Applications of Laguerre Polynomials for Bazilevič and θ-Pseudo-Starlike Bi-Univalent Functions Associated with Sakaguchi-Type Functions

Abstract

The aim of the present article is to introduce and investigate a new family ${\mathcal{L}}_{\Sigma }(\delta ,\eta ,\theta ,t;h)$ of normalized holomorphic and bi-univalent functions that involve the Sakaguchi-type Bazilevič functions and Sakaguchi-type $\theta$-pseudo-starlike functions associated with Laguerre polynomials. We obtain estimates on the initial Taylor–Maclaurin coefficients and the Fekete–Szegö problem for functions in this family. Properties of symmetry can be studied for this newly family of functions.

1. Introduction

Let $\mathcal{A}$ denote the set of all holomorphic functions $\mathfrak{s}\left(\varsigma \right)$ in $\mathfrak{D}$ = $\{\varsigma \in \mathbb{C}:|\varsigma |<1\}$ with the series representation

$$\mathfrak{s}\left(\varsigma \right)=\varsigma +\sum _{\mathit{j}=2}^{\infty }{a}_j{\varsigma }^j.$$

(1)

Further, let $\mathcal{S}$ be the sub-collection of $\mathcal{A}$ containing functions satisfying (1) that are univalent in $\mathfrak{D}$.

A function $\mathfrak{s}\in \mathcal{A}$ is said to be a Bazilevič function in $\mathfrak{D}$ if (see [1])

$$\Re \left(\frac{{\varsigma }^{1-\eta }{\mathfrak{s}}^{\prime }\left(\varsigma \right)}{{\left(\mathfrak{s}\left(\varsigma \right)\right)}^{1-\eta }}\right)>0,(\varsigma \in \mathfrak{D};\eta \ge 0).$$

A function $\mathfrak{s}\in \mathcal{A}$ is said to be a $\theta$-pseudo-starlike function in $\mathfrak{D}$ if (see [2])

$$\Re \left(\frac{\varsigma {\left({\mathfrak{s}}^{\prime }\left(\varsigma \right)\right)}^{\theta }}{\mathfrak{s}\left(\varsigma \right)}\right)>0,(\varsigma \in \mathfrak{D};\theta \ge 1).$$

Frasin [3] introduced and studied the family $S(\rho ,s,t)$ consisting of functions $\mathfrak{s}\in \mathcal{A}$ that satisfy the condition

$$\Re \left\{\frac{(s-t)\varsigma {\mathfrak{s}}^{\prime }\left(\varsigma \right)}{\mathfrak{s}\left(s\varsigma \right)-\mathfrak{s}\left(t\varsigma \right)}\right\}>\rho ,$$

where $0\le \rho <1;t,s\in \mathbb{C}\mathrm{with}t\ne s;1\ge \left|t\right|$ and $\varsigma \in \mathfrak{D}$.

According to the Koebe one-quarter theorem (see [4]), every function $\mathfrak{s}\in \mathcal{S}$ has an inverse ${\mathfrak{s}}^{-1}$ such that $\varsigma ={\mathfrak{s}}^{-1}\left(\mathfrak{s}\left(\varsigma \right)\right),$$(\varsigma \in \mathfrak{D})$ and $w=\mathfrak{s}\left({\mathfrak{s}}^{-1}\left(w\right)\right)$, $(\left|w\right|<r_0\left(\mathfrak{s}\right),r_0\left(\mathfrak{s}\right)\ge \frac{1}{4})$. If $\mathfrak{s}$ is of the form (1), then

$$\mathfrak{g}\left(w\right)={\mathfrak{s}}^{-1}\left(w\right)=w-a_2{w}^2+\left(2a_2^2-a_3\right)w^3-\left(5a_2^3-5a_2{a}_3+a_4\right)w^4+\cdots ,\left|w\right|<r_0\left(\mathfrak{s}\right).$$

(2)

A function $\mathfrak{s}\in \mathcal{A}$ is said to be bi-univalent in $\mathfrak{D}$ if both $\mathfrak{s}$ and ${\mathfrak{s}}^{-1}$ are univalent in $\mathfrak{D}$. We denote by $\Sigma$ the set of bi-univalent functions in $\mathfrak{D}$.

