8}.$$

(24)

Adding (20) and (22), using (24), we obtain

$$a_2^2=\frac{{\left[B_1\left(x\right)\right]}^3(r_2+s_2)}{4({\left[B_1\left(x\right)\right]}^2-B_2\left(x\right))}.$$

(25)

Using relation (5), from (16) for $r_2$ and $s_2$, we get (10).

Using (23) and (24), by subtracting (22) from relation (20), we get

$$\begin{array}{ccc}\hfill a_3& =& \frac{B_1\left(x\right)\left(r_2-s_2\right)+\frac{B_2\left(x\right)}{2!}(r_1^2-s_1^2)}{4}+a_2^2\hfill \\ & =& \frac{B_1\left(x\right)\left(r_2-s_2\right)+\frac{B_2\left(x\right)}{2!}(r_1^2-s_1^2)}{4}+\frac{{\left[B_1\left(x\right)\right]}^2\left(r_1^2+s_1^2\right)}{8}.\hfill \end{array}$$

(26)

Once again applying (23) and using (5), for the coefficients $r_1,$$s_1,$$r_2,$$s_2,$ we deduce (11). □

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

We obtain the Fekete–Szegö inequality for the class $S_s^{\mathrm{\Sigma }}\left(x\right)$ due to the result of Zaprawa; see [19].

Theorem 2. 

If $l$given by (1) is in the class $S_s^{\mathrm{\Sigma }}\left(x\right)$ where $\mu \in \mathbb{R},$then we have

$$\left|a_3-\mu a_2^2\right|\le \left\{\begin{array}{ccc}\frac{B_1\left(x\right)}{2},& if& \left|h\left(\mu \right)\right|\le \frac{1}{4},\\ 2B_1\left(x\right)\left|h\left(\mu \right)\right|,& if& \left|h\left(\mu \right)\right|\ge \frac{1}{4},\end{array}\right.$$

where

$$h\left(\mu \right)=3(1-\mu ){\left[B_1\left(x\right)\right]}^2.$$

Proof. 

If l$\in S_s^{\mathrm{\Sigma }}\left(x\right)$ is given by (1), from (25) and (26), we have

$$\begin{array}{ccc}\hfill a_3-\mu a_2^2& =& \frac{B_1\left(x\right)\left(r_2-s_2\right)}{4}+(1-\mu )a_2^2\hfill \\ & =& \frac{B_1\left(x\right)\left(r_2-s_2\right)}{4}+\frac{(1-\mu ){\left[B_1\left(x\right)\right]}^3(r_2+s_2)}{4({\left[B_1\left(x\right)\right]}^2-B_2\left(x\right))}\hfill \\ & =& B_1\left(x\right)\left[\frac{r_2}{4}-\frac{s_2}{4}+\frac{\left(1-\mu \right){\left[B_1\left(x\right)\right]}^2{r}_2}{4({\left[B_1\left(x\right)\right]}^2-B_2\left(x\right))}+\frac{\left(1-\mu \right){\left[B_1\left(x\right)\right]}^2{s}_2}{4({\left[B_1\left(x\right)\right]}^2-B_2\left(x\right))}\right]\hfill \\ & =& B_1\left(x\right)\left[\left(h\left(\mu \right)+\frac{1}{4}\right)r_2+\left(h\left(\mu \right)-\frac{1}{4}\right)s_2\right],\hfill \end{array}$$

where

$$h\left(\mu \right)=\frac{(1-\mu ){\left[B_1\left(x\right)\right]}^2}{4({\left[B_1\left(x\right)\right]}^2-B_2\left(x\right))}$$

Now, by using (5)

$$a_3-\mu a_2^2=\left(x-\frac{1}{2}\right)\left[\left(h\left(\mu \right)+\frac{1}{4}\right)r_2+\left(h\left(\mu \right)-\frac{1}{4}\right)s_2\right],$$

where

$$h\left(\mu \right)=3(1-\mu ){\left(x-\frac{1}{2}\right)}^2.$$

Therefore, given (5) and (16), we conclude that the necessary inequality holds. □

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

We will obtain upper bounds of $\left|a_2\right|$and$\left|a_3\right|$ for the functions belonging to a class $C_S^{\mathrm{\Sigma }}\left(x\right).$

Theorem 3. 

