Applications Laguerre Polynomials for Families of Bi-Univalent Functions Defined with (p,q)-Wanas Operator

Abstract

In current manuscript, using Laguerre polynomials and $(p-q)$-Wanas operator, we identify upper bounds $\left|a_2\right|$ and $\left|a_3\right|$ which are first two Taylor-Maclaurin coefficients for a specific bi-univalent functions classes ${\mathcal{W}}_{\sum }(\eta ,\delta ,\lambda ,\sigma ,\theta ,\alpha ,\beta ,p,q;h)$ and ${\mathcal{K}}_{\sum }(\xi ,\rho ,\sigma ,\theta ,\alpha ,\beta ,p,q;h)$ which cover the convex and starlike functions. Also, we discuss Fekete-Szegö type inequality for defined class.

1. Introduction

Denote by $\mathcal{A}$ function collections that have the style:

$$f\left(z\right)=z+\sum _{n=2}^{\infty }{a}_n{z}^n,z\in \mathbb{D},$$

(1)

holomorphic in $\mathbb{D}=\left\{z:\left|z\right|<1\right\}$ in the complex plane $\mathbb{C}$.

Further, present by $\mathcal{S}$ the sub-set of $\mathcal{A}$ including of univalent functions in $\mathbb{D}$ fullfiling (1). Taking account the Koebe $\frac{1}{4}$ theorem (see [1]), each $f\in \mathcal{S}$ has an inverse $f^{-1}$ with the properties $f^{-1}\left(f\left(z\right)\right)=z$, for $z\in \mathbb{D}$ and $f\left(f^{-1}\left(w\right)\right)=w$, with $\left|w\right|<r_0\left(f\right),$ where $r_0\left(f\right)\ge \frac{1}{4}$. If f is of the style (1), then

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

(2)

When f and $f^{-1}$ are univalent functions, $f\in \mathcal{A}$ is bi-univalent in $\mathbb{D}$. The set of bi-univalent functions can be expressed by $\sum$. The work on bi-univalent functions have been brightened by Srivastava et al. [2] in recent years. The following functions can be examplified for functions in the set of bi-univalent.

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

Although Koebe function is not an element of bi-univalent set of functions, the $\sum$ is not null set.

Later, such studies continued by Ali et al. [3], Bulut et al. [4], Srivastava et al. [5] and others (see, for example, [6,7,8,9,10,11,12,13,14,15,16,17,18]). However, non decisive predictions of the $|a_2|$ and $|a_3|$ coefficients in given by (1) were declared in different studies. Generalized inequalities on Taylor-Maclaurin coefficients

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

for $f\in \sum$ has not been totally solved yet for several subfamilies of the $\sum$.

$\left|a_3-\mu a_2^2\right|$ of the Fekete-Szegö function for $f\in \mathcal{S}$ is well-known in the Geometric Function Theory.

Its origin lies in the refutation of the Littlewood-Paley conjecture by Fekete-Szegö [19]. In that case, the coefficients of odd (single-valued) univalent functions are bounded by unity.

Functions have received much attention since then, especially in the investigation of many subclasses of the single-valued function family.

This topic has become very interesting for Geometric Function Theorists (see for example [20,21,22,23,24,25]).

The generator function for Laguerre polynomial $L_n^{\gamma }\left(\tau \right)$ is the polynomial answer $\varphi \left(\tau \right)$ of the differential equation ([26])

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

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

The generating function of generator function for Laguerre polynomial $L_n^{\gamma }\left(\tau \right)$ is expressed as below:

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

(3)

where $\tau \in \mathbb{R}$ and $z\in \mathbb{D}$. The generator function for Laguerre polynomial can also be expressed given below:

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

with the initial terms

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

(4)

Simply, when $\gamma =0$ the generator function for Laguerre polynomial leads to the simply Laguerre polynomial, $L_n^0\left(\tau \right)=L_n\left(\tau \right)$.

Let f and g be holomorphic in $\mathbb{D}$, it is clear that f is subordinate to g, if there occurs a holomorphic function w in $\mathbb{D}$ such that $w\left(0\right)=0$, and $\left|w\left(z\right)\right|<1$, for $z\in \mathbb{D}$ so that $f\left(z\right)=g\left(w\left(z\right)\right)$. This subordination is indicated by $f\prec g$. Moreover, if g is univalent in $\mathbb{D}$, then we have the balance (see [27]), given by $f\left(z\right)\prec g\left(z\right)⟺f\left(\mathbb{D}\right)\subset g\left(\mathbb{D}\right)and$ $f\left(0\right)=g\left(0\right)$.

The $(p,q)$-derivative operator or $(p,q)$-difference operator ($0<q<p\le 1$), for a function f is stated by

$$D_{p,q}f\left(z\right)=\frac{f\left(pz\right)-f\left(qz\right)}{(p-q)z}(z\in {\mathbb{D}}^{*}=\mathbb{D}\setminus \left\{0\right\}),$$

and

$$D_{p,q}f\left(0\right)=f^{\prime }\left(0\right).$$

More information on the subject of $(p,q)$-calculus are founded in [28,29,30,31,32,33].