In fact, Srivastava et al. [5] have actually revived the study of holomorphic and bi-univalent functions in recent years. This was followed by such works as those by Bulut et al. [6], Ali et al. [7], Srivastava and et al. [8] and others; see for example [9,10,11,12,13,14].

The research of analytical and bi-univalent functions is reintroduced in [5]; previous studies include those of [15,16,17,18]. Several authors introduced new subclasses of bi-univalent functions and obtained bounds for the initial coefficients (see [15,16,17,19,20,21,22,23,24,25]).

It is known that (see [26]) the generalized Laguerre polynomial $L_n^{\gamma }\left(\tau \right)$ is the polynomial solution $\varphi \left(\tau \right)$ of the differential equation:

$$0=\tau {\varphi }^{″}+(\gamma +1-\tau ){\varphi }^{\prime }+\varphi n,$$

n is non-negative integers and $\gamma >-1$.

The generating function of generalized Laguerre polynomial $L_n^{\gamma }\left(\tau \right)$ is given by the following relations:

$$H_{\gamma }\left(\tau ,\varsigma \right)=\sum _{n=0}^{\infty }{L}_n^{\gamma }\left(\tau \right){\varsigma }^n=\frac{e^{-\frac{\tau \varsigma }{1-\varsigma }}}{{\left(1-\varsigma \right)}^{\gamma +1}},$$

(3)

$\varsigma \in \mathfrak{D}$ and $\tau \in \mathbb{R}$. The generalized Laguerre polynomials can be given by the following relations:

$$L_{n+1}^{\gamma }\left(\tau \right)=\frac{\gamma +1-\tau +2n}{1+n}{L}_n^{\gamma }\left(\tau \right)-\frac{\gamma +n}{1+n}{L}_{n-1}^{\gamma }\left(\tau \right)(n\ge 1),$$

with the initial conditions

$$1=L_0^{\gamma }\left(\tau \right),\gamma 1-\tau =L_1^{\gamma }\left(\tau \right)\mathrm{and}{L}_2^{\gamma }\left(\tau \right)=\frac{{\tau }^2}{2}-\tau (\gamma +2)+\frac{(2+\gamma )(1+\gamma )}{2}.$$

(4)

Clearly, if $\gamma =0$, the generalized Laguerre polynomial leads to the simply Laguerre polynomial, i.e., $L_n\left(\tau \right)=L_n^0\left(\tau \right)$.

K.O. Babalola [2] explained the class $L_{\lambda }\left(\beta \right)$ of $\lambda$-pseudo-starlike functions of order $\beta$ and detected that all pseudo starlike functions are Bazilevič of kind $(1-\frac{1}{\lambda })$ order ${\beta }^{\frac{1}{\lambda }}$ and univalent in open unit disk $\mathfrak{D}.$

If we consider the holomorphic functions $\mathfrak{s}$ and $\mathfrak{g}$ in $\mathfrak{D}$, we know that the function $\mathfrak{s}$ is subordinate to $\mathfrak{g}$, if there exists a function w, holomorphic in $\mathfrak{D}$ with $w\left(0\right)=0$, and $\left|w\left(\varsigma \right)\right|<1$, $(\varsigma \in \mathfrak{D})$ such that $\mathfrak{s}\left(\varsigma \right)=\mathfrak{g}\left(w\left(\varsigma \right)\right)$. This subordination is indicated by $\mathfrak{s}\prec \mathfrak{g}$ or $\mathfrak{s}\left(\varsigma \right)\prec \mathfrak{g}\left(\varsigma \right)$$(\varsigma \in \mathfrak{D})$. Furthermore, if the function $\mathfrak{g}$ is univalent in $\mathfrak{D}$, then we have the following equivalence (see [27]), $\mathfrak{s}\left(\varsigma \right)\prec \mathfrak{g}\left(\varsigma \right)⟺\mathfrak{s}\left(0\right)=\mathfrak{g}\left(0\right)and\mathfrak{s}\left(\mathfrak{D}\right)\subset \mathfrak{g}\left(\mathfrak{D}\right)$.

