Fekete–Szegö inequality for bi-univalent functions by means of Horadam polynomials

Abstract

In this paper, make use of the Horadam polynomials, we introduce a comprehensive subclass of analytic and bi-univalent functions. For functions belonging to this class we derive coefficient inequalities and the Fekete–Szegö inequalities. Also, variety observations of the results presented here are also discussed.

1 Introduction

Let ${\mathcal {A}}$ denote the class of analytic functions f in the open unit disk ${\mathbb {U}}=\{z\in {\mathbb {C}}:\left| z\right| <1\}$ and normalized by the conditions $f(0)=0$ and $f^{\prime }(0)=1$. Thus, each $f\in {\mathcal {A}}$ has a Maclaurin series expansion of the form:

$$\begin{aligned} f(z)=z+\sum \limits _{n=2}^{\infty }a_{n}z^{n},\quad (z\in {\mathbb {U}}). \end{aligned}$$

(1)

Further, let ${\mathcal {S}}$ denote the class of all functions $f\in {\mathcal {A}}$ which are univalent in ${\mathbb {U}}$ (see [8]).

It is well-known that if f(z) is analytic and univalent from a domain ${\mathbb {D}}_{1}$ onto a domain ${\mathbb {D}}_{2}$, then the inverse function g(z) defined by:

$$\begin{aligned} g\left( f(z)\right) =z,\quad (z\in {\mathbb {D}}_{1}), \end{aligned}$$

is an analytic and univalent mapping from ${\mathbb {D}}_{2}$ to ${\mathbb {D}}_{1}$. Moreover, by the familiar Koebe one-quarter theorem every function $f\in {\mathcal {S}}$ has an inverse map $f^{-1}$ that satisfies the following conditions (for details, see [8]):

$$\begin{aligned} f^{-1}(f(z))=z\quad {\text {and}}\; f\left( f^{-1}(w)\right) = w \quad \left( z\in {\mathbb {U}}\text {,}|w|<r_{0}(f);r_{0}(f)\ge \frac{1}{4}\right) . \end{aligned}$$

where

$$\begin{aligned} f^{-1}(w)=w-a_{2}w^{2}+(2a_{2}^{2}-a_{3})w^{3}-(5a_{2}^{3}-5a_{2}a_{3} +a_{4})w^{4}+\cdots . \end{aligned}$$

(2)

A function $f\in {\mathcal {A}}$ is said to be bi-univalent in ${\mathbb {U}}$ if both f and $f^{-1}$ are univalent in ${\mathbb {U}}$. Let $\Sigma$ denote the class of bi-univalent functions in ${\mathbb {U}}$ given by (1).

Several authors have introduced and investigated subclasses of bi-univalent functions and obtained bounds for the initial coefficients (see [1, 9,10,11, 17, 18, 20, 22]).

In 1967, Lewin [16] showed that $|a_{2}|<1.51$ for bi-univalent function class $\Sigma$. Subsequently, Brannan and Clunie [7] conjectured that $|a_{2}|\le \sqrt{2}$. For each $f\in \Sigma$ given by (1), the coefficient estimate for $|a_{n}|$ $(n\in \{3,4,5,\cdots \})$ is still an open problem.

Horzum and Kocer [14] considered the Horadam polynomials $h_{n}(x)$, which are given by:

$$\begin{aligned} h_{n}(x)=pxh_{n-1}(x)+qh_{n-2}(x),\quad (n\in \left\{ 3,4,5,\cdots \right\} ), \end{aligned}$$

(3)

with

$$\begin{aligned} h_{1}(x)=a\text {, }h_{2}(x)=bx,\text { and }h_{3}(x)=pbx^{2}+aq \end{aligned}$$

(4)

for the constants a, b, p,$q\in {\mathbb {R}} .$

Remark 1.1

For particular values of a, b, p and q, the Horadam polynomials $h_{n}(x)$ lead to various polynomials (see [14] and [13]), for example:

• (1) If $a=b=p=q=1$, then we get the Fibonacci polynomials $F_{n}(x)$;

• (2) If $a=2$ and $b=p=q=1$, then we get the Lucas polynomials $L_{n}(x)$;

• (3) If $a=b=1$, $p=2$ and $q=-1$, then we get the Chebyshev polynomials $T_{n}(x)$ of the first kind;

• (4) If $a=1$, $b=p=2$ and $q=-1$, then we get the Chebyshev polynomials $U_{n}(x)$ of the second kind;

• (5) If $a=q=1$ and $b=p=2$, then we get the Pell polynomials $P_{n}(x)$;

• (6) If $a=$ $b=p=2$ and $q=1,$ then we get the Pell-Lucas polynomials $Q_{n}(x)$ of the first kind.

