Applications of Laguerre Polynomials on a New Family of Bi-Prestarlike Functions

Abstract

In this article, we establish the bounds for the initial Taylor–Maclaurin coefficients $|a_2|$ and $|a_3|$ for a new family ${\mathcal{G}}_{\Sigma }(\delta ,\xi ,\lambda ;h)$ of holormorphic and bi-univalent functions which involve the prestarlike functions. Furthermore, for the family functions ${\mathcal{G}}_{\Sigma }(\delta ,\xi ,\lambda ;h)$ we investigate the Fekete–Szegö type inequality, special cases and consequences.

1. Introduction

We indicate by $\mathcal{A}$ the collection of all holomorphic functions of the type

$$\mathfrak{F}\left(z\right)=z+\sum _{n=2}^{\infty }{a}_n{z}^n$$

(1)

in the open unit disc $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$. Further, by $\mathcal{S}$ we shall denote the family of all functions in $\mathcal{A}$ which are univalent in $\mathbb{D}$.

The famous Koebe one-quarter theorem [1] ensures that the image of $\mathbb{D}$ under each univalent function $\mathfrak{F}\in \mathcal{A}$ contain a disk of radius $\frac{1}{4}$. Furthermore, each function $\mathfrak{F}\in \mathcal{S}$ has an inverse ${\mathfrak{F}}^{-1}$ defined by ${\mathfrak{F}}^{-1}\left(\mathfrak{F}\left(z\right)\right)=z$ and

$$\mathfrak{F}\left({\mathfrak{F}}^{-1}\left(w\right)\right)=w,\left(\left|w\right|<r_0\left(\mathfrak{F}\right),r_0\left(\mathfrak{F}\right)\geqq \frac{1}{4}\right)$$

where

$${\mathfrak{F}}^{-1}\left(w\right)=w-a_2{w}^2+(2a_2^2-a_3)w^3-(5a_2^3-5a_2{a}_3+a_4)w^4+\cdots .$$

(2)

A function $\mathfrak{F}\in \mathcal{A}$ is named bi-univalent in $\mathbb{D}$ if both $\mathfrak{F}$ and ${\mathfrak{F}}^{-1}$ are univalent in $\mathbb{D}$. The family of all bi-univalent functions in $\mathbb{D}$ is denoted by $\Sigma$.

In fact, Srivastava et al. [2] have actually revived the study of analytic and bi-univalent functions in recent years, it was followed by such works as those by Ali et al. [3], Bulut et al. [4], Srivastava and et al. [5] and others (see, for example, [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]). From the work of Srivastava et al. [2], we choose to recall the following examples of functions in the family $\Sigma$:

$$\frac{z}{1-z},-log(1-z)\mathrm{and}\frac{1}{2}log\left(\frac{1+z}{1-z}\right).$$

We notice that the family $\Sigma$ is not empty. However, the Koebe function is not a member of $\Sigma .$

The problem to obtain the general coefficient bounds on the Taylor–Maclaurin coefficients

$$|a_n|(n\in \mathbb{N};n\geqq 3)$$

for functions $\mathfrak{F}\in \Sigma$ is still not completely addressed for many of the subfamilies of $\Sigma$. The Fekete–Szegö functional $\left|a_3-\eta a_2^2\right|$ for $\mathfrak{F}\in \mathcal{S}$ is well known for its rich history in the field of Geometric Function Theory. Its origin was in the disproof by Fekete and Szegö [23] of the Littlewood–Paley conjecture that the coefficients of odd univalent functions are bounded by unity. In recent years, many authors obtained Fekete–Szegö inequalities for different classes of functions (see [24,25,26,27,28,29,30]).

Ruscheweyh [31] studied and investigated the family of prestarlike functions of order $\mu$, which are the function $\mathfrak{F}$ such that $\mathfrak{F}\ast I_{\mu }$ is a starlike function of order $\mu$, where

$$I_{\mu }\left(z\right)=\frac{z}{{\left(1-z\right)}^{2(1-\mu )}}(0\leqq \mu <1;z\in \mathbb{D}),$$

and ∗ stands the "Hadamard product". The function $I_{\mu }$ can be written in the form:

$$I_{\mu }\left(z\right)=z+\sum _{n=2}^{\infty }{\phi }_n\left(\mu \right)z^n,$$

where

$${\phi }_n\left(\mu \right)=\frac{{\prod }_{i=2}^n\left(i-2\mu \right)}{(n-1)!},n\geqq 2.$$