For $f\in \mathcal{A}$, we conclude that

$$D_{p,q}f\left(z\right)=1+\sum _{n=2}^{\infty }{\left[n\right]}_{p,q}{a}_n{z}^{n-1},$$

where the $(p,q)$-bracket number or twin-basic ${\left[n\right]}_{p,q}$ is showed by

$${\left[n\right]}_{p,q}=\frac{p^n-q^n}{p-q}=p^{n-1}+p^{n-2}q+p^{n-3}{q}^2+\cdots +p{q}^{n-2}+q^{n-1}(p\ne q),$$

which is a native generator number for q, namely is, we get (see [34,35])

$$\underset{p\to 1^{-}}{lim}{\left[n\right]}_{p,q}={\left[n\right]}_q=\frac{1-q^n}{1-q}.$$

Obviously, the impression ${\left[n\right]}_{p,q}$ is symmetric, namely,

$${\left[n\right]}_{p,q}={\left[n\right]}_{q,p}.$$

Wanas and Cotîrlǎ [36] presented $W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }:\mathcal{A}⟶\mathcal{A}$ known as $(p-q)$-Wanas operator showed by

$$W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)=z+\sum _{n=2}^{\infty }{\left(\frac{{\left[{\Psi }_n(\sigma ,\alpha ,\beta )\right]}_{p,q}}{{\left[{\Psi }_1(\sigma ,\alpha ,\beta )\right]}_{p,q}}\right)}^{\theta }{a}_n{z}^n=z+\sum _{n=2}^{\infty }\frac{{\left[{\Psi }_n(\sigma ,\alpha ,\beta )\right]}_{p,q}^{\theta }}{{\left[{\Psi }_1(\sigma ,\alpha ,\beta )\right]}_{p,q}^{\theta }}{a}_n{z}^n,$$

where

$${\Psi }_n(\sigma ,\alpha ,\beta )=\sum _{\tau =1}^{\sigma }\left(\begin{array}{c}\sigma \\ \tau \end{array}\right){(-1)}^{\tau +1}\left({\alpha }^{\tau }+n{\beta }^{\tau }\right),{\Psi }_1(\sigma ,\alpha ,\beta )=\sum _{\tau =1}^{\sigma }\left(\begin{array}{c}\sigma \\ \tau \end{array}\right){(-1)}^{\tau +1}\left({\alpha }^{\tau }+{\beta }^{\tau }\right),$$

and

$$\alpha \in \mathbb{R},\beta \in {\mathbb{R}}_0^{+}\mathrm{with}\alpha +\beta >0,n-1\in \mathbb{N},\sigma \in \mathbb{N},\theta \in {\mathbb{N}}_0,0<q<p\le 1\mathrm{and}z\in \mathbb{D}.$$

Remark 1.

The operator $W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }$ is a generalized form of several operators given in previous researches for some values of parameters which are mentioned below.

1.

For $p=\sigma =\beta =1$, $\theta =-\nu$, $\Re \left(\nu \right)>1$ and $\alpha \in \mathbb{C}\setminus {\mathbb{Z}}_0^{-}$, the operator $W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }$ decreases to the q-Srivastava Attiya operator $J_{q,\alpha }^{\nu }$ [37].

2.

For $p=\sigma =\beta =1$, $\theta =-1$ and $\alpha >-1$, the operator $W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }$ decreases to the q-Bernardi operator [38].

3.

For $p=\sigma =\alpha =\beta =1$ and $\theta =-1$, the operator $W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }$ decreases to the q-Libera operator [38].

4.

For $\alpha =0$ and $p=\sigma =\beta =1$, the operator $W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }$ decreases to the q-Sălăgean operator [39].

5.

For $q⟶1^{-}$ and $p=\sigma =1$, the operator $W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }$ decreases to the operator $I_{\alpha ,\beta }^{\theta }$ was presented and studied by Swamy [40].

6.

For $q⟶1^{-}$, $p=\sigma =\beta =1$, $\theta =-\nu$, $\Re \left(\nu \right)>1$ and $s\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-}$, the operator $W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }$ decreases to the operator $J_{\alpha }^{\nu }$ was presented by Srivastava and Attiya [41]. The operator $J_s^{\nu }$ is well-known as Srivastava-Attiya operator by researchers.

7.

For $q⟶1^{-}$, $p=\sigma =\beta =1$ and $\alpha >-1$, the operator $W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }$, decreases to the operator $I_{\alpha }^{\theta }$ was presented by Cho and Srivastava [42].

8.

For $q⟶1^{-}$, $p=\sigma =\alpha =\beta =1$, the operator $W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }$ decreases to the operator $I^{\theta }$ was presented by Uralegaddi and Somanatha [43].

9.

For $q⟶1^{-}$, $p=\sigma =\alpha =\beta =1$, $\theta =-\xi$ and $\xi >0$, the operator $W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }$ decreases to the operator $I^{\xi }$ was presented by Jung et al. [44]. The operator $I^{\xi }$ is the Jung-Kim-Srivastava integral operator.

10.

For $q⟶1^{-}$, $p=\sigma =\beta =1$, $\theta =-1$ and $\alpha >-1$, the operator $W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }$ decreases to the Bernardi operator [45].

11.

For $q⟶1^{-}$, $\alpha =0$, $p=\sigma =\beta =1$ and $\theta =-1$, the operator $W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }$ decreases to the Alexander operator [46].

12.

For $q⟶1^{-}$, $p=\sigma =1$, $\alpha =1-\beta$ and $t\ge 0$, the operator $W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }$ decreases to the operator $D_{\beta }^{\theta }$ was presented by Al-Oboudi [19].

13.

For $q⟶1^{-}$, $p=\sigma =1$, $\alpha =0$ and $\beta =1$, the operator $W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }$ decreases to the operator $S^{\theta }$ was presented by Sălăgean [47].

2. Main Results

Firstly, We start to present the classes ${\mathcal{W}}_{\sum }(\eta ,\delta ,\lambda ,\sigma ,\theta ,\alpha ,\beta ,p,q;h)$ and ${\mathcal{K}}_{\sum }(\xi ,\rho ,\sigma ,\theta ,$$\alpha ,\beta ,p,q;h)$ given below:

Definition 1.