The coefficient estimates and Fekete–Szegö inequality are found for bi-univalent functions associated with certain polynomials like the Fibonacci polynomials, Lucas polynomials, Chebyshev polynomials and the Horadam polynomials. We also note that the above polynomials and other special polynomials are potentially important in the mathematical, physical, statistical and engineering sciences. These polynomials have been studied in several papers (see [2,3,4,5,6, 12, 15, 19, 21, 23]).

Theorem 1.2

( [14]) Let $\Phi (x,z)$ be the generating function of the Horadam polynomials $h_{n}(x)$. Then,

$$\begin{aligned} \Phi (x,z)=\sum \limits _{n=1}^{\infty }h_{n}(x)z^{n-1}=\frac{(b-ap)xz+a}{1-pxz-qz^{2}}. \end{aligned}$$

(5)

In this paper, we define the following subclass of $\Sigma$ by making use of the Horadam polynomials, which are given by the recurrence relation (3) and the generating function (5).

Definition 1.3

For $\alpha ,\mu \ge 1,$ $\sigma \in {\mathbb {C}}$ and ${\text {Re}}(\sigma )\ge 0$, a function $f\in \Sigma$ given by (1) is said to be in the class ${\mathbf {b}}_{\Sigma }^{a}(\mu ,\alpha ,\sigma )$ if the following subordinations are satisfied:

$$\begin{aligned} (1-\mu )f^{\prime }(z)+\mu \left( f^{\prime }(z)\right) ^{\alpha }\left( \frac{f(z)}{z}\right) ^{\sigma -1}\prec \Phi (x,z)-a+1 \end{aligned}$$

and

$$\begin{aligned} (1-\mu )g^{\prime }(w)+\mu \left( g^{\prime }(w)\right) ^{\alpha }\left( \frac{g(w)}{w}\right) ^{\sigma -1}\prec \Phi (x,w)-a+1, \end{aligned}$$

where g is given by (2).

2 Coefficient estimates for the subclass ${\mathbf {b}}_{\Sigma }^{a} (\mu ,\alpha ,\sigma )$

In this section, we estimates the coefficients $\left| a_{2}\right|$ and $\left| a_{3}\right|$ for functions in the subclass ${\mathbf {b}}_{\Sigma }^{a}(\mu ,\alpha ,\sigma )$, which we introduced in Definition 1.3. Also, the Fekete–Szegö problem for this subclass is solved.

Theorem 2.1

For $\alpha ,\mu \ge 1,$ $\sigma \in {\mathbb {C}}$ and ${\text {Re}}(\sigma )\ge 0$, let $f\in {\mathcal {A}}$ belongs to the class ${\mathbf {b}}_{\Sigma }^{a}(\mu ,\alpha ,\sigma ).$ Then

$$\begin{aligned} \left| a_{2}\right| \le \frac{\left| bx\right| \sqrt{2\left| bx\right| }}{\sqrt{ \begin{array} {l} \left| \left[ \mu \left( \left( \sigma +2\right) \left( \sigma -3\right) +4\alpha \left( \sigma -1\right) +2\alpha (2\alpha +1)\right) +6\right] b^{2}x^{2}\right. \\ \left. -2\left( \mu \left( 2\alpha +\sigma -3\right) +2\right) ^{2}\left( pbx^{2}+aq\right) \right| \end{array} }}, \\ \left| a_{3}\right| \le \frac{b^{2}x^{2}}{\left| \mu \left( 2\alpha +\sigma -3\right) +2\right| ^{2}}+\frac{\left| bx\right| }{\left| \mu \left( 3\alpha +\sigma -4\right) +3\right| }, \end{aligned}$$

and

$$ \left| a_{3}-\eta a_{2}^{2}\right| \le \left\{ \begin{array} {l} \frac{\left| bx\right| }{\mu \left( 3\alpha +\sigma -4\right) +3},\\ \\ \frac{2\left| bx\right| \left| 1-\eta \right| }{\left| \kappa (x)\right| }, \end{array} \right. \begin{array} {ll} \left| \eta -1\right| \le \frac{\left| \kappa (x)\right| }{2\mu \left( 3\alpha +\sigma -4\right) +6}\\ \\ \left| \eta -1\right| \ge \frac{\left| \kappa (x)\right| }{2\mu \left( 3\alpha +\sigma -4\right) +6}, \end{array} $$

where

$$ \kappa (x)= \begin{array} {l} \left[ \mu \left( \left( \sigma +2\right) \left( \sigma -3\right) +4\alpha \left( \sigma -1\right) +2\alpha (2\alpha +1)\right) +6\right] \\ -2\left( \mu \left( 2\alpha +\sigma -3\right) +2\right) ^{2}\frac{\left( pbx^{2}+aq\right) }{b^{2}x^{2}}. \end{array} $$