We note that ${\phi }_n\left(\mu \right)$ is a decreasing function in $\mu$ and satisfies

$$\underset{n\to \infty }{lim}{\phi }_n\left(\mu \right)=\left\{\begin{array}{c}\infty ,if\mu <\frac{1}{2}\hfill \\ 1,if\mu =\frac{1}{2}\hfill \\ 0,if\mu >\frac{1}{2}\hfill \end{array}\right..$$

With a view to recalling the principle of subordination between holomorphic functions, let the functions $\mathfrak{F}$ and $\zeta$ be holomorphic in $\mathbb{D}$. The function $\mathfrak{F}$ is subordinate to $\zeta$, if there exists a Schwarz function $\omega$, which is analytic in $\mathbb{D}$ with

$$\omega \left(0\right)=0\mathrm{and}\left|\omega \right(z\left)\right|<1(z\in \mathbb{D}),$$

such that

$$\mathfrak{F}\left(z\right)=\zeta \left(\omega \left(z\right)\right).$$

This subordination is denoted by

$$\mathfrak{F}\prec \zeta \mathrm{or}\mathfrak{F}\left(z\right)\prec \zeta \left(z\right)(z\in \mathbb{D}).$$

It is well known that (see [32]), if the function $\zeta$ is univalent in $\mathbb{D}$, then

$$\mathfrak{F}\prec \zeta (z\in \mathbb{D})⟺\mathfrak{F}\left(0\right)=\zeta \left(0\right)\mathrm{and}\mathfrak{F}\left(\mathbb{D}\right)\subseteq \zeta \left(\mathbb{D}\right).$$

The generalized Laguerre polynomial $L_n^{\gamma }\left(\tau \right)$ is the polynomial solution $\varphi \left(\tau \right)$ of the differential Equation (see [33])

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

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

The generating function of generalized Laguerre polynomial $L_n^{\gamma }\left(\tau \right)$ is defined by

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

(3)

where $\tau \in \mathbb{R}$ and $z\in \mathbb{D}$. Generalized Laguerre polynomials can also be defined by the following recurrence relations:

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

with the initial conditions

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

(4)

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

2. Main Results

Indicate by $h\left(z\right)$ the holomorphic function with positive real part in $\mathbb{D}$ such that

$$h\left(0\right)=1,h^{\prime }\left(0\right)>0$$

and $h\left(\mathbb{D}\right)$ is symmetric with respect to real axis, which is of the type:

$$h\left(z\right)=1+e_{1}z+e_2{z}^2+e_3{z}^3+\cdots ,$$

where $h_1>0$.

We now define the family ${\mathcal{G}}_{\Sigma }(\delta ,\xi ,\lambda ;h)$ as follows:

Definition 1.

Assume that$0\leqq \delta \leqq 1$, $0\leqq \xi \leqq 1$, $0\leqq \lambda \leqq 1$and h is analytic in$\mathbb{D}$, $h\left(0\right)=1$. The function$\mathfrak{F}\in \Sigma$is in the family${\mathcal{G}}_{\Sigma }(\delta ,\xi ,\lambda ;h)$if it fulfills the subordinations:

$${\left(\frac{z{\left(\mathfrak{F}\ast I_{\mu }\right)}^{\prime }\left(z\right)}{\left(\mathfrak{F}\ast I_{\mu }\right)\left(z\right)}\right)}^{\delta }{\left[(1-\xi )\frac{z{\left(\mathfrak{F}\ast I_{\mu }\right)}^{\prime }\left(z\right)}{\left(\mathfrak{F}\ast I_{\mu }\right)\left(z\right)}+\xi \left(1+\frac{z{\left(\mathfrak{F}\ast I_{\mu }\right)}^{\prime \prime }\left(z\right)}{{\left(\mathfrak{F}\ast I_{\mu }\right)}^{\prime }\left(z\right)}\right)\right]}^{\lambda }\prec h\left(z\right)$$

and

$${\left(\frac{w{\left({\mathfrak{F}}^{-1}\ast I_{\mu }\right)}^{\prime }\left(w\right)}{\left({\mathfrak{F}}^{-1}\ast I_{\mu }\right)\left(w\right)}\right)}^{\delta }{\left[(1-\xi )\frac{w{\left({\mathfrak{F}}^{-1}\ast I_{\mu }\right)}^{\prime }\left(w\right)}{\left({\mathfrak{F}}^{-1}\ast I_{\mu }\right)\left(w\right)}+\xi \left(1+\frac{w{\left({\mathfrak{F}}^{-1}\ast I_{\mu }\right)}^{\prime \prime }\left(w\right)}{{\left({\mathfrak{F}}^{-1}\ast I_{\mu }\right)}^{\prime }\left(w\right)}\right)\right]}^{\lambda }\prec h\left(w\right),$$

where${\mathfrak{F}}^{-1}$is given by (2).