Suppose that $0\le \eta \le 1$, $0\le \lambda \le 1$, $0\le \delta \le 1$ and h is analytic in $\mathbb{D}$, $h\left(0\right)=1$. $f\in \sum$ is in the class ${\mathcal{W}}_{\sum }(\eta ,\delta ,\lambda ,\sigma ,\theta ,\alpha ,\beta ,p,q;h)$ if it provides the subordinations:

$${\left(\frac{z{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)\right)}^{\prime }}{W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)}\right)}^{\eta }{\left[(1-\delta )\frac{z{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)\right)}^{\prime }}{W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)}+\delta \left(1+\frac{z{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)\right)}^{″}}{{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)\right)}^{\prime }}\right)\right]}^{\lambda }\prec h\left(z\right)$$

and

$${\left(\frac{w{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)\right)}^{\prime }}{W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)}\right)}^{\eta }{\left[(1-\delta )\frac{w{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)\right)}^{\prime }}{W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)}+\delta \left(1+\frac{w{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)\right)}^{″}}{{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)\right)}^{\prime }}\right)\right]}^{\lambda }\prec h\left(w\right),$$

where $f^{-1}$ is given by (2).

Definition 2.

Suppose that $0\le \xi \le 1$, $0\le \rho <1$ and h is analytic in $\mathbb{D}$, $h\left(0\right)=1$. $f\in \sum$ is in the class ${\mathcal{K}}_{\sum }(\xi ,\rho ,\sigma ,\theta ,\alpha ,\beta ,p,q;h)$ if it provides the subordinations:

$$(1-\xi )\frac{z{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)\right)}^{\prime }}{(1-\rho )W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)+\rho z{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)\right)}^{\prime }}$$

$$+\xi \left(\frac{{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)\right)}^{\prime }+z{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)\right)}^{″}}{{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)\right)}^{\prime }+\rho z{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)\right)}^{″}}\right)\prec h\left(z\right)$$

and

$$(1-\xi )\frac{w{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)\right)}^{\prime }}{(1-\rho )W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)+\rho w{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)\right)}^{\prime }}$$

$$+\xi \left(\frac{{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)\right)}^{\prime }+w{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)\right)}^{″}}{{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)\right)}^{\prime }+\rho w{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)\right)}^{″}}\right)\prec h\left(w\right),$$

where $f^{-1}$ is given by (2).

Theorem 1.

Suppose that $0\le \eta \le 1$, $0\le \lambda \le 1$ and $0\le \delta \le 1$. If $f\in \sum$ of the style (1) be an element of class ${\mathcal{W}}_{\sum }(\eta ,\delta ,\lambda ,\sigma ,\theta ,\alpha ,\beta ,p,q;h)$, with $h\left(z\right)=1+e_{1}z+e_2{z}^2+\cdots$, then

$$\left|a_2\right|\le \frac{\left(\eta +\lambda (\delta +1)\right){\left[{\Psi }_2(\sigma ,\alpha ,\beta )\right]}_{p,q}^{\theta }\left|e_1\right|}{{\left[{\Psi }_1(\sigma ,\alpha ,\beta )\right]}_{p,q}^{\theta }}=\frac{|e_1|}{\Omega }$$

and

$$\left|a_3\right|\le min\left\{max\left\{\left|\frac{e_1}{\Delta }\right|,\left|\frac{e_2}{\Delta }-\frac{\phi e_1^2}{{\Omega }^2\Delta }\right|\right\},max\left\{\left|\frac{e_1}{\Delta }\right|,\left|\frac{e_2}{\Delta }-\frac{(2\Delta +\phi )e_1^2}{{\Omega }^2\Delta }\right|\right\}\right\},$$

(5)

where

$$\begin{array}{cc}\Omega \hfill & =\frac{\left(\eta +\lambda (\delta +1)\right){\left[{\Psi }_2(\sigma ,\alpha ,\beta )\right]}_{p,q}^{\theta }}{{\left[{\Psi }_1(\sigma ,\alpha ,\beta )\right]}_{p,q}^{\theta }},\hfill \\ \hfill & \hfill & \\ \Delta \hfill & =\frac{2\left(\eta +\lambda (2\delta +1)\right){\left[{\Psi }_3(\sigma ,\alpha ,\beta )\right]}_{p,q}^{\theta }}{{\left[{\Psi }_1(\sigma ,\alpha ,\beta )\right]}_{p,q}^{\theta }},\hfill \\ \hfill & \hfill & \\ \phi \hfill & =\frac{\left[\eta (\eta -1)+\lambda (\delta +1)\left(2\eta +(\lambda -1)(\delta +1)\right)-2\left(\eta +\lambda (3\delta +1)\right)\right]{\left[{\Psi }_2(\sigma ,\alpha ,\beta )\right]}_{p,q}^{2\theta }}{2{\left[{\Psi }_1(\sigma ,\alpha ,\beta )\right]}_{p,q}^{2\theta }}.\hfill \end{array}$$

(6)

Proof. 