Proof

Let $f\in {\mathbf {b}}_{\Sigma }^{a}(\mu ,\alpha ,\sigma ).$ From Definition 1.3, for some analytic functions $\Upsilon ,$ $\digamma$ such that $\Upsilon \left( 0\right) =\digamma \left( 0\right) =0$ and $\left| \Upsilon \left( z\right) \right| <1,$ $\left| \digamma \left( w\right) \right| <1$ for all $z,w\in {\mathbb {U}},$ then we can write

$$\begin{aligned}&(1-\mu )f^{\prime }(z)+\mu \left( f^{\prime }(z)\right) ^{\alpha }\left( \frac{f(z)}{z}\right) ^{\sigma -1}\nonumber \\&=1+h_{1}(x)-a+h_{2}(x)\Upsilon \left( z\right) +h_{3}(x)\Upsilon ^{2}\left( z\right) +\cdots \end{aligned}$$

(6)

and

$$\begin{aligned}&(1-\mu )g^{\prime }(w)+\mu \left( g^{\prime }(w)\right) ^{\alpha }\left( \frac{g(w)}{w}\right) ^{\sigma -1}\nonumber \\&=1+h_{1}(x)-a+h_{2}(x)\digamma \left( w\right) +h_{3}(x)\digamma ^{2}\left( w\right) +\cdots . \end{aligned}$$

(7)

From the equalities (6) and (7), we obtain

$$\begin{aligned}&(1-\mu )f^{\prime }(z)+\mu \left( f^{\prime }(z)\right) ^{\alpha }\left( \frac{f(z)}{z}\right) ^{\sigma -1}\nonumber \\&=1+h_{2}(x)r_{1}z+\left[ h_{2}(x)r_{2}+h_{3}(x)r_{1} ^{2}\right] z^{2}+\cdots \end{aligned}$$

(8)

and

$$\begin{aligned}&(1-\mu )g^{\prime }(w)+\mu \left( g^{\prime }(w)\right) ^{\alpha }\left( \frac{g(w)}{w}\right) ^{\sigma -1}\\&=1+h_{2}(x)s_{1}w+\left[ h_{2}(x)s_{2}+h_{3}(x)s_{1} ^{2}\right] w^{2}+\cdots .\nonumber \end{aligned}$$

(9)

It is well-known that if

$$\begin{aligned} \left| \Upsilon \left( z\right) \right| =\left| s_{1}z+s_{2} z^{2}+s_{3}z^{3}+\cdots \right| <1, \quad (z\in {\mathbb {U}}) \end{aligned}$$

and

$$\begin{aligned} \left| \digamma \left( w\right) \right| =\left| t_{1}w+t_{2} w^{2}+t_{3}w^{3}+\cdots \right| <1,\quad (w\in {\mathbb {U}}), \end{aligned}$$

then

$$\begin{aligned} \left| s_{j}\right|<1\text { and }\left| t_{j}\right| <1\text { for }j\in {\mathbb {N}} . \end{aligned}$$

(10)

Thus, comparing the coefficients in (8) and (9), we have:

$$\begin{aligned}&\left( \mu \left( 2\alpha +\sigma -3\right) +2\right) a_{2}=h_{2}(x)s_{1}, \end{aligned}$$

(11)

$$\begin{aligned}&\left[ \mu \left( \frac{\left( \sigma -1\right) \left( \sigma -2\right) }{2}+2\alpha \left( \sigma -1\right) +2\alpha (\alpha -1)\right) \right] a_{2}^{2}+\left[ \mu \left( 3\alpha +\sigma -4\right) +3\right] a_{3}\nonumber \\&=h_{2}(x)s_{2}+h_{3}(x)s_{1}^{2}, \end{aligned}$$

(12)

$$\begin{aligned}&-\left( \mu \left( 2\alpha +\sigma -3\right) +2\right) a_{2}=h_{2}(x)t_{1}, \end{aligned}$$

(13)

and

$$\begin{aligned}&\left[ \mu \left( \frac{\left( \sigma -2\right) \left( \sigma +3\right) }{2}+2\alpha \left( \sigma -1\right) +2\alpha (\alpha +2)-4\right) +6\right] a_{2}^{2}\nonumber \\&\quad -\left[ \mu \left( 3\alpha +\sigma -4\right) +3\right] a_{3}\nonumber \\&=h_{2}(x)t_{2}+h_{3}(x)t_{1}^{2}. \end{aligned}$$

(14)