The Fekete–Szego problem is the problem of maximizing the absolute value of the functional $|a_3-\mu a_2^2|.$ The Fekete–Szego inequalities introduced in 1933 (see [28]) preoccupied researchers regarding different classes of univalent functions [29,30,31]. Hence, it is obvious that such inequalities were also obtained regarding bi-univalent functions, and very recently published papers can be cited to support the assertion that the topic still provides interesting results.

2. Main Results

We define in this section a family of functions, denoted by ${\mathcal{L}}_{\Sigma }(\delta ,\eta ,\theta ,t;\mathfrak{h})$ as follows:

Definition 1.

Let $0\leqq \delta \le 1$, $\eta \ge 0$, $\theta \ge 1$, $t\in \mathbb{C}$, $\left|t\right|\le 1$ and $\mathfrak{h}$ be analytic in $\mathfrak{D}$, $\mathfrak{h}\left(0\right)=1$. The family ${\mathcal{L}}_{\Sigma }(\delta ,\eta ,\theta ,t;\mathfrak{h})$ contains all the functions $\mathfrak{s}\in \Sigma$ if it fulfills the subordinations

$$\left(1-\delta \right)\frac{{\left(\varsigma (1-t)\right)}^{1-\eta }{\mathfrak{s}}^{\prime }\left(\varsigma \right)}{{\left(\mathfrak{s}\left(\varsigma \right)-\mathfrak{s}\left(t\varsigma \right)\right)}^{1-\eta }}+\delta \frac{\varsigma (1-t){\left({\mathfrak{s}}^{\prime }\left(\varsigma \right)\right)}^{\theta }}{\mathfrak{s}\left(\varsigma \right)-\mathfrak{s}\left(t\varsigma \right)}\prec \mathfrak{h}\left(\varsigma \right)$$

and

$$\left(1-\delta \right)\frac{{\left((-t+1)w\right)}^{1-\eta }{\mathfrak{g}}^{\prime }\left(w\right)}{{\left(\mathfrak{g}\left(w\right)-\mathfrak{g}\left(tw\right)\right)}^{1-\eta }}+\delta \frac{(-t+1)w{\left({\mathfrak{g}}^{\prime }\left(w\right)\right)}^{\theta }}{\mathfrak{g}\left(w\right)-\mathfrak{g}\left(tw\right)}\prec \mathfrak{h}\left(w\right),$$

where $\mathfrak{g}$ is given by (2).

Remark 1.

The family ${\mathcal{L}}_{\Sigma }(\delta ,\eta ,\theta ,t;\mathfrak{h})$ is a generalization of several known families considered in earlier investigations, which are recalled below.

1.

For $\mathfrak{h}\left(\varsigma \right)={\left(\frac{1+\varsigma }{1-\varsigma }\right)}^{\alpha }$, ${\mathcal{L}}_{\Sigma }(\delta ,\eta ,\theta ,t;\mathfrak{h})$ becomes ${\mathcal{Z}}_{\Sigma }(\delta ,\eta ,\theta ,t;\alpha )$, which was introduced recently by Wanas and Radhi [32].

2.

For $\mathfrak{h}\left(\varsigma \right)=\frac{1+(1-2\beta )\varsigma }{1-\varsigma }$, ${\mathcal{L}}_{\Sigma }(\delta ,\eta ,\theta ,t;\mathfrak{h})$ becomes ${\mathcal{Z}}_{\Sigma }^{*}(\delta ,\eta ,\theta ,t;\beta )$ which was studied by Wanas and Radhi [32].

3.