Remark 1.

The family${\mathcal{G}}_{\Sigma }(\delta ,\xi ,\lambda ;h)$is a generalization of several known families considered in earlier investigations which are being recalled below.

1. 

For$\delta =0$, $\lambda =1$and$\mu =\frac{1}{2}$, we have

$${\mathcal{G}}_{\Sigma }(0,\xi ,1;h)=:{\mathcal{M}}_{\Sigma }(\xi ;h),$$

where the family${\mathcal{M}}_{\Sigma }(\xi ;h)$introduced by Ali et al. [3].

2. 

For$\delta =0$, $\lambda =1$, $\mu =\frac{1}{2}$and$h\left(z\right)={\left(\frac{1+z}{1-z}\right)}^{\alpha }$, $0<\alpha \leqq 1$, we obtain

$${\mathcal{G}}_{\Sigma }\left(0,\xi ,1;{\left(\frac{1+z}{1-z}\right)}^{\alpha }\right)=:M_{\Sigma }(\alpha ,\xi ),$$

where the family$M_{\Sigma }(\alpha ,\xi )$considered by Liu and Wang [34].

3. 

For$\delta =0$, $\lambda =1$, $\mu =\frac{1}{2}$and$h\left(z\right)=\frac{1+(1-2\beta )z}{1-z}$, $0\leqq \beta <1$, we have

$${\mathcal{G}}_{\Sigma }\left(0,\xi ,1;\frac{1+(1-2\beta )z}{1-z}\right)=:M_{\Sigma }\left(\xi \right),$$

where the family$M_{\Sigma }\left(\xi \right)$studied by Liu and Wang [34].

4. 

For$\delta =\xi =0$, $\lambda =1$, $\mu =\frac{1}{2}$and$h\left(z\right)={\left(\frac{1+z}{1-z}\right)}^{\alpha }$, $0<\alpha \leqq 1$, we get

$${\mathcal{G}}_{\Sigma }\left(0,0,1;{\left(\frac{1+z}{1-z}\right)}^{\alpha }\right)=:S_{\Sigma }^{\ast }\left(\alpha \right),$$

where the family$S_{\Sigma }^{\ast }\left(\alpha \right)$considered by Brannan and Taha [35].

5. 

For$\delta =\xi =0$, $\lambda =1$, $\mu =\frac{1}{2}$and$h\left(z\right)=\frac{1+(1-2\beta )z}{1-z}$, $0\leqq \beta <1$, we obtain

$${\mathcal{G}}_{\Sigma }\left(0,0,1;\frac{1+(1-2\beta )z}{1-z}\right)=:S_{\Sigma }^{\ast }\left(\beta \right),$$

where the family $S_{\Sigma }^{\ast }\left(\beta \right)$investigated by Brannan and Taha [35].

6. 

For$\delta =0$, $\lambda =1$, $\mu =\frac{1}{2}$and$h\left(z\right)=\frac{a+(b-a{p}_1)rz}{1-p_{1}rz-q_1{z}^2}+1-a$, $r\in \mathbb{R}$, $a,b,p_1$and$q_1$are real constant, we have

$${\mathcal{G}}_{\Sigma }\left(0,\xi ,1;\frac{a+(b-a{p}_1)rz}{1-p_{1}rz-q_1{z}^2}+1-a\right)=:M_{\Sigma }(\xi ,r),$$

where the family$M_{\Sigma }(\xi ,r)$studied by Abirami et al. [24].

7. 

For$\delta =0$, $\lambda =\xi =1$, $\mu =\frac{1}{2}$and$h\left(z\right)=\frac{a+(b-a{p}_1)rz}{1-p_{1}rz-q_1{z}^2}+1-a$, $r\in \mathbb{R}$, $a,b,p_1$and$q_1$are real constant, we obtain

$${\mathcal{G}}_{\Sigma }\left(0,1,1;\frac{a+(b-a{p}_1)rz}{1-p_{1}rz-q_1{z}^2}+1-a\right)=:{\mathcal{K}}_{\Sigma }\left(r\right),$$

where the family${\mathcal{K}}_{\Sigma }\left(r\right)$introduced by Abirami et al. [24].

8. 