It follows from (11) and (13) that

$$\begin{aligned} s_{1}=-t_{1} \end{aligned}$$

(15)

and

$$\begin{aligned} 2\left( \mu \left( 2\alpha +\sigma -3\right) +2\right) ^{2}a_{2}^{2} =h_{2}^{2}(x)\left( s_{1}^{2}+t_{1}^{2}\right) . \end{aligned}$$

(16)

If we add (12) and (14), we get

$$\begin{aligned}&\left[ \mu \left( \left( \sigma +2\right) \left( \sigma -3\right) +4\alpha \left( \sigma -1\right) +2\alpha (2\alpha +1)\right) +6\right] a_{2}^{2}\nonumber \\&=h_{2}(x)\left( s_{2}+t_{2}\right) +h_{3}(x)\left( s_{1}^{2}+t_{1} ^{2}\right) . \end{aligned}$$

(17)

Substituting the value of $\left( s_{1}^{2}+t_{1}^{2}\right)$ from (16) in (17), we have:

$$\begin{aligned}&\left\{ \left[ \mu \left( \left( \sigma +2\right) \left( \sigma -3\right) +4\alpha \left( \sigma -1\right) +2\alpha (2\alpha +1)\right) +6\right] h_{2}^{2}(x)\right. \nonumber \\&\left. \left[ -2\left( \mu \left( 2\alpha +\sigma -3\right) +2\right) ^{2}h_{3}(x)\right] \right\} a_{2}^{2}\nonumber \\&=h_{2}^{3}(x)\left( s_{2}+t_{2}\right) . \end{aligned}$$

(18)

Moreover, using (4) and (10) in (18), we find that:

$$ \left| a_{2}\right| \le \frac{\left| bx\right| \sqrt{2\left| bx\right| }}{\sqrt{ \begin{array} {l} \left| \left[ \mu \left( \left( \sigma +2\right) \left( \sigma -3\right) +4\alpha \left( \sigma -1\right) +2\alpha (2\alpha +1)\right) +6\right] b^{2}x^{2}\right. \\ \left. -2\left( \mu \left( 2\alpha +\sigma -3\right) +2\right) ^{2}\left( pbx^{2}+aq\right) \right| \end{array} }}. $$

Next, if we subtract (14) from (12), we obtain:

$$\begin{aligned} 2\left[ \mu \left( 3\alpha +\sigma -4\right) +3\right] \left( a_{3} -a_{2}^{2}\right) =h_{2}(x)\left( s_{2}-t_{2}\right) +h_{3}(x)\left( s_{1}^{2}-t_{1}^{2}\right) . \end{aligned}$$

(19)

Then, in view of (15) and (16), Eq. (19) becomes:

$$\begin{aligned} a_{3}=\frac{h_{2}^{2}(x)}{2\left( \mu \left( 2\alpha +\sigma -3\right) +2\right) ^{2}}\left( s_{1}^{2}+t_{1}^{2}\right) +\frac{h_{2}(x)}{2\left[ \mu \left( 3\alpha +\sigma -4\right) +3\right] }\left( s_{2}-t_{2}\right) . \end{aligned}$$

Thus, applying (4), we conclude that:

$$\begin{aligned} \left| a_{3}\right| \le \frac{b^{2}x^{2}}{\left| \mu \left( 2\alpha +\sigma -3\right) +2\right| ^{2}}+\frac{\left| bx\right| }{\left| \mu \left( 3\alpha +\sigma -4\right) +3\right| }. \end{aligned}$$

From (18) and (19)

$$\begin{aligned} a_{3}-\eta a_{2}^{2}&=\frac{\left( 1-\eta \right) \left( s_{2} +t_{2}\right) h_{2}^{3}(x)}{ \begin{array} {l} \left[ \mu \left( \left( \sigma +2\right) \left( \sigma -3\right) +4\alpha \left( \sigma -1\right) +2\alpha (2\alpha +1)\right) +6\right] h_{2}^{2}(x)\\ -2\left( \mu \left( 2\alpha +\sigma -3\right) +2\right) ^{2}h_{3}(x) \end{array} }\\&+\frac{\left( s_{2}-t_{2}\right) h_{2}(x)}{2\left[ \mu \left( 3\alpha +\sigma -4\right) +3\right] }\\& =h_{2}(x)\left( \left[ \varphi (\eta ,x)+\frac{1}{2\left[ \mu \left( 3\alpha +\sigma -4\right) +3\right] }\right] s_{2}\right. \\&\left. +\left[ \varphi (\eta ,x)-\frac{1}{2\left[ \mu \left( 3\alpha +\sigma -4\right) +3\right] }\right] t_{2}\right) , \end{aligned}$$

where

$$ \varphi (\eta ,x)=\frac{h_{2}^{2}(x)\left( 1-\eta \right) }{ \begin{array} {l} \left[ \mu \left( \left( \sigma +2\right) \left( \sigma -3\right) +4\alpha \left( \sigma -1\right) +2\alpha (2\alpha +1)\right) +6\right] h_{2}^{2}(x)\\ -2\left( \mu \left( 2\alpha +\sigma -3\right) +2\right) ^{2}h_{3}(x) \end{array} }, $$