For $t=0$ and $\mathfrak{h}\left(\varsigma \right)={\left(\frac{1+\varsigma }{1-\varsigma }\right)}^{\alpha }$, ${\mathcal{L}}_{\Sigma }(\delta ,\eta ,\theta ,t;\mathfrak{h})$ becomes ${\mathcal{T}}_{\Sigma }(\delta ,\eta ,\theta ;\alpha ),$ defined recently by Srivastava et al. [33].

4.

For $t=0$ and $\mathfrak{h}\left(\varsigma \right)=\frac{1+(1-2\beta )\varsigma }{1-\varsigma }$, ${\mathcal{L}}_{\Sigma }(\delta ,\eta ,\theta ,t;\mathfrak{h})$ becomes ${\mathcal{T}}_{\Sigma }^{*}(\delta ,\eta ,\theta ;\beta ),$ studied by Srivastava et al. [33].

5.

For $\delta =t=0$ and $\mathfrak{h}\left(\varsigma \right)={\left(\frac{1+\varsigma }{1-\varsigma }\right)}^{\alpha }$, ${\mathcal{L}}_{\Sigma }(\delta ,\eta ,\theta ,t;\mathfrak{h})$ reduces to $P_{\Sigma }(\alpha ,\eta )$ which was investigated recently by Prema and Keerthi [34].

6.

For $\delta =t=0$ and $\mathfrak{h}\left(\varsigma \right)=\frac{1+(1-2\beta )\varsigma }{1-\varsigma }$, ${\mathcal{L}}_{\Sigma }(\delta ,\eta ,\theta ,t;\mathfrak{h})$ reduces to $P_{\Sigma }(\beta ,\eta )$ which was defined recently by Prema and Keerthi [34].

7.

For $\delta =1$, $t=0$ and $\mathfrak{h}\left(\varsigma \right)={\left(\frac{1+\varsigma }{1-\varsigma }\right)}^{\alpha }$, ${\mathcal{L}}_{\Sigma }(\delta ,\eta ,\theta ,t;\mathfrak{h})$ reduces to $\mathcal{L}{B}_{\Sigma }^{\theta }\left(\alpha \right),$ studied by Joshi et al. [35].

8.

For $\delta =1$, $t=0$ and $\mathfrak{h}\left(\varsigma \right)=\frac{1+(1-2\beta )\varsigma }{1-\varsigma }$, ${\mathcal{L}}_{\Sigma }(\delta ,\eta ,\theta ,t;\mathfrak{h})$ reduces to $\mathcal{L}{B}_{\Sigma }(\theta ,\beta ),$ studied recently by Joshi et al. [35].

9.

For $\delta =\eta =t=0$ and $\mathfrak{h}\left(\varsigma \right)={\left(\frac{1+\varsigma }{1-\varsigma }\right)}^{\alpha }$, ${\mathcal{L}}_{\Sigma }(\delta ,\eta ,\theta ,t;\mathfrak{h})$ reduces to $S_{\Sigma }^{*}\left(\alpha \right),$ investigated recently by Brannan and Taha [15].

10.

For $\delta =t=\eta =0$ and $\mathfrak{h}\left(\varsigma \right)=\frac{1+(1-2\beta )\varsigma }{1-\varsigma }$, ${\mathcal{L}}_{\Sigma }(\delta ,\eta ,\theta ,t;\mathfrak{h})$ reduces to $S_{\Sigma }^{*}\left(\beta \right)$ defined recently by Brannan and Taha [15].

11.

For $\eta =1,$$\delta =t=0$ and $\mathfrak{h}\left(\varsigma \right)={\left(\frac{1+\varsigma }{1-\varsigma }\right)}^{\alpha }$, ${\mathcal{L}}_{\Sigma }(\delta ,\eta ,\theta ,t;\mathfrak{h})$ reduces to ${\mathcal{H}}_{\Sigma }^{\alpha }$ introduced recently by Srivastava et al. [5].

12.

For $\delta =t=0$, $\eta =1$ and $\mathfrak{h}\left(\varsigma \right)=\frac{1+(1-2\beta )\varsigma }{1-\varsigma }$, ${\mathcal{L}}_{\Sigma }(\delta ,\eta ,\theta ,t;\mathfrak{h})$ reduces to ${\mathcal{H}}_{\Sigma }\left(\beta \right),$ introduced recently by Srivastava et al. [5].