For$\delta =\xi =0$, $\lambda =1$, $\mu =\frac{1}{2}$and$h\left(z\right)=\frac{a+(b-a{p}_1)rz}{1-p_{1}rz-q_1{z}^2}+1-a$, $r\in \mathbb{R}$, $a,b,p_1$and$q_1$are real constant, we obtain

$${\mathcal{G}}_{\Sigma }\left(0,0,1;\frac{a+(b-a{p}_1)rz}{1-p_{1}rz-q_1{z}^2}+1-a\right)=:{\mathcal{W}}_{\Sigma }\left(r\right),$$

where the family ${\mathcal{W}}_{\Sigma }\left(r\right)$defined by Srivastava et al. [17].

9. 

For$\delta =0$, $\lambda =1$, $\mu =\frac{1}{2}$and$h\left(z\right)=\frac{1}{1-2tz+z^2}$, $t\in (\frac{\sqrt{2}}{2},1]$, we have

$${\mathcal{G}}_{\Sigma }\left(0,\xi ,1;\frac{1}{1-2tz+z^2}\right)=:H_{\Sigma }(\xi ,t),$$

where the family$H_{\Sigma }(\xi ,t)$studied by Altınkaya and Yalçin [25].

10. 

For$\delta =\xi =0$, $\lambda =1$, $\mu =\frac{1}{2}$and$h\left(z\right)=\frac{1}{1-2tz+z^2}$, $t\in (\frac{1}{2},1]$, we obtain

$${\mathcal{G}}_{\Sigma }\left(0,0,1;\frac{1}{1-2tz+z^2}\right)=:S_{\Sigma }^{\ast }\left(t\right),$$

where the family $S_{\Sigma }^{\ast }\left(t\right)$introduced by Bulut et al. [4].

Theorem 1.

Assume that$0\leqq \delta \leqq 1$, $0\leqq \xi \leqq 1$and$0\leqq \lambda \leqq 1$. If$\mathfrak{F}\in \Sigma$of the form (1) is in the family${\mathcal{G}}_{\Sigma }(\delta ,\xi ,\lambda ;h)$, with$h\left(z\right)=1+e_{1}z+e_2{z}^2+\cdots$, then

$$\left|a_2\right|\leqq \frac{|e_1|}{2(1-\mu )\left(\delta +\lambda (\xi +1)\right)}$$

(5)

and

$$\begin{array}{c}\left|a_3\right|\leqq min\left\{max\left\{\left|\frac{e_1}{2(1-\mu )(3-2\mu )\left(\delta +\lambda (2\xi +1)\right)}\right|,\right.\right.\hfill \\ \left.\left|\frac{2{\left(\delta +\lambda (\xi +1)\right)}^2{e}_2-{\rm Y}(\delta ,\xi ,\lambda )e_1^2}{4(1-\mu )(3-2\mu )\left(\delta +\lambda (2\xi +1)\right){\left(\delta +\lambda (\xi +1)\right)}^2}\right|\right\},max\left\{\left|\frac{e_1}{2(1-\mu )(3-2\mu )\left(\delta +\lambda (2\xi +1)\right)}\right|,\right.\\ \hfill \left.\left.\left|\frac{2(1-\mu ){\left(\delta +\lambda (\xi +1)\right)}^2{e}_2-\left[2(3-2\mu )\left(\delta +\lambda (2\xi +1)\right)+(1-\mu ){\rm Y}(\delta ,\xi ,\lambda )\right]e_1^2}{4{\left(1-\mu \right)}^2(3-2\mu )\left(\delta +\lambda (2\xi +1)\right){\left(\delta +\lambda (\xi +1)\right)}^2}\right|\right\}\right\},\end{array}$$

(6)

where

$${\rm Y}(\delta ,\xi ,\lambda )=\delta (\delta -1)+\lambda (\xi +1)\left(2\delta +(\lambda -1)(\xi +1)\right)-2\left(\delta +\lambda (3\xi +1)\right).$$

(7)

Proof. 

Suppose that $\mathfrak{F}\in {\mathcal{G}}_{\Sigma }(\delta ,\xi ,\lambda ;h)$. Then there exists two holomorphic functions $\phi ,\chi :\mathbb{D}⟶\mathbb{D}$ given by

$$\phi \left(z\right)=r_{1}z+r_2{z}^2+r_3{z}^3+\cdots (z\in \mathbb{D})$$

(8)

and

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

(9)

with $\varphi \left(0\right)=\psi \left(0\right)=0$, $\left|\phi \left(z\right)\right|<1$, $\left|\chi \left(w\right)\right|<1$, $z,w\in \mathbb{D}$ such that