Then, in view of (4), we get:

$$ \left| a_{3}-\eta a_{2}^{2}\right| \le \left\{ \begin{array} {ll} \frac{\left| h_{2}(x)\right| }{\mu \left( 3\alpha +\sigma -4\right) +3}\\ \\ 2\left| h_{2}(x)\right| \left| \varphi (\eta ,x)\right| \end{array} \right. \begin{array} {ll} 0\le \left| \varphi (\eta ,x)\right| \le \frac{1}{2\mu \left( 3\alpha +\sigma -4\right) +6},\\ \\ \left| \varphi (\eta ,x)\right| \ge \frac{1}{2\mu \left( 3\alpha +\sigma -4\right) +6}. \end{array} $$

So, the proof of Theorem 2.1 is complete. $\square$

By setting $\alpha =\mu =1$ and $\sigma =0$ in Theorem 2.1, we obtain the following consequence.

Corollary 2.2

Let $f\in {\mathcal {A}}$ belongs to the class ${\mathbf {b}}_{\Sigma }^{a}(1,1,0).$ Then

$$\begin{aligned} \left| a_{2}\right| \le \frac{\left| bx\right| \sqrt{\left| bx\right| }}{\sqrt{\left| (b-p)bx^{2}-aq\right| }}, \\ \left| a_{3}\right| \le b^{2}x^{2}+\frac{\left| bx\right| }{2}, \end{aligned}$$

and

$$ \left| a_{3}-\eta a_{2}^{2}\right| \le \left\{ \begin{array} {ll} \frac{\left| bx\right| }{2},\\ \\ \frac{\left| bx\right| ^{3}\left| 1-\eta \right| }{\left| (b-p)bx^{2}-aq\right| }, \end{array} \right. \begin{array} {ll} \left| \eta -1\right| \le \frac{\left| (b-p)bx^{2}-aq\right| }{2b^{2}x^{2}}\\ \\ \left| \eta -1\right| \ge \frac{\left| (b-p)bx^{2}-aq\right| }{2b^{2}x^{2}}, \end{array} . $$

In view of Remark 1.1, Theorem 2.1 can be shown to yield the following interesting observations.

By setting Fibonacci polynomials $F_{n}(x)$ instead of Horadam polynomials $h_{n}(x)$ in Theorem 2.1, we obtain the following corollary.

Corollary 2.3

For $\alpha ,\mu \ge 1,$ $\sigma \in {\mathbb {C}}$ and ${\text {Re}}(\sigma )\ge 0$, let $f\in {\mathcal {A}}$ belongs to the class ${\mathbf {b}}_{\Sigma }^{a}(\mu ,\alpha ,\sigma ).$ Then

$$\begin{aligned} \left| a_{2}\right| \le \frac{\left| x\right| \sqrt{2\left| x\right| }}{\sqrt{ \begin{array} {ll} \left| \left[ \mu \left( \left( \sigma +2\right) \left( \sigma -3\right) +4\alpha \left( \sigma -1\right) +2\alpha (2\alpha +1)\right) +6\right] x^{2}\right. \\ \left. -2\left( \mu \left( 2\alpha +\sigma -3\right) +2\right) ^{2}\left( x^{2}+1\right) \right| \end{array} }}, \\ \left| a_{3}\right| \le \frac{x^{2}}{\left| \mu \left( 2\alpha +\sigma -3\right) +2\right| ^{2}}+\frac{\left| x\right| }{\left| \mu \left( 3\alpha +\sigma -4\right) +3\right| }, \end{aligned}$$

and

$$\begin{aligned} \left| a_{3}-\eta a_{2}^{2}\right| \le \left\{ \begin{array} {ll} \frac{\left| x\right| }{\mu \left( 3\alpha +\sigma -4\right) +3},\\ \\ \frac{2\left| x\right| \left| 1-\eta \right| }{\left| \kappa (x)\right| }, \end{array} \right. \begin{array} {ll} \left| \eta -1\right| \le \frac{\left| \kappa (x)\right| }{2\mu \left( 3\alpha +\sigma -4\right) +6}\\ \\ \left| \eta -1\right| \ge \frac{\left| \kappa (x)\right| }{2\mu \left( 3\alpha +\sigma -4\right) +6}, \end{array} \end{aligned}$$

where

$$\begin{aligned} \begin{array} {ll} \kappa (x)=\left[ \mu \left( \left( \sigma +2\right) \left( \sigma -3\right) +4\alpha \left( \sigma -1\right) +2\alpha (2\alpha +1)\right) +6\right] \\ -2\left( \mu \left( 2\alpha +\sigma -3\right) +2\right) ^{2}\frac{\left( x^{2}+1\right) }{x^{2}}. \end{array} \end{aligned}$$