If $l\in C_s^{\mathrm{\Sigma }}\left(x\right),$ then

$$\left|a_2\right|\le \frac{\left|B_1\left(x\right)\right|\sqrt{\left|B_1\left(x\right)\right|}}{\sqrt{\left|6{\left[B_1\left(x\right)\right]}^2-8B_2\left(x\right)\right|}},$$

(27)

and

$$\left|a_3\right|\le \frac{B_1\left(x\right)}{6}+\frac{{\left[B_1\left(x\right)\right]}^2}{16}.$$

(28)

Proof. 

Let l∈$C_s^{\mathrm{\Sigma }}\left(x\right)$ and $m=l^{-1}$. From (8) and (9), we get

$$\frac{2{\left[z{l}^{\prime }\left(z\right)\right]}^{\prime }}{{\left[l\left(z\right)-l(-z)\right]}^{\prime }}=F(x,\phi \left(z\right)),$$

(29)

and

$$\frac{2{\left[\omega m^{\prime }\left(\omega \right)\right]}^{\prime }}{{\left[m\left(\omega \right)-m(-\omega )\right]}^{\prime }}=F(x,\chi \left(\omega \right))$$

(30)

where $\phi$and$\chi$ are analytic functions in U given by

$$\phi \left(z\right)=r_{1}z+r_2{z}^2+\cdots ,$$

(31)

$$\chi \left(\omega \right)=s_1\omega +s_2{\omega }^2+\cdots ,$$

(32)

where $\phi \left(0\right)=\chi \left(0\right)=0,$ and $\left|\phi \left(z\right)\right|<1,$$\left|\chi \left(\omega \right)\right|<1,$$z,\omega$∈$U.$

As a result of Lemma 1,

$$\left|r_k\right|\le 1\mathrm{and}\left|s_k\right|\le 1,k\in \mathbb{N}.$$

(33)

If we replace (31) and (32) in (29) and (30), respectively, we obtain

$$\frac{2{\left[z{l}^{\prime }\left(z\right)\right]}^{\prime }}{{\left[l\left(z\right)-l(-z)\right]}^{\prime }}=B_0\left(x\right)+B_1\left(x\right)\phi \left(z\right)+\frac{B_2\left(x\right)}{2!}{\phi }^2\left(z\right)+\cdots ,$$

(34)

and

$$\frac{2{\left[\omega m^{\prime }\left(\omega \right)\right]}^{\prime }}{{\left[m\left(\omega \right)-m(-\omega )\right]}^{\prime }}=B_0\left(x\right)+B_1\left(x\right)\chi \left(\omega \right)+\frac{B_2\left(x\right)}{2!}{\chi }^2\left(\omega \right)+\cdots .$$

(35)

In view of (1) and (2), from (34) and (35), we obtain

$$1+4a_{2}z+6a_3{z}^2+\cdots =1+B_1\left(x\right)r_{1}z+\left[B_1\left(x\right)r_2+\frac{B_2\left(x\right)}{2!}{r_1}^2\right]z^2+\cdots$$

and

$$1-4a_2\omega +\left(12{a_2}^2-6a_3\right){\omega }^2+\cdots =1+B_1\left(x\right)s_1\omega +\left[B_1\left(x\right)s_2+\frac{B_2\left(x\right)}{2!}{s_1}^2\right]{\omega }^2+\cdots ,$$

which yields the following relations:

$$4a_2=B_1\left(x\right)r_1,$$

(36)

$$6a_3=B_1\left(x\right)r_2+\frac{B_2\left(x\right)}{2!}{r}_1^2,$$

(37)

and

$$-4a_2=B_1\left(x\right)s_1,$$

(38)

$$12a_2^2-6a_3=B_1\left(x\right)s_2+\frac{B_2\left(x\right)}{2!}{s}_1^2.$$

(39)

From (36) and (38), it follows that

$$r_1=-s_1,$$

(40)

and

$$32a_2^2={\left[B_1\left(x\right)\right]}^2\left(r_1^2+s_1^2\right)$$

$$a_2^2=\frac{{\left[B_1\left(x\right)\right]}^2\left(r_1^2+s_1^2\right)}{32}.$$

(41)

Adding (37) and (39), using (41), we obtain

$$a_2^2=\frac{{\left[B_1\left(x\right)\right]}^3(r_2+s_2)}{4(3{\left[B_1\left(x\right)\right]}^2-4B_2\left(x\right))}.$$

(42)