$$\begin{array}{c}{\left(\frac{z{\left(\mathfrak{F}\ast I_{\mu }\right)}^{\prime }\left(z\right)}{\left(\mathfrak{F}\ast I_{\mu }\right)\left(z\right)}\right)}^{\delta }{\left[(1-\xi )\frac{z{\left(\mathfrak{F}\ast I_{\mu }\right)}^{\prime }\left(z\right)}{\left(\mathfrak{F}\ast I_{\mu }\right)\left(z\right)}+\xi \left(1+\frac{z{\left(\mathfrak{F}\ast I_{\mu }\right)}^{\prime \prime }\left(z\right)}{{\left(\mathfrak{F}\ast I_{\mu }\right)}^{\prime }\left(z\right)}\right)\right]}^{\lambda }\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=1+e_1\phi \left(z\right)+e_2{\phi }^2\left(z\right)+\cdots \end{array}$$

(10)

and

$$\begin{array}{c}{\left(\frac{w{\left({\mathfrak{F}}^{-1}\ast I_{\mu }\right)}^{\prime }\left(w\right)}{\left({\mathfrak{F}}^{-1}\ast I_{\mu }\right)\left(w\right)}\right)}^{\delta }{\left[(1-\xi )\frac{w{\left({\mathfrak{F}}^{-1}\ast I_{\mu }\right)}^{\prime }\left(w\right)}{\left({\mathfrak{F}}^{-1}\ast I_{\mu }\right)\left(w\right)}+\xi \left(1+\frac{w{\left({\mathfrak{F}}^{-1}\ast I_{\mu }\right)}^{\prime \prime }\left(w\right)}{{\left({\mathfrak{F}}^{-1}\ast I_{\mu }\right)}^{\prime }\left(w\right)}\right)\right]}^{\lambda }\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=1+e_1\chi \left(w\right)+e_2{\chi }^2\left(w\right)+\cdots .\end{array}$$

(11)

Combining (8)–(11), yield

$$\begin{array}{c}{\left(\frac{z{\left(\mathfrak{F}\ast I_{\mu }\right)}^{\prime }\left(z\right)}{\left(\mathfrak{F}\ast I_{\mu }\right)\left(z\right)}\right)}^{\delta }{\left[(1-\xi )\frac{z{\left(\mathfrak{F}\ast I_{\mu }\right)}^{\prime }\left(z\right)}{\left(\mathfrak{F}\ast I_{\mu }\right)\left(z\right)}+\xi \left(1+\frac{z{\left(\mathfrak{F}\ast I_{\mu }\right)}^{\prime \prime }\left(z\right)}{{\left(\mathfrak{F}\ast I_{\mu }\right)}^{\prime }\left(z\right)}\right)\right]}^{\lambda }\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=1+e_1{r}_{1}z+\left[e_1{r}_2+e_2{r}_1^2\right]z^2+\cdots \end{array}$$

(12)

and

$$\begin{array}{c}{\left(\frac{w{\left({\mathfrak{F}}^{-1}\ast I_{\mu }\right)}^{\prime }\left(w\right)}{\left({\mathfrak{F}}^{-1}\ast I_{\mu }\right)\left(w\right)}\right)}^{\delta }{\left[(1-\xi )\frac{w{\left({\mathfrak{F}}^{-1}\ast I_{\mu }\right)}^{\prime }\left(w\right)}{\left({\mathfrak{F}}^{-1}\ast I_{\mu }\right)\left(w\right)}+\xi \left(1+\frac{w{\left({\mathfrak{F}}^{-1}\ast I_{\mu }\right)}^{\prime \prime }\left(w\right)}{{\left({\mathfrak{F}}^{-1}\ast I_{\mu }\right)}^{\prime }\left(w\right)}\right)\right]}^{\lambda }\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=1+e_1{s}_{1}w+\left[e_1{s}_2+e_2{s}_1^2\right]w^2+\cdots .\end{array}$$

(13)

It is quite well-known that if $\left|\phi \left(z\right)\right|<1$ and $\left|\chi \left(w\right)\right|<1$, $z,w\in \mathbb{D}$, we get

$$\left|r_j\right|\leqq 1\mathrm{and}\left|s_j\right|\leqq 1(j\in \mathbb{N}).$$

(14)

In the light of (12) and (13), after simplifying, we find that

$$2(1-\mu )\left(\delta +\lambda (\xi +1)\right)a_2=e_1{r}_1,$$

(15)

$$\begin{array}{c}2(1-\mu )(3-2\mu )\left(\delta +\lambda (2\xi +1)\right)a_3\hfill \\ \hfill +2{\left(1-\mu \right)}^2\left[\delta (\delta -1)+\lambda (\xi +1)\left(2\delta +(\lambda -1)(\xi +1)\right)-2\left(\delta +\lambda (3\xi +1)\right)\right]a_2^2=e_1{r}_2+e_2{r}_1^2,\end{array}$$