By setting Lucas polynomials $L_{n}(x)$ instead of Horadam polynomials $h_{n}(x)$ in Theorem 2.1, we obtain the following corollary.

Corollary 2.4

For $\alpha ,\mu \ge 1,$ $\sigma \in {\mathbb {C}}$ and ${\text {Re}}(\sigma )\ge 0$, let $f\in {\mathcal {A}}$ belongs to the class ${\mathbf {b}}_{\Sigma }^{a}(\mu ,\alpha ,\sigma ).$ Then

$$\begin{aligned} \left| a_{2}\right| \le \frac{\left| x\right| \sqrt{2\left| x\right| }}{\sqrt{ \begin{array} {ll} \left| \left[ \mu \left( \left( \sigma +2\right) \left( \sigma -3\right) +4\alpha \left( \sigma -1\right) +2\alpha (2\alpha +1)\right) +6\right] x^{2}\right. \\ \left. -2\left( \mu \left( 2\alpha +\sigma -3\right) +2\right) ^{2}\left( x^{2}+2\right) \right| \end{array} }}, \\ \left| a_{3}\right| \le \frac{x^{2}}{\left| \mu \left( 2\alpha +\sigma -3\right) +2\right| ^{2}}+\frac{\left| x\right| }{\left| \mu \left( 3\alpha +\sigma -4\right) +3\right| }, \end{aligned}$$

and

$$\begin{aligned} \left| a_{3}-\eta a_{2}^{2}\right| \le \left\{ \begin{array} {ll} \frac{\left| x\right| }{\mu \left( 3\alpha +\sigma -4\right) +3},\\ \\ \frac{2\left| x\right| \left| 1-\eta \right| }{\left| \kappa (x)\right| }, \end{array} \right. \begin{array} {ll} \left| \eta -1\right| \le \frac{\left| \kappa (x)\right| }{2\mu \left( 3\alpha +\sigma -4\right) +6}\\ \\ \left| \eta -1\right| \ge \frac{\left| \kappa (x)\right| }{2\mu \left( 3\alpha +\sigma -4\right) +6}, \end{array} \end{aligned}$$

where

$$\begin{aligned} \kappa (x)= \begin{array} {ll} \left[ \mu \left( \left( \sigma +2\right) \left( \sigma -3\right) +4\alpha \left( \sigma -1\right) +2\alpha (2\alpha +1)\right) +6\right] \\ -2\left( \mu \left( 2\alpha +\sigma -3\right) +2\right) ^{2}\frac{\left( x^{2}+2\right) }{x^{2}}. \end{array} \end{aligned}$$

By setting Chebyshev polynomials $T_{n}(x)$ of the first kind instead of Horadam polynomials $h_{n}(x)$ in Theorem 2.1, we obtain the following corollary.

Corollary 2.5

For $\alpha ,\mu \ge 1,$ $\sigma \in {\mathbb {C}}$ and ${\text {Re}}(\sigma )\ge 0$, let $f\in {\mathcal {A}}$ belongs to the class ${\mathbf {b}}_{\Sigma }^{a}(\mu ,\alpha ,\sigma ).$ Then

$$\begin{aligned} \left| a_{2}\right| \le \frac{\left| x\right| \sqrt{2\left| x\right| }}{\sqrt{ \begin{array} {ll} \left| \left[ \mu \left( \left( \sigma +2\right) \left( \sigma -3\right) +4\alpha \left( \sigma -1\right) +2\alpha (2\alpha +1)\right) +6\right] x^{2}\right. \\ \left. -2\left( \mu \left( 2\alpha +\sigma -3\right) +2\right) ^{2}\left( 2x^{2}-1\right) \right| \end{array} }}, \\ \left| a_{3}\right| \le \frac{x^{2}}{\left| \mu \left( 2\alpha +\sigma -3\right) +2\right| ^{2}}+\frac{\left| x\right| }{\left| \mu \left( 3\alpha +\sigma -4\right) +3\right| }, \end{aligned}$$