Assume that $f\in {\mathcal{W}}_{\sum }(\eta ,\delta ,\lambda ,\sigma ,\theta ,\alpha ,\beta ,p,q;e_1;e_2)$. Then there consists two holomorphic functions $\varphi ,\psi :\mathbb{D}⟶\mathbb{D}$ showed by

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

(7)

and

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

(8)

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

$$\begin{array}{cc}\hfill & {\left(\frac{z{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)\right)}^{\prime }}{W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)}\right)}^{\eta }{\left[(1-\delta )\frac{z{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)\right)}^{\prime }}{W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)}+\delta \left(1+\frac{z{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)\right)}^{″}}{{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)\right)}^{\prime }}\right)\right]}^{\lambda }\hfill \\ \hfill & =1+e_1\varphi \left(z\right)+e_2{\varphi }^2\left(z\right)+\cdots \hfill \end{array}$$

(9)

and

$$\begin{array}{cc}\hfill & {\left(\frac{w{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)\right)}^{\prime }}{W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)}\right)}^{\eta }{\left[(1-\delta )\frac{w{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)\right)}^{\prime }}{W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)}+\delta \left(1+\frac{w{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)\right)}^{″}}{{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)\right)}^{\prime }}\right)\right]}^{\lambda }\hfill \\ \hfill & =1+e_1\psi \left(w\right)+e_2{\psi }^2\left(w\right)+\cdots .\hfill \end{array}$$

(10)

Unification of (7), (8), (9) and (10), yield

$$\begin{array}{cc}\hfill & {\left(\frac{z{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)\right)}^{\prime }}{W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)}\right)}^{\eta }{\left[(1-\delta )\frac{z{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)\right)}^{\prime }}{W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)}+\delta \left(1+\frac{z{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)\right)}^{″}}{{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)\right)}^{\prime }}\right)\right]}^{\lambda }\hfill \\ \hfill & =1+e_1{r}_{1}z+\left[e_1{r}_2+e_2{r}_1^2\right]z^2+\cdots \hfill \end{array}$$

(11)

and

$$\begin{array}{cc}\hfill & {\left(\frac{w{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)\right)}^{\prime }}{W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)}\right)}^{\eta }{\left[(1-\delta )\frac{w{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)\right)}^{\prime }}{W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)}+\delta \left(1+\frac{w{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)\right)}^{″}}{{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)\right)}^{\prime }}\right)\right]}^{\lambda }\hfill \\ \hfill & =1+e_1{s}_{1}w+\left[e_1{s}_2+e_2{s}_1^2\right]w^2+\cdots .\hfill \end{array}$$

(12)

It is clear that if $\left|\varphi \left(z\right)\right|<1$ and $\left|\psi \left(w\right)\right|<1$, $z,w\in \mathbb{D}$, we obtain

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

Taking into account (11) and (12), after simplifying, we find that

$$\frac{\left(\eta +\lambda (\delta +1)\right){\left[{\Psi }_2(\sigma ,\alpha ,\beta )\right]}_{p,q}^{\theta }}{{\left[{\Psi }_1(\sigma ,\alpha ,\beta )\right]}_{p,q}^{\theta }}{a}_2=e_1{r}_1,$$

(13)

$$\begin{array}{cc}\hfill & \frac{2\left(\eta +\lambda (2\delta +1)\right){\left[{\Psi }_3(\sigma ,\alpha ,\beta )\right]}_{p,q}^{\theta }}{{\left[{\Psi }_1(\sigma ,\alpha ,\beta )\right]}_{p,q}^{\theta }}{a}_3\hfill \\ \hfill & +\frac{\left[\eta (\eta -1)+\lambda (\delta +1)\left(2\eta +(\lambda -1)(\delta +1)\right)-2\left(\eta +\lambda (3\delta +1)\right)\right]{\left[{\Psi }_2(\sigma ,\alpha ,\beta )\right]}_{p,q}^{2\theta }}{2{\left[{\Psi }_1(\sigma ,\alpha ,\beta )\right]}_{p,q}^{2\theta }}{a}_2^2\hfill \\ \hfill & =e_1{r}_2+e_2{r}_1^2,\hfill \end{array}$$

(14)

$$-\frac{\left(\eta +\lambda (\delta +1)\right){\left[{\Psi }_2(\sigma ,\alpha ,\beta )\right]}_{p,q}^{\theta }}{{\left[{\Psi }_1(\sigma ,\alpha ,\beta )\right]}_{p,q}^{\theta }}{a}_2=e_1{s}_1$$

(15)

and

$$\begin{array}{cc}\hfill & \frac{2\left(\eta +\lambda (2\delta +1)\right){\left[{\Psi }_3(\sigma ,\alpha ,\beta )\right]}_{p,q}^{\theta }}{{\left[{\Psi }_1(\sigma ,\alpha ,\beta )\right]}_{p,q}^{\theta }}\left(2a_2^2-a_3\right)\hfill \\ \hfill & +\frac{\left[\eta (\eta -1)+\lambda (\delta +1)\left(2\eta +(\lambda -1)(\delta +1)\right)-2\left(\eta +\lambda (3\delta +1)\right)\right]{\left[{\Psi }_2(\sigma ,\alpha ,\beta )\right]}_{p,q}^{2\theta }}{2{\left[{\Psi }_1(\sigma ,\alpha ,\beta )\right]}_{p,q}^{2\theta }}{a}_2^2\hfill \\ \hfill & =e_1{s}_2+e_2{s}_1^2.\hfill \end{array}$$

(16)

If we implement notation (6), then (13) and (14) becomes

$$\Omega a_2=e_1{r}_1,\Delta a_3+\phi a_2^2=e_1{r}_2+e_2{r}_1^2.$$

(17)

This gives

$$\frac{\Delta }{e_1}{a}_3=r_2+\left(\frac{e_2}{e_1}-\frac{\phi e_1}{{\Omega }^2}\right)r_1^2,$$

(18)

and on using the given certain result ([48], p. 10):