(16)

$$-2(1-\mu )\left(\delta +\lambda (\xi +1)\right)a_2=e_1{s}_1$$

(17)

and

$$\begin{array}{c}2(1-\mu )(3-2\mu )\left(\delta +\lambda (2\xi +1)\right)\left(2a_2^2-a_3\right)\hfill \\ \hfill +2{\left(1-\mu \right)}^2\left[\delta (\delta -1)+\lambda (\xi +1)\left(2\delta +(\lambda -1)(\xi +1)\right)-2\left(\delta +\lambda (3\xi +1)\right)\right]a_2^2=e_1{s}_2+e_2{s}_1^2.\end{array}$$

(18)

Inequality (5) follows from (15) and (17). In view of (15) and (16), we conclude that

$$\frac{2(1-\mu )(3-2\mu )\left(\delta +\lambda (2\xi +1)\right)}{e_1}{a}_3=r_2+\left(\frac{e_2}{e_1}-\frac{2{\left(1-\mu \right)}^2{\rm Y}(\delta ,\xi ,\lambda )e_1}{4{\left(1-\mu \right)}^2{\left(\delta +\lambda (\xi +1)\right)}^2}\right)r_1^2,$$

(19)

where ${\rm Y}(\delta ,\xi ,\lambda )$ is given by (7). By using the known sharp result ([36], p. 10):

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

(20)

for all $\mu \in \mathbb{C}$, we obtain

$$\left|\frac{2(1-\mu )(3-2\mu )\left(\delta +\lambda (2\xi +1)\right)}{e_1}\right|\left|a_3\right|\leqq max\left\{1,\left|\frac{e_2}{e_1}-\frac{2{\left(1-\mu \right)}^2{\rm Y}(\delta ,\xi ,\lambda )e_1}{4{\left(1-\mu \right)}^2{\left(\delta +\lambda (\xi +1)\right)}^2}\right|\right\}.$$

(21)

It follows from (17) and (18), we deduce that

$$\begin{array}{c}-\frac{2(1-\mu )(3-2\mu )\left(\delta +\lambda (2\xi +1)\right)}{e_1}{a}_3\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=s_2+\left(\frac{e_2}{e_1}-\frac{\left[2(3-2\mu )\left(\delta +\lambda (2\xi +1)\right)+(1-\mu ){\rm Y}(\delta ,\xi ,\lambda )\right]e_1}{2(1-\mu ){\left(\delta +\lambda (\xi +1)\right)}^2}\right)s_1^2.\end{array}$$

(22)

Applying (20), we obtain

$$\begin{array}{c}\left|\frac{2(1-\mu )(3-2\mu )\left(\delta +\lambda (2\xi +1)\right)}{e_1}\right|\left|a_3\right|\leqq max\left\{1,\left|\frac{e_2}{e_1}\right.\right.\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left.\left.-\frac{\left[2(3-2\mu )\left(\delta +\lambda (2\xi +1)\right)+(1-\mu ){\rm Y}(\delta ,\xi ,\lambda )\right]e_1}{2(1-\mu ){\left(\delta +\lambda (\xi +1)\right)}^2}\right|\right\}.\end{array}$$

(23)

Inequality (6) follows from (21) and (23). □

If we take the generating function (3) of the generalized Laguerre polynomials $L_n^{\gamma }\left(\tau \right)$ as $h\left(z\right)$, then from (4), we have $e_1=1+\gamma -\tau$ and $e_2=\frac{{\tau }^2}{2}-(\gamma +2)\tau +\frac{(\gamma +1)(\gamma +2)}{2}$, and Theorem 1 becomes the following corollary

Corollary 1.

If$\mathfrak{F}\in \Sigma$of the form (1) is in the family${\mathcal{G}}_{\Sigma }(\delta ,\xi ,\lambda ;H_{\gamma }\left(\tau ,z\right))$, then

$$\left|a_2\right|\leqq \frac{|1+\gamma -\tau |}{2(1-\mu )\left(\delta +\lambda (\xi +1)\right)}$$