and

$$\begin{aligned} \left| a_{3}-\eta a_{2}^{2}\right| \le \left\{ \begin{array} {ll} \frac{\left| x\right| }{\mu \left( 3\alpha +\sigma -4\right) +3},\\ \\ \frac{2\left| x\right| \left| 1-\eta \right| }{\left| \kappa (x)\right| }, \end{array} \right. \begin{array} {ll} \left| \eta -1\right| \le \frac{\left| \kappa (x)\right| }{2\mu \left( 3\alpha +\sigma -4\right) +6}\\ \\ \left| \eta -1\right| \ge \frac{\left| \kappa (x)\right| }{2\mu \left( 3\alpha +\sigma -4\right) +6}, \end{array} \end{aligned}$$

where

$$\begin{aligned} \kappa (x)= \begin{array} {ll} \left[ \mu \left( \left( \sigma +2\right) \left( \sigma -3\right) +4\alpha \left( \sigma -1\right) +2\alpha (2\alpha +1)\right) +6\right] \\ -2\left( \mu \left( 2\alpha +\sigma -3\right) +2\right) ^{2}\frac{\left( 2x^{2}-1\right) }{x^{2}}. \end{array} \end{aligned}$$

By setting Chebyshev polynomials $U_{n}(x)$ of the second kind instead of Horadam polynomials $h_{n}(x)$ in Theorem 2.1, we obtain the following corollary.

Corollary 2.6

For $\alpha ,\mu \ge 1,$ $\sigma \in {\mathbb {C}}$ and ${\text {Re}}(\sigma )\ge 0$, let $f\in {\mathcal {A}}$ belongs to the class ${\mathbf {b}}_{\Sigma }^{a}(\mu ,\alpha ,\sigma ).$ Then

$$\begin{aligned} \left| a_{2}\right| \le \frac{4\left| x\right| \sqrt{\left| x\right| }}{\sqrt{ \begin{array} {ll} \left| 4\left[ \mu \left( \left( \sigma +2\right) \left( \sigma -3\right) +4\alpha \left( \sigma -1\right) +2\alpha (2\alpha +1)\right) +6\right] x^{2}\right. \\ \left. -2\left( \mu \left( 2\alpha +\sigma -3\right) +2\right) ^{2}\left( 4x^{2}-1\right) \right| \end{array} }}, \\ \left| a_{3}\right| \le \frac{4x^{2}}{\left| \mu \left( 2\alpha +\sigma -3\right) +2\right| ^{2}}+\frac{2\left| x\right| }{\left| \mu \left( 3\alpha +\sigma -4\right) +3\right| }, \end{aligned}$$

and

$$\begin{aligned} \left| a_{3}-\eta a_{2}^{2}\right| \le \left\{ \begin{array} {ll} \frac{\left| 2x\right| }{\mu \left( 3\alpha +\sigma -4\right) +3},\\ \\ \frac{4\left| x\right| \left| 1-\eta \right| }{\left| \kappa (x)\right| }, \end{array} \right. \begin{array} {ll} \left| \eta -1\right| \le \frac{\left| \kappa (x)\right| }{2\mu \left( 3\alpha +\sigma -4\right) +6}\\ \\ \left| \eta -1\right| \ge \frac{\left| \kappa (x)\right| }{2\mu \left( 3\alpha +\sigma -4\right) +6}, \end{array} \end{aligned}$$

where

$$\begin{aligned} \kappa (x)= \begin{array} {ll} \left[ \mu \left( \left( \sigma +2\right) \left( \sigma -3\right) +4\alpha \left( \sigma -1\right) +2\alpha (2\alpha +1)\right) +6\right] \\ -\left( \mu \left( 2\alpha +\sigma -3\right) +2\right) ^{2}\frac{\left( 4x^{2}-1\right) }{2x^{2}}. \end{array} \end{aligned}$$

By setting Pell polynomials $P_{n}(x)$ instead of Horadam polynomials $h_{n}(x)$ in Theorem 2.1, we obtain the following corollary.

Corollary 2.7

For $\alpha ,\mu \ge 1,$ $\sigma \in {\mathbb {C}}$ and ${\text {Re}}(\sigma )\ge 0$, let $f\in {\mathcal {A}}$ belongs to the class ${\mathbf {b}}_{\Sigma }^{a}(\mu ,\alpha ,\sigma ).$ Then