Theorem 1.

Let $0\leqq \delta \le 1$, $\eta \ge 0$, $\theta \ge 1$, $t\in \mathbb{C}$ and $\left|t\right|\le 1$. If $\mathfrak{s}\in \Sigma$ defined by (1) is in the class ${\mathcal{L}}_{\Sigma }(\delta ,\eta ,\theta ,t;\mathfrak{h})$ where $\mathfrak{h}\left(\varsigma \right)=1+e_1\varsigma +e_2{\varsigma }^2+\cdots ,$ then

$$\left|a_2\right|\le \frac{|e_1|}{(1-\delta )(2-(1-\eta \left)\right(t+1\left)\right)+\delta (2\theta -t-1)}$$

(5)

and

$$\left|a_3\right|\le min\left\{max\left\{\left|\frac{e_1}{M}\right|,\left|\frac{e_2}{M}-\frac{N{e}_1^2}{E^{2}M}\right|\right\},max\left\{\left|\frac{e_1}{M}\right|,\left|\frac{e_2}{M}-\frac{(2M+N)e_1^2}{E^{2}M}\right|\right\}\right\},$$

(6)

where

$$\begin{array}{cc}E\hfill & =(1-\delta )(2-(1-\eta \left)\right(t+1\left)\right)+\delta (2\theta -t-1),\hfill \\ \\ M\hfill & =(1-\delta )\left(3-(1-\eta )\left(t^2+t+1\right)\right)+\delta \left(3\theta -t^2-t-1\right),\hfill \\ \\ N\hfill & =(1-\delta )(1-\eta )(1+t)\left(\frac{1}{2}(1+t)(2-\eta )-2\right)+\delta \left({\left(1+t\right)}^2-2\theta (2+t-\theta )\right).\hfill \end{array}$$

(7)

Proof. 

Suppose that $\mathfrak{s}\in {\mathcal{L}}_{\Sigma }(\delta ,\eta ,\theta ,t;e_1;e_2)$. Then, there exist $\varphi ,\psi :\mathfrak{D}⟶\mathfrak{D}$ two holomorphic functions, where

$$\varphi \left(\varsigma \right)=r_1\varsigma +r_2{\varsigma }^2+r_3{\varsigma }^3+\cdots (\varsigma \in \mathfrak{D})$$

(8)

and

$$\psi \left(w\right)=s_{1}w+s_2{w}^2+s_3{w}^3+\cdots (w\in \mathfrak{D}),$$

(9)

where $0=\varphi \left(0\right)=\psi \left(0\right)$, $1>\left|\psi \left(w\right)\right|$, $1>\left|\varphi \left(\varsigma \right)\right|$, $\varsigma ,w\in \mathfrak{D}$ such that

$$\left(-\delta +1\right)\frac{{\left((-t+1)\varsigma \right)}^{1-\eta }{\mathfrak{s}}^{\prime }\left(\varsigma \right)}{{\left(\mathfrak{s}\left(\varsigma \right)-\mathfrak{s}\left(t\varsigma \right)\right)}^{1-\eta }}+\delta \frac{\varsigma (-t+1){\left({\mathfrak{s}}^{\prime }\left(\varsigma \right)\right)}^{\theta }}{\mathfrak{s}\left(\varsigma \right)-\mathfrak{s}\left(t\varsigma \right)}=1+e_1\varphi \left(\varsigma \right)+e_2{\varphi }^2\left(\varsigma \right)+\cdots$$

(10)

and

$$\left(1-\delta \right)\frac{{\left((-t+1)w\right)}^{-\eta +1}{\mathfrak{g}}^{\prime }\left(w\right)}{{\left(\mathfrak{g}\left(w\right)-\mathfrak{g}\left(tw\right)\right)}^{-\eta +1}}+\delta \frac{(-t+1)w{\left({\mathfrak{g}}^{\prime }\left(w\right)\right)}^{\theta }}{\mathfrak{g}\left(w\right)-\mathfrak{g}\left(tw\right)}=1+e_1\psi \left(w\right)+e_2{\psi }^2\left(w\right)+\cdots .$$