and

$$\begin{array}{c}\left|a_3\right|\leqq min\left\{max\left\{\left|\frac{1+\gamma -\tau }{2(1-\mu )(3-2\mu )\left(\delta +\lambda (2\xi +1)\right)}\right|,\right.\right.\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left.\left|\frac{2{\left(\delta +\lambda (\xi +1)\right)}^2\left(\frac{{\tau }^2}{2}-(\gamma +2)\tau +\frac{(\gamma +1)(\gamma +2)}{2}\right)-{\rm Y}(\delta ,\xi ,\lambda ){\left(1+\gamma -\tau \right)}^2}{4(1-\mu )(3-2\mu )\left(\delta +\lambda (2\xi +1)\right){\left(\delta +\lambda (\xi +1)\right)}^2}\right|\right\},\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}max\left\{\left|\frac{1+\gamma -\tau }{2(1-\mu )(3-2\mu )\left(\delta +\lambda (2\xi +1)\right)}\right|,\right.\\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left.\left.\left|\frac{2(1-\mu ){\left(\delta +\lambda (\xi +1)\right)}^2\left(\frac{{\tau }^2}{2}-(\gamma +2)\tau +\frac{(\gamma +1)(\gamma +2)}{2}\right)-\left[2(3-2\mu )\left(\delta +\lambda (2\xi +1)\right)+(1-\mu ){\rm Y}(\delta ,\xi ,\lambda )\right]{\left(1+\gamma -\tau \right)}^2}{4{\left(1-\mu \right)}^2(3-2\mu )\left(\delta +\lambda (2\xi +1)\right){\left(\delta +\lambda (\xi +1)\right)}^2}\right|\right\}\right\},\end{array}$$

for all$\delta ,\xi ,\lambda ,\gamma ,\mu ,\tau$such that$0\leqq \delta \leqq 1$, $0\leqq \xi \leqq 1$, $0\leqq \lambda \leqq 1$, $\gamma >-1$, $0\leqq \mu <1$and$\tau \in \mathbb{R}$, where$H_{\gamma }\left(\tau ,z\right)$is given by (3).

In the next theorem, we provide the Fekete–Szegö type inequality for the functions of the family ${\mathcal{G}}_{\Sigma }(\delta ,\xi ,\lambda ;h)$.

Theorem 2.

If$\mathfrak{F}\in \Sigma$of the form (1) is in the family${\mathcal{G}}_{\Sigma }(\delta ,\xi ,\lambda ;h)$, then

$$\begin{array}{c}\left|a_3-\eta a_2^2\right|\leqq \left|\frac{e_1}{2(1-\mu )(3-2\mu )\left(\delta +\lambda (2\xi +1)\right)}\right|\times \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times min\left\{max\left\{1,\left|\frac{e_2}{e_1}-\frac{\left[\left(1-\mu \right){\rm Y}(\delta ,\xi ,\lambda )-\eta (3-2\mu )\left(\delta +\lambda (2\xi +1)\right)\right]e_1}{2(1-\mu ){\left(\delta +\lambda (\xi +1)\right)}^2}\right|\right\},\right.\\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left.max\left\{1,\left|\frac{e_2}{e_1}-\frac{\left[\left(1-\mu \right){\rm Y}(\delta ,\xi ,\lambda )-(\eta -2)(3-2\mu )\left(\delta +\lambda (2\xi +1)\right)\right]e_1}{2(1-\mu ){\left(\delta +\lambda (\xi +1)\right)}^2}\right|\right\}\right\},\end{array}$$

(24)

for all$\delta ,\xi ,\lambda ,\mu ,\eta$such that$0\leqq \delta \leqq 1$, $0\leqq \xi \leqq 1$, $0\leqq \lambda \leqq 1$, $0\leqq \mu <1$and$\eta \in \mathbb{C}$.

Proof. 

We apply the notations from the proof of Theorem 1. From (15), (16) and (19), we have

$$\begin{array}{c}{a}_3-\eta a_2^2=\frac{e_1}{2(1-\mu )(3-2\mu )\left(\delta +\lambda (2\xi +1)\right)}\times \hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times \left(r_2+\left(\frac{e_2}{e_1}-\frac{\left[\left(1-\mu \right){\rm Y}(\delta ,\xi ,\lambda )-\eta (3-2\mu )\left(\delta +\lambda (2\xi +1)\right)\right]e_1}{2(1-\mu ){\left(\delta +\lambda (\xi +1)\right)}^2}\right)r_1^2\right)\end{array}$$

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

$$\begin{array}{c}|a_3-\eta a_2^2|\leqq |\frac{e_1}{2(1-\mu )(3-2\mu )\left(\delta +\lambda (2\xi +1)\right)}|\times \hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times max\left\{1,\left|\frac{e_2}{e_1}-\frac{\left[\left(1-\mu \right){\rm Y}(\delta ,\xi ,\lambda )-\eta (3-2\mu )\left(\delta +\lambda (2\xi +1)\right)\right]e_1}{2(1-\mu ){\left(\delta +\lambda (\xi +1)\right)}^2}\right|\right\}.\end{array}$$