$$\begin{aligned} \left| a_{2}\right| \le \frac{4\left| x\right| \sqrt{\left| x\right| }}{\sqrt{ \begin{array} {ll} \left| 4\left[ \mu \left( \left( \sigma +2\right) \left( \sigma -3\right) +4\alpha \left( \sigma -1\right) +2\alpha (2\alpha +1)\right) +6\right] x^{2}\right. \\ \left. -2\left( \mu \left( 2\alpha +\sigma -3\right) +2\right) ^{2}\left( 4x^{2}+1\right) \right| \end{array} }}, \\ \left| a_{3}\right| \le \frac{4x^{2}}{\left| \mu \left( 2\alpha +\sigma -3\right) +2\right| ^{2}}+\frac{\left| 2x\right| }{\left| \mu \left( 3\alpha +\sigma -4\right) +3\right| }, \end{aligned}$$

and

$$\begin{aligned} \left| a_{3}-\eta a_{2}^{2}\right| \le \left\{ \begin{array} {ll} \frac{\left| 2x\right| }{\mu \left( 3\alpha +\sigma -4\right) +3},\\ \\ \frac{4\left| x\right| \left| 1-\eta \right| }{\left| \kappa (x)\right| }, \end{array} \right. \begin{array} {ll} \left| \eta -1\right| \le \frac{\left| \kappa (x)\right| }{2\mu \left( 3\alpha +\sigma -4\right) +6}\\ \\ \left| \eta -1\right| \ge \frac{\left| \kappa (x)\right| }{2\mu \left( 3\alpha +\sigma -4\right) +6}, \end{array} \end{aligned}$$

where

$$\begin{aligned} \kappa (x)= \begin{array} {ll} \left[ \mu \left( \left( \sigma +2\right) \left( \sigma -3\right) +4\alpha \left( \sigma -1\right) +2\alpha (2\alpha +1)\right) +6\right] \\ -2\left( \mu \left( 2\alpha +\sigma -3\right) +2\right) ^{2}\frac{\left( 4x^{2}+1\right) }{4x^{2}}. \end{array} \end{aligned}$$

By setting Pell-Lucas polynomials $Q_{n}(x)$ instead of Horadam polynomials $h_{n}(x)$ in Theorem 2.1, we obtain the following corollary.

Corollary 2.8

For $\alpha ,\mu \ge 1,$ $\sigma \in {\mathbb {C}}$ and ${\text {Re}}(\sigma )\ge 0$, let $f\in {\mathcal {A}}$ belongs to the class ${\mathbf {b}}_{\Sigma }^{a}(\mu ,\alpha ,\sigma ).$ Then

$$\begin{aligned} \left| a_{2}\right| \le \frac{2\left| x\right| \sqrt{\left| x\right| }}{\sqrt{ \begin{array} {ll} \left| \left[ \mu \left( \left( \sigma +2\right) \left( \sigma -3\right) +4\alpha \left( \sigma -1\right) +2\alpha (2\alpha +1)\right) +6\right] x^{2}\right. \\ \left. -\left( \mu \left( 2\alpha +\sigma -3\right) +2\right) ^{2}\left( 2x^{2}+1\right) \right| \end{array} }}, \\ \left| a_{3}\right| \le \frac{4x^{2}}{\left| \mu \left( 2\alpha +\sigma -3\right) +2\right| ^{2}}+\frac{\left| 2x\right| }{\left| \mu \left( 3\alpha +\sigma -4\right) +3\right| }, \end{aligned}$$

and

$$\begin{aligned} \left| a_{3}-\eta a_{2}^{2}\right| \le \left\{ \begin{array} {ll} \frac{\left| 2x\right| }{\mu \left( 3\alpha +\sigma -4\right) +3},\\ \\ \frac{4\left| x\right| \left| 1-\eta \right| }{\left| \kappa (x)\right| }, \end{array} \right. \begin{array} {ll} \left| \eta -1\right| \le \frac{\left| \kappa (x)\right| }{2\mu \left( 3\alpha +\sigma -4\right) +6}\\ \\ \left| \eta -1\right| \ge \frac{\left| \kappa (x)\right| }{2\mu \left( 3\alpha +\sigma -4\right) +6}, \end{array} \end{aligned}$$

where

$$\begin{aligned} \kappa (x)= \left[ \mu \left( \left( \sigma +2\right) \left( \sigma -3\right) +4\alpha \left( \sigma -1\right) +2\alpha (2\alpha +1)\right) +6\right] \\ -2\left( \mu \left( 2\alpha +\sigma -3\right) +2\right) ^{2}\frac{\left( 4x^{2}+1\right) }{4x^{2}}. \end{aligned}$$

3 Conclusions

In this paper, using the concept of subordination, we have introduced a new subclass of bi-univalent functions in the open unit disk ${\mathbb {U}}$ associated with Horadam polynomials. We have then derived the initial coefficient estimations and also Fekete–Szegö inequalities for functions belonging to this class. Our main results are obtained in Theorem 2.1. Further by specializing the parameters, several consequences of these new families are mentioned.