(11)

Combining (8)–(11) yields

$$\left(1-\delta \right)\frac{{\left((-t+1)\varsigma \right)}^{1-\eta }{\mathfrak{s}}^{\prime }\left(\varsigma \right)}{{\left(\mathfrak{s}\left(\varsigma \right)-\mathfrak{s}\left(t\varsigma \right)\right)}^{1-\eta }}+\delta \frac{(1-t)\varsigma {\left({\mathfrak{s}}^{\prime }\left(\varsigma \right)\right)}^{\theta }}{\mathfrak{s}\left(\varsigma \right)-\mathfrak{s}\left(t\varsigma \right)}=1+e_1{r}_1\varsigma +\left[e_1{r}_2+e_2{r}_1^2\right]{\varsigma }^2+\cdots$$

(12)

and

$$\left(1-\delta \right)\frac{{\left((1-t)w\right)}^{1-\eta }{\mathfrak{g}}^{\prime }\left(w\right)}{{\left(\mathfrak{g}\left(w\right)-\mathfrak{g}\left(tw\right)\right)}^{1-\eta }}+\delta \frac{(1-t)w{\left({\mathfrak{g}}^{\prime }\left(w\right)\right)}^{\theta }}{\mathfrak{g}\left(w\right)-\mathfrak{g}\left(tw\right)}=1+e_1{s}_{1}w+\left[e_1{s}_2+e_2{s}_1^2\right]w^2+\cdots .$$

(13)

We know that if $\varsigma ,w\in \mathfrak{D},$$1>\left|\psi \left(w\right)\right|$ and $1>\left|\varphi \left(\varsigma \right)\right|$, we obtain

$$\left|r_j\right|\le 1\mathrm{and}\left|s_j\right|\le 1,j\in \mathbb{N}.$$

(14)

From (12) and (13), after simplifying, we obtain

$$\left[(1-\delta )(2-(1-\eta \left)\right(t+1\left)\right)+\delta (2\theta -t-1)\right]a_2=e_1{r}_1,$$

(15)

$$\begin{array}{cc}\hfill & \left[(1-\delta )\left(3-(1-\eta )\left(1+t^2+t\right)\right)+\delta \left(3\theta -t^2-t-1\right)\right]a_3\hfill \\ \hfill & +\left[(1-\delta )(1+t)(1-\eta )\left(\frac{1}{2}(2-\eta )(1+t)-2\right)+\delta \left({\left(1+t\right)}^2-2\theta (2+t-\theta )\right)\right]a_2^2\hfill \\ \hfill & =e_1{r}_2+e_2{r}_1^2,\hfill \end{array}$$

(16)

$$-\left[(1-\delta )(2-(1-\eta \left)\right(t+1\left)\right)+\delta (2\theta -t-1)\right]a_2=e_1{s}_1$$

(17)

and

$$\begin{array}{cc}\hfill & \left[(1-\delta )\left(3-(1-\eta )\left(1+t^2+t\right)\right)+\delta \left(3\theta -t^2-t-1\right)\right]\left(2a_2^2-a_3\right)\hfill \\ \hfill & +\left[(1-\delta )(t+1)(1-\eta )\left(\frac{1}{2}(2-\eta )(1+t)-2\right)+\delta \left({\left(1+t\right)}^2-2\theta (t-\theta +2)\right)\right]a_2^2\hfill \\ \hfill & =e_1{s}_2+e_2{s}_1^2.\hfill \end{array}$$

(18)

Inequality (5) follows from (15) and (17). If we apply notation (7), then (15) and (16) becomes

$$E{a}_2=e_1{r}_1,M{a}_3+N{a}_2^2=e_1{r}_2+e_2{r}_1^2.$$

(19)