Using relation (5), from (33) for $r_2$ and $s_2$, we get (27). Using (40) and (41), by subtracting (39) from relation (37), we get

$$\begin{array}{ccc}\hfill a_3& =& \frac{B_1\left(x\right)\left(r_2-s_2\right)+\frac{B_2\left(x\right)}{2!}\left(r_1^2-s_1^2\right)}{12}+a_2^2\hfill \\ & =& \frac{B_1\left(x\right)\left(r_2-s_2\right)+\frac{B_2\left(x\right)}{2!}\left(r_1^2-s_1^2\right)}{12}+\frac{{\left[B_1\left(x\right)\right]}^2\left(r_1^2+s_1^2\right)}{32}.\hfill \end{array}$$

(43)

Once again applying (40) and using (5), for the coefficients $r_1,$$s_1$, $r_2$, $s_2$, we deduce (28). □

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

We obtain the Fekete–Szegö inequality for the class $C_s^{\mathrm{\Sigma }}\left(x\right)$ due to the result of Zaprawa; see [19].

Theorem 4. 

If $l$given by (1) is in the class $C_s^{\mathrm{\Sigma }}\left(x\right)$ where $\mu \in \mathbb{R},$then, we have

$$\left|a_3-\mu a_2^2\right|\le \left\{\begin{array}{ccc}\frac{B_1\left(x\right)}{6},& if& \left|h\left(\mu \right)\right|\le \frac{1}{12},\\ 2B_1\left(x\right)\left|h\left(\mu \right)\right|,& if& \left|h\left(\mu \right)\right|\ge \frac{1}{12},\end{array}\right.$$

where

$$h\left(\mu \right)=\frac{(1-\mu ){\left[B_1\left(x\right)\right]}^2}{4({3\left[B_1\left(x\right)\right]}^2-{4B}_2\left(x\right))}.$$

Proof. 

If l$\in C_s^{\mathrm{\Sigma }}\left(x\right)$ is given by (1), from (42) and (43), we have

$$\begin{array}{ccc}\hfill a_3-\mu a_2^2& =& \frac{B_1\left(x\right)\left(r_2-s_2\right)}{12}+(1-\mu )a_2^2\hfill \\ & =& \frac{B_1\left(x\right)\left(r_2-s_2\right)}{12}+\frac{(1-\mu ){\left[B_1\left(x\right)\right]}^3(r_2+s_2)}{4({{3B}_1\left(x\right)}^2-{4B}_2\left(x\right))}\hfill \\ & =& B_1\left(x\right)\left[\frac{r_2-s_2}{12}+\frac{\left(1-\mu \right){\left[B_1\left(x\right)\right]}^2{r}_2}{4({3\left[B_1\left(x\right)\right]}^2-{4B}_2\left(x\right))}+\frac{\left(1-\mu \right){\left[B_1\left(x\right)\right]}^2{s}_2}{4({3\left[B_1\left(x\right)\right]}^2-{4B}_2\left(x\right))}\right]\hfill \\ & =& B_1\left(x\right)\left[\left(h\left(\mu \right)+\frac{1}{12}\right)r_2+\left(h\left(\mu \right)-\frac{1}{12}\right)s_2\right],\hfill \end{array}$$

where

$$h\left(\mu \right)=\frac{(1-\mu ){\left[B_1\left(x\right)\right]}^2}{4({3\left[B_1\left(x\right)\right]}^2-{4B}_2\left(x\right))}.$$

Now, by using (5)

$$a_3-\mu a_2^2=\left(x-\frac{1}{2}\right)\left[\left(h\left(\mu \right)+\frac{1}{12}\right)r_2+\left(h\left(\mu \right)-\frac{1}{12}\right)s_2\right],$$

where

$$h\left(\mu \right)=\frac{(1-\mu ){\left[x-\frac{1}{2}\right]}^2}{4({3\left(x-\frac{1}{2}\right)}^2-4{(x}^2-x+\frac{1}{6}))}.$$

Therefore, given (5) and (33), we conclude that the required inequality holds. □

6. Conclusions

We introduce and investigate new subclasses of bi-univalent functions in U associated with Bernoulli polynomials and satisfying subordination conditions. Moreover, we obtain upper bounds for the initial Taylor–Maclaurin coeffcients $\left|a_2\right|,$$\left|a_3\right|$ and Fekete–Szegö problem $\left|a_3-\mu a_2^2\right|$ for functions in these subclasses.

The approach employed here has also been extended to generate new bi-univalent function subfamilies using the other special functions. The researchers may carry out the linked outcomes in practice.