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

(19)

for every $\mu \in \mathbb{C}$, we get

$$\left|\frac{\Delta }{e_1}\right|\left|a_3\right|\le max\left\{1,\left|\frac{e_2}{e_1}-\frac{\phi e_1}{{\Omega }^2}\right|\right\}.$$

(20)

In the same way, (15) and (16) becomes

$$-\Omega a_2=e_1{s}_1,\Delta (2a_2^2-a_3)+\phi a_2^2=e_1{s}_2+e_2{s}_1^2.$$

(21)

This gives

$$-\frac{\Delta }{e_1}{a}_3=s_2+\left(\frac{e_2}{e_1}-\frac{(2\Delta +\phi )e_1}{{\Omega }^2}\right)s_1^2.$$

(22)

Applying (19), we obtain

$$\left|\frac{\Delta }{e_1}\right|\left|a_3\right|\le max\left\{1,\left|\frac{e_2}{e_1}-\frac{(2\Delta +\phi )e_1}{{\Omega }^2}\right|\right\}.$$

(23)

Inequality (5) follows from (20) and (23). □

If we take the generating function $L_n^{\gamma }\left(\tau \right)$ given by (3) common generalized Laguerre polynomials as $h\left(z\right)$, then from the equalities given (4), we get $e_1=1+\gamma -\tau$ and $e_2=\frac{{\tau }^2}{2}-(\gamma +2)\tau +\frac{(\gamma +1)(\gamma +2)}{2}$. We obtain following corollary from Theorem 1.

Corollary 1.

If $f\in \sum$ given by style (1) is in the family ${\mathcal{W}}_{\sum }(\eta ,\delta ,\lambda ,\sigma ,\theta ,\alpha ,\beta ,p,q;H_{\gamma }\left(\tau ,z\right))$, then

$$\left|a_2\right|\le \frac{\left(\eta +\lambda (\delta +1)\right){\left[{\Psi }_2(\sigma ,\alpha ,\beta )\right]}_{p,q}^{\theta }|1+\gamma -\tau |}{{\left[{\Psi }_1(\sigma ,\alpha ,\beta )\right]}_{p,q}^{\theta }}=\frac{|1+\gamma -\tau |}{\Omega }$$

and

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

for all $\eta ,\lambda ,\delta$ so that $0\le \eta \le 1$, $0\le \lambda \le 1$ and $0\le \delta \le 1$, where $\Omega ,\Delta ,\phi$ are given by (6) and $H_{\gamma }\left(\tau ,z\right)$ is given by (3).

Theorem 2.

Suppose that $0\le \xi \le 1$ and $0\le \rho <1$. If $f\in \sum$ of the style (1) be an element of the class ${\mathcal{K}}_{\sum }(\xi ,\rho ,\sigma ,\theta ,\alpha ,\beta ,p,q;h)$, with $h\left(z\right)=1+e_{1}z+e_2{z}^2+\cdots$, then

$$\left|a_2\right|\le \frac{(\xi +1)(1-\rho ){\left[{\Psi }_2(\sigma ,\alpha ,\beta )\right]}_{p,q}^{\theta }\left|e_1\right|}{{\left[{\Psi }_1(\sigma ,\alpha ,\beta )\right]}_{p,q}^{\theta }}=\frac{|e_1|}{{\rm Y}}$$

and

$$\left|a_3\right|\le min\left\{max\left\{\left|\frac{e_1}{\Phi }\right|,\left|\frac{e_2}{\Phi }-\frac{\chi e_1^2}{{{\rm Y}}^2\Phi }\right|\right\},max\left\{\left|\frac{e_1}{\Phi }\right|,\left|\frac{e_2}{\Phi }-\frac{(2\Phi +\chi )e_1^2}{{{\rm Y}}^2\Phi }\right|\right\}\right\},$$

(24)

where

$$\begin{array}{cc}{\rm Y}\hfill & =\frac{(\xi +1)(1-\rho ){\left[{\Psi }_2(\sigma ,\alpha ,\beta )\right]}_{p,q}^{\theta }}{{\left[{\Psi }_1(\sigma ,\alpha ,\beta )\right]}_{p,q}^{\theta }},\hfill \\ \hfill & \hfill & \\ \Phi \hfill & =\frac{2(2\xi +1)(1-\rho ){\left[{\Psi }_3(\sigma ,\alpha ,\beta )\right]}_{p,q}^{\theta }}{{\left[{\Psi }_1(\sigma ,\alpha ,\beta )\right]}_{p,q}^{\theta }},\hfill \\ \hfill & \hfill & \\ \chi \hfill & =\frac{(2\xi +1)({\rho }^2-1){\left[{\Psi }_2(\sigma ,\alpha ,\beta )\right]}_{p,q}^{2\theta }}{{\left[{\Psi }_1(\sigma ,\alpha ,\beta )\right]}_{p,q}^{2\theta }}.\hfill \end{array}$$

(25)

Proof. 