This gives

$$\frac{M}{e_1}{a}_3=r_2+\left(\frac{e_2}{e_1}-\frac{N{e}_1}{E^2}\right)r_1^2,$$

(20)

and using the sharp result ([36], p. 10):

$$|r_2-\mu r_1^2|\le max\left\{1,\left|\mu \right|\right\}$$

(21)

where $\mu \in \mathbb{C}$, we get

$$\left|\frac{M}{e_1}\right|\left|a_3\right|\le max\left\{1,\left|\frac{e_2}{e_1}-\frac{N{e}_1}{E^2}\right|\right\}.$$

(22)

In the same way, (17) and (18) becomes

$$-E{a}_2=e_1{s}_1,M(2a_2^2-a_3)+N{a}_2^2=e_1{s}_2+e_2{s}_1^2.$$

(23)

This gives

$$-\frac{M}{e_1}{a}_3=s_2+\left(\frac{e_2}{e_1}-\frac{(2M+N)e_1}{E^2}\right)s_1^2.$$

(24)

Applying (21), we obtain

$$\left|\frac{M}{e_1}\right|\left|a_3\right|\le max\left\{1,\left|\frac{e_2}{e_1}-\frac{(2M+N)e_1}{E^2}\right|\right\}.$$

(25)

Inequality (6) follows from (22) and (25). □

If we consider the generating function (3) of the generalized Laguerre polynomials $L_n^{\gamma }\left(\tau \right)$ as $h\left(\varsigma \right)$, then by (4), we get $e_1=-\tau +1+\gamma ,$$e_2=\frac{{\tau }^2}{2}-(2+\gamma )\tau +\frac{(2+\gamma )(1+\gamma )}{2}$, and Theorem 1 becomes the next corollary.

Corollary 1.

If ${\mathcal{L}}_{\Sigma }(\delta ,\eta ,\theta ,t;H_{\gamma }\left(\tau ,\varsigma \right))$ contains all $\mathfrak{s}\in \Sigma$ given by (1), then

$$\left|a_2\right|\le \frac{|-\tau +1+\gamma |}{(1-\delta )(2-(1-\eta \left)\right(1+t\left)\right)+\delta (-t+2\theta -1)}$$

and

$$\begin{array}{ccc}\hfill \left|a_3\right|& \le & min\left\{max\left\{\left|\frac{-\tau +1+\gamma }{M}\right|,\left|\frac{\frac{{\tau }^2}{2}-\tau (2+\gamma )+\frac{(2+\gamma )(1+\gamma )}{2}}{M}-\frac{N{\left(-\tau +1+\gamma \right)}^2}{E^{2}M}\right|\right\},\right.\hfill \\ & & \left.max\left\{\left|\frac{-\tau +1+\gamma }{M}\right|,\left|\frac{\frac{{\tau }^2}{2}-\tau (2+\gamma )+\frac{(1+\gamma )\left(2\gamma \right)}{2}}{M}-\frac{(N+2M){\left(-\tau +1+\gamma \right)}^2}{E^{2}M}\right|\right\}\right\},\hfill \end{array}$$

for all $\delta ,\eta ,\theta ,t$ such that $0\leqq \delta \le 1$, $\eta \ge 0$, $\theta \ge 1$, $t\in \mathbb{C}$ and $\left|t\right|\le 1$, where $E,M,N$ are given by (7) and $H_{\gamma }\left(\tau ,\varsigma \right)$ is given by (3).

We discuss in the following theorem the Fekete–Szegö problem for ${\mathcal{L}}_{\Sigma }(\delta ,\eta ,\theta ,t;\mathfrak{h})$.

Theorem 2.