(25)

In the same way, from (17), (18) and (22), we conclude that

$$\begin{array}{c}{a}_3-\eta a_2^2=-\frac{e_1}{2(1-\mu )(3-2\mu )\left(\delta +\lambda (2\xi +1)\right)}\left(s_2+\left(\frac{e_2}{e_1}\right.\right.\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left.\left.-\frac{\left[\left(1-\mu \right){\rm Y}(\delta ,\xi ,\lambda )-(\eta -2)(3-2\mu )\left(\delta +\lambda (2\xi +1)\right)\right]e_1}{2(1-\mu ){\left(\delta +\lambda (\xi +1)\right)}^2}\right)s_1^2\right)\end{array}$$

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

$$\begin{array}{c}|a_3-\eta a_2^2|\leqq |\frac{e_1}{2(1-\mu )(3-2\mu )\left(\delta +\lambda (2\xi +1)\right)}|max\left\{1,\left|\frac{e_2}{e_1}\right.\right.\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left.\left.-\frac{\left[\left(1-\mu \right){\rm Y}(\delta ,\xi ,\lambda )-(\eta -2)(3-2\mu )\left(\delta +\lambda (2\xi +1)\right)\right]e_1}{2(1-\mu ){\left(\delta +\lambda (\xi +1)\right)}^2}\right|\right\}.\end{array}$$

(26)

Inequality (24) follows from (25) and (26). □

Corollary 2.

If$\mathfrak{F}\in \Sigma$of the form (1) is in the family${\mathcal{G}}_{\Sigma }(\delta ,\xi ,\lambda ;H_{\gamma }\left(\tau ,z\right))$, then

$$\begin{array}{c}\left|a_3-\eta a_2^2\right|\leqq \left|\frac{1+\gamma -\tau }{2(1-\mu )(3-2\mu )\left(\delta +\lambda (2\xi +1)\right)}\right|\times \hfill \\ min\left\{max\left\{1,\left|\frac{\frac{{\tau }^2}{2}-(\gamma +2)\tau +\frac{(\gamma +1)(\gamma +2)}{2}}{1+\gamma -\tau }-\right.\right.\right.\\ \left.\left.\frac{\left[\left(1-\mu \right){\rm Y}(\delta ,\xi ,\lambda )-\eta (3-2\mu )\left(\delta +\lambda (2\xi +1)\right)\right](1+\gamma -\tau )}{2(1-\mu ){\left(\delta +\lambda (\xi +1)\right)}^2}\right|\right\},\\ max\left\{1,\left|\frac{\frac{{\tau }^2}{2}-(\gamma +2)\tau +\frac{(\gamma +1)(\gamma +2)}{2}}{1+\gamma -\tau }-\right.\right.\\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left.\left.\left.\frac{\left[\left(1-\mu \right){\rm Y}(\delta ,\xi ,\lambda )-(\eta -2)(3-2\mu )\left(\delta +\lambda (2\xi +1)\right)\right](1+\gamma -\tau )}{2(1-\mu ){\left(\delta +\lambda (\xi +1)\right)}^2}\right|\right\}\right\},\end{array}$$

for all$\delta ,\xi ,\lambda ,\gamma ,\mu ,\eta ,\tau$such that$0\leqq \delta \leqq 1$, $0\leqq \xi \leqq 1$, $0\leqq \lambda \leqq 1$, $\gamma >-1$, $0\leqq \mu <1$, $\eta \in \mathbb{C}$and$\tau \in \mathbb{R}$, where$H_{\gamma }\left(\tau ,z\right)$is given by (3).

3. Conclusions

The purpose of our present work is to create a new family ${\mathcal{G}}_{\Sigma }(\delta ,\xi ,\lambda ;h)$ of holormorphic and bi-univalent functions which involve the prestarlike functions and also using the generalized Laguerre polynomials $L_n^{\gamma }\left(\tau \right)$, which are given by the recurrence relation (4) and generating function $H_{\gamma }\left(\tau ,z\right)$ in (3). We derived initial Taylor–Maclaurin coefficient inequalities for functions belonging to this newly introduced bi-univalent function family ${\mathcal{G}}_{\Sigma }(\delta ,\xi ,\lambda ;h)$ and viewed the famous Fekete–Szegö problem.

Symmetry properties for this family of holormorphic and bi-univalent functions can be investigated in the future.