Assume that $f\in {\mathcal{K}}_{\sum }(\xi ,\rho ,\sigma ,\theta ,\alpha ,\beta ,p,q;e_1;e_2)$. Then there consists two holomorphic functions $\varphi ,\psi :\mathbb{D}⟶\mathbb{D}$ such that

$$\begin{array}{cc}\hfill & (1-\xi )\frac{z{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)\right)}^{\prime }}{(1-\rho )W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)+\rho z{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)\right)}^{\prime }}+\xi \left(\frac{{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)\right)}^{\prime }+z{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)\right)}^{″}}{{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)\right)}^{\prime }+\rho z{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)\right)}^{″}}\right)\hfill \\ \hfill & =1+e_1\varphi \left(z\right)+e_2{\varphi }^2\left(z\right)+\cdots \hfill \end{array}$$

(26)

and

$$\begin{array}{cc}\hfill & (1-\xi )\frac{w{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)\right)}^{\prime }}{(1-\rho )W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)+\rho w{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)\right)}^{\prime }}+\xi \left(\frac{{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)\right)}^{\prime }+w{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)\right)}^{″}}{{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)\right)}^{\prime }+\rho w{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)\right)}^{″}}\right)\hfill \\ \hfill & =1+e_1\psi \left(w\right)+e_2{\psi }^2\left(w\right)+\cdots ,\hfill \end{array}$$

(27)

where $\varphi$ and $\psi$ given by the style (7) and (8). Unification of (26) and (27), serve

$$\begin{array}{cc}\hfill & (1-\xi )\frac{z{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)\right)}^{\prime }}{(1-\rho )W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)+\rho z{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)\right)}^{\prime }}+\xi \left(\frac{{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)\right)}^{\prime }+z{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)\right)}^{″}}{{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)\right)}^{\prime }+\rho z{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }f\left(z\right)\right)}^{″}}\right)\hfill \\ \hfill & =1+e_1{r}_{1}z+\left[e_1{r}_2+e_2{r}_1^2\right]z^2+\cdots \hfill \end{array}$$

(28)

and

$$\begin{array}{cc}\hfill & (1-\xi )\frac{w{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)\right)}^{\prime }}{(1-\rho )W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)+\rho w{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)\right)}^{\prime }}+\xi \left(\frac{{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)\right)}^{\prime }+w{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)\right)}^{″}}{{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)\right)}^{\prime }+\rho w{\left(W_{\alpha ,\beta ,p,q}^{\sigma ,\theta }{f}^{-1}\left(w\right)\right)}^{″}}\right)\hfill \\ \hfill & =1+e_1{s}_{1}w+\left[e_1{s}_2+e_2{s}_1^2\right]w^2+\cdots .\hfill \end{array}$$

(29)

It is clear that if $\left|\varphi \left(z\right)\right|<1$ and $\left|\psi \left(w\right)\right|<1$, $z,w\in \mathbb{D}$, we obtain

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

Taking into account (28) and (29), after simplifying, we find that

$$\frac{(\xi +1)(1-\rho ){\left[{\Psi }_2(\sigma ,\alpha ,\beta )\right]}_{p,q}^{\theta }}{{\left[{\Psi }_1(\sigma ,\alpha ,\beta )\right]}_{p,q}^{\theta }}{a}_2=e_1{r}_1,$$

(30)

$$\frac{2(2\xi +1)(1-\rho ){\left[{\Psi }_3(\sigma ,\alpha ,\beta )\right]}_{p,q}^{\theta }}{{\left[{\Psi }_1(\sigma ,\alpha ,\beta )\right]}_{p,q}^{\theta }}{a}_3+\frac{(2\xi +1)({\rho }^2-1){\left[{\Psi }_2(\sigma ,\alpha ,\beta )\right]}_{p,q}^{2\theta }}{{\left[{\Psi }_1(\sigma ,\alpha ,\beta )\right]}_{p,q}^{2\theta }}{a}_2^2=e_1{r}_2+e_2{r}_1^2,$$

(31)

$$-\frac{(\xi +1)(1-\rho ){\left[{\Psi }_2(\sigma ,\alpha ,\beta )\right]}_{p,q}^{\theta }}{{\left[{\Psi }_1(\sigma ,\alpha ,\beta )\right]}_{p,q}^{\theta }}{a}_2=e_1{s}_1$$

(32)

and

$$\begin{array}{cc}\hfill & \frac{2(2\xi +1)(1-\rho ){\left[{\Psi }_3(\sigma ,\alpha ,\beta )\right]}_{p,q}^{\theta }}{{\left[{\Psi }_1(\sigma ,\alpha ,\beta )\right]}_{p,q}^{\theta }}\left(2a_2^2-a_3\right)+\frac{(2\xi +1)({\rho }^2-1){\left[{\Psi }_2(\sigma ,\alpha ,\beta )\right]}_{p,q}^{2\theta }}{{\left[{\Psi }_1(\sigma ,\alpha ,\beta )\right]}_{p,q}^{2\theta }}{a}_2^2\hfill \\ \hfill & =e_1{s}_2+e_2{s}_1^2.\hfill \end{array}$$

(33)

If we implement notation (25), then (30) and (31) becomes

$${\rm Y}{a}_2=e_1{r}_1,\Phi a_3+\chi a_2^2=e_1{r}_2+e_2{r}_1^2.$$

(34)

This gives

$$\frac{\Phi }{e_1}{a}_3=r_2+\left(\frac{e_2}{e_1}-\frac{\chi e_1}{{{\rm Y}}^2}\right)r_1^2,$$

(35)

and on using the given certain result ([48], p. 10):

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

(36)

for every $\mu \in \mathbb{C}$, we get

$$\left|\frac{\Phi }{e_1}\right|\left|a_3\right|\le max\left\{1,\left|\frac{e_2}{e_1}-\frac{\chi e_1}{{{\rm Y}}^2}\right|\right\}.$$

(37)

In the same way, (32) and (33) becomes

$$-{\rm Y}{a}_2=e_1{s}_1,\Phi (2a_2^2-a_3)+\chi a_2^2=e_1{s}_2+e_2{s}_1^2.$$

(38)