If ${\mathcal{L}}_{\Sigma }(\delta ,\eta ,\theta ,t;\mathfrak{h})$ contains all $\mathfrak{s}\in \Sigma$ of the form (1), then

$$\left|a_3-\xi a_2^2\right|\le |\frac{e_1}{M}|min\left\{max\left\{1,\left|\frac{e_2}{e_1}-\frac{(N-\xi M)e_1}{E^2}\right|\right\},max\left\{1,\left|\frac{e_2}{e_1}-\frac{(2M+N-\xi M)e_1}{E^2}\right|\right\}\right\},$$

(26)

for all $\delta ,\eta ,\theta ,t$ such that $\eta \ge 0$, $0\leqq \delta \le 1$, $\theta \ge 1$, $t\in \mathbb{C}$, $\left|t\right|\le 1,$$\xi \in \mathbb{C}$, where $E,M,N$ are given by (7).

Proof. 

We use the notations given in the proof of Theorem 1. By (19) and (20), we obtain

$$a_3-\xi a_2^2=|\frac{e_1}{M}|\left(r_2+\left(\frac{e_2}{e_1}+\frac{(N+\xi M)e_1}{E^2}\right)r_1^2\right)$$

(27)

and on using the known sharp result $|r_2-\mu r_1^2|\le max\left\{1,\left|\mu \right|\right\}$, we get

$$|a_3-\xi a_2^2|\le |\frac{e_1}{M}|max\left\{1,\left|\frac{e_2}{e_1}-\frac{(N-\xi M)e_1}{E^2}\right|\right\}.$$

(28)

In the same way, from (23) and from (24), we have

$$a_3-\xi a_2^2=-\frac{e_1}{M}\left(s_2+\left(\frac{e_2}{e_1}-\frac{(2M+N-\xi M)e_1}{E^2}\right)s_1^2\right)$$

(29)

and on using $|s_2-\mu s_1^2|\le max\left\{1,\left|\mu \right|\right\}$, we get

$$|a_3-\xi a_2^2|\le |\frac{e_1}{M}|max\left\{1,\left|\frac{e_2}{e_1}-\frac{(2M+N-\xi M)e_1}{E^2}\right|\right\}.$$

(30)

Inequality (26) follows from (28) and (30). □

Corollary 2.

If ${\mathcal{L}}_{\Sigma }(\delta ,\eta ,\theta ,t;H_{\gamma }\left(\tau ,\varsigma \right))$ contains all $\mathfrak{s}\in \Sigma$ given by (1), then

$$\begin{array}{ccc}& & \left|a_3-\xi a_2^2\right|\hfill \\ & \le & \frac{|-\tau +1+\gamma |}{M}min\left\{max\left\{1,\left|\frac{\frac{{\tau }^2}{2}-\tau (2+\gamma )+\frac{(2+\gamma )(1+\gamma )}{2}}{-\tau +1+\gamma }-\frac{(N-\xi M)(-\tau +1+\gamma )}{E^2}\right|\right\},\right.\hfill \\ & & \left.max\left\{1,\left|\frac{\frac{{\tau }^2}{2}-\tau (2+\gamma )+\frac{(2+\gamma )(1+\gamma )}{2}}{-\tau +1+\gamma }-\frac{(N+2M-\xi M)(-\tau +1+\gamma )}{E^2}\right|\right\}\right\},\hfill \end{array}$$

for all $\delta ,\eta ,\theta ,t$ such that $0\leqq \delta \le 1$, $\eta \ge 0$, $\theta \ge 1$, $t\in \mathbb{C}$, $\left|t\right|\le 1$ and $\xi \in \mathbb{C}$, where $E,M,N$ are given by (7) and $H_{\gamma }\left(\tau ,\varsigma \right)$ is given by (3).

3. Conclusions

The purpose of this paper was to introduce and investigate a new family of normalized holomorphic and bi-univalent functions that involve the Sakaguchi type Bazilevič functions and Sakaguchi-type $\theta$-pseudo-starlike functions associated with Laguerre polynomials. This family was denoted by ${\mathcal{L}}_{\Sigma }(\delta ,\eta ,\theta ,t;\mathfrak{h})$. We obtained estimates on the initial Taylor–Maclaurin coefficients and the Fekete–Szegö problem for the functions in this newly defined class. We believe that this paper will motivate a number of researchers to extend this idea for another class of bi-univalent functions.