This gives

$$-\frac{\Phi }{e_1}{a}_3=s_2+\left(\frac{e_2}{e_1}-\frac{(2\Phi +\chi )e_1}{{{\rm Y}}^2}\right)s_1^2.$$

(39)

Applying (36), we obtain

$$\left|\frac{\Phi }{e_1}\right|\left|a_3\right|\le max\left\{1,\left|\frac{e_2}{e_1}-\frac{(2\Phi +\chi )e_1}{{{\rm Y}}^2}\right|\right\}.$$

(40)

Inequality (24) follows from (37) and (40). □

If we take the generating function $L_n^{\gamma }\left(\tau \right)$ given by (3) common generalized Laguerre polynomials as $h\left(z\right)$, then from the equalities given (4), we get $e_1=1+\gamma -\tau$ and $e_2=\frac{{\tau }^2}{2}-(\gamma +2)\tau +\frac{(\gamma +1)(\gamma +2)}{2}$. We obtain following corollary from Theorem 2.

Corollary 2.

If $f\in \sum$ of the style (1) be an element of the class ${\mathcal{K}}_{\sum }(\xi ,\rho ,\sigma ,\theta ,\alpha ,\beta ,p,q;H_{\gamma }\left(\tau ,z\right))$, then

$$\left|a_2\right|\le \frac{(\xi +1)(1-\rho ){\left[{\Psi }_2(\sigma ,\alpha ,\beta )\right]}_{p,q}^{\theta }|1+\gamma -\tau |}{{\left[{\Psi }_1(\sigma ,\alpha ,\beta )\right]}_{p,q}^{\theta }}=\frac{|1+\gamma -\tau |}{{\rm Y}}$$

and

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

for all $\xi ,\rho$ so that $0\le \xi \le 1$ and $0\le \rho <1$, where ${\rm Y},\Phi ,\chi$ are introduced by (25) and $H_{\gamma }\left(\tau ,z\right)$ is given by (3).

We investigate the “Fekete-Szegö Inequalities” for the families ${\mathcal{W}}_{\sum }(\eta ,\delta ,\lambda ,\sigma ,\theta ,\alpha ,\beta ,p,q;h)$ and ${\mathcal{K}}_{\sum }(\xi ,\rho ,\sigma ,\theta ,\alpha ,\beta ,p,q;h)$ in next theorems.

Theorem 3.

If $f\in \sum$ of the style (1) be an element of family ${\mathcal{W}}_{\sum }(\eta ,\delta ,\lambda ,\sigma ,\theta ,\alpha ,\beta ,p,q;h)$, then

$$\left|a_3-\zeta a_2^2\right|\le \frac{|e_1|}{\Delta }min\left\{max\left\{1,\left|\frac{e_2}{e_1}+\frac{(\zeta \Delta -\phi )e_1}{{\Omega }^2}\right|\right\},max\left\{1,\left|\frac{e_2}{e_1}-\frac{(2\Delta +\phi -\zeta \Delta )e_1}{{\Omega }^2}\right|\right\}\right\},$$

for all $\zeta ,\eta ,\lambda ,\delta$ such that $\zeta \in \mathbb{R}$, $0\le \eta \le 1$, $0\le \lambda \le 1$ and $0\le \delta \le 1$, where $\Omega ,\Delta ,\phi$ are given by (6) and $e_1$, $e_2$, $a_2$ and $a_3$ as defined in Theorem 1.

Proof. 

We implement the impressions from the Theorem 1’s proof. From (17) and from (18), we get

$$a_3-\zeta a_2^2=\frac{e_1}{\Delta }\left(r_2+\left(\frac{e_2}{e_1}+\frac{(\zeta \Delta -\phi )e_1}{{\Omega }^2}\right)r_1^2\right)$$

by using the certain result $|r_2-\mu r_1^2|\le max\left\{1,\left|\mu \right|\right\}$, we get

$$|a_3-\zeta a_2^2|\le \frac{|e_1|}{\Delta }max\left\{1,\left|\frac{e_2}{e_1}+\frac{(\zeta \Delta -\phi )e_1}{{\Omega }^2}\right|\right\}.$$

In the same way, from (21) and from (22), we get

$$a_3-\zeta a_2^2=-\frac{e_1}{\Delta }\left(s_2+\left(\frac{e_2}{e_1}-\frac{(2\Delta +\phi -\zeta \Delta )e_1}{{\Omega }^2}\right)s_1^2\right)$$

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

$$|a_3-\zeta a_2^2|\le \frac{|e_1|}{\Delta }max\left\{1,\left|\frac{e_2}{e_1}-\frac{(2\Delta +\phi -\zeta \Delta )e_1}{{\Omega }^2}\right|\right\}.$$

□

Corollary 3.

If $f\in \sum$ of the style (1) be an element of ${\mathcal{W}}_{\sum }(\eta ,\delta ,\lambda ,\sigma ,\theta ,\alpha ,\beta ,p,q;H_{\gamma }\left(\tau ,z\right))$, then

$$\begin{array}{ccc}& & \left|a_3-\zeta a_2^2\right|\hfill \\ & \le & \frac{|1+\gamma -\tau |}{\Delta }min\left\{max\left\{1,\left|\frac{\frac{{\tau }^2}{2}-(\gamma +2)\tau +\frac{(\gamma +1)(\gamma +2)}{2}}{1+\gamma -\tau }+\frac{(\zeta \Delta -\phi )(1+\gamma -\tau )}{{\Omega }^2}\right|\right\},\right.\hfill \\ & & \left.max\left\{1,\left|\frac{\frac{{\tau }^2}{2}-(\gamma +2)\tau +\frac{(\gamma +1)(\gamma +2)}{2}}{1+\gamma -\tau }-\frac{(2\Delta +\phi -\zeta \Delta )(1+\gamma -\tau )}{{\Omega }^2}\right|\right\}\right\},\hfill \end{array}$$

for each $\zeta ,\eta ,\lambda ,\delta$ such that $\zeta \in \mathbb{R}$, $0\le \eta \le 1$, $0\le \lambda \le 1$ and $0\le \delta \le 1$, where $\Omega ,\Delta ,\phi$ are given by (6) and $H_{\gamma }\left(\tau ,z\right)$ is presented by (3).

Theorem 4.

If $f\in \sum$ of the style (1) is in the family ${\mathcal{K}}_{\sum }(\xi ,\rho ,\sigma ,\theta ,\alpha ,\beta ,p,q;h)$, then

$$\left|a_3-\zeta a_2^2\right|\le \frac{|e_1|}{\Phi }min\left\{max\left\{1,\left|\frac{e_2}{e_1}+\frac{(\zeta \Phi -\chi )e_1}{{{\rm Y}}^2}\right|\right\},max\left\{1,\left|\frac{e_2}{e_1}-\frac{(2\Phi +\chi -\zeta \Phi )e_1}{{{\rm Y}}^2}\right|\right\}\right\},$$

for all $\zeta ,\xi ,\rho$ such that $\zeta \in \mathbb{R}$, $0\le \xi \le 1$ and $0\le \rho <1$, where ${\rm Y},\Phi ,\chi$ are given by (25) and $e_1$, $e_2$, $a_2$ and $a_3$ as defined in Theorem 2.

Proof. 

We implement the impressions from the Theorem 2’s proof. From (34) and from (35), we get

$$a_3-\zeta a_2^2=\frac{e_1}{\Phi }\left(r_2+\left(\frac{e_2}{e_1}+\frac{(\zeta \Phi -\chi )e_1}{{{\rm Y}}^2}\right)r_1^2\right)$$

by using the certain result $|r_2-\mu r_1^2|\le max\left\{1,\left|\mu \right|\right\}$, we get

$$|a_3-\zeta a_2^2|\le \frac{|e_1|}{\Phi }max\left\{1,\left|\frac{e_2}{e_1}+\frac{(\zeta \Phi -\chi )e_1}{{{\rm Y}}^2}\right|\right\}.$$

In the same way, from (38) and from (39), we get

$$a_3-\zeta a_2^2=-\frac{e_1}{\Phi }\left(s_2+\left(\frac{e_2}{e_1}-\frac{(2\Phi +\chi -\zeta \Phi )e_1}{{{\rm Y}}^2}\right)s_1^2\right)$$

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

$$|a_3-\zeta a_2^2|\le \frac{|e_1|}{\Phi }max\left\{1,\left|\frac{e_2}{e_1}-\frac{(2\Phi +\chi -\zeta \Phi )e_1}{{{\rm Y}}^2}\right|\right\}.$$

□

Corollary 4.

If $f\in \sum$ of the style (1) be an element of ${\mathcal{K}}_{\sum }(\xi ,\rho ,\sigma ,\theta ,\alpha ,\beta ,p,q;H_{\gamma }\left(\tau ,z\right))$, then

$$\begin{array}{ccc}& & \left|a_3-\zeta a_2^2\right|\hfill \\ & \le & \frac{|1+\gamma -\tau |}{\Phi }min\left\{max\left\{1,\left|\frac{\frac{{\tau }^2}{2}-(\gamma +2)\tau +\frac{(\gamma +1)(\gamma +2)}{2}}{1+\gamma -\tau }+\frac{(\zeta \Phi -\chi )(1+\gamma -\tau )}{{{\rm Y}}^2}\right|\right\},\right.\hfill \\ & & \left.max\left\{1,\left|\frac{\frac{{\tau }^2}{2}-(\gamma +2)\tau +\frac{(\gamma +1)(\gamma +2)}{2}}{1+\gamma -\tau }-\frac{(2\Phi +\chi -\zeta \Phi )(1+\gamma -\tau )}{{{\rm Y}}^2}\right|\right\}\right\},\hfill \end{array}$$

for each $\zeta ,\xi ,\rho$ such that $\zeta \in \mathbb{R}$, $0\le \xi \le 1$ and $0\le \rho <1$, where ${\rm Y},\Phi ,\chi$ are given by (25) and $H_{\gamma }\left(\tau ,z\right)$ is presented by (3).

3. Conclusions

The main aim of this study was to constitute a new classes ${\mathcal{W}}_{\sum }(\eta ,\delta ,\lambda ,\sigma ,\theta ,\alpha ,\beta ,p,q;h)$ and ${\mathcal{K}}_{\sum }(\xi ,\rho ,\sigma ,\theta ,\alpha ,\beta ,p,q;h)$ of bi-univalent functions described through $(p-q)$-Wanas operator and also utilization of the generator function for Laguerre polynomial $L_n^{\gamma }\left(\tau \right)$, presented by the equalities in (4) and the producing function $H_{\gamma }\left(\tau ,z\right)$ given by (3). The initial Taylor-Maclaurin coefficient estimates for functions of these freshly presented bi-univalent function classes ${\mathcal{W}}_{\sum }(\eta ,\delta ,\lambda ,\sigma ,\theta ,\alpha ,\beta ,p,q;h)$ and ${\mathcal{K}}_{\sum }(\xi ,\rho ,\sigma ,\theta ,\alpha ,\beta ,p,q;h)$ were produced and the well-known Fekete-Szegö inequalities were examined.