Subclasses of Bi-Univalent Functions Connected with Integral Operator Based upon Lucas Polynomial

Abstract

In the contemporary paper, we introduce new subclasses of analytic and bi-univalent functions involving integral operator based upon Lucas polynomial. Furthermore, we find estimates on the first two Taylor-Maclaurin coefficients $\left|a_2\right|$ and $\left|a_3\right|$ for functions in these subclasses and obtain Fekete-Szegö problem for these subclasses.

1. Introduction, Definitions and Preliminaries

Let $\mathsf{\Lambda }$ signify the class of analytic functions of the form

$$\mathsf{{\rm Y}}\left(\xi \right)=\xi +\sum _{k=2}^{\infty }{a}_k{\xi }^k,\xi \in \mathsf{\Omega }:=\{\xi \in \mathbb{C}:|\xi |<1\},$$

(1)

and S be the subclass of $\mathsf{\Lambda }$ which are univalent functions in $\mathsf{\Omega }$.

If $ϝ\in \mathsf{\Gamma }$ is given by

$$ϝ\left(\xi \right)=\xi +\sum _{k=2}^{\infty }{b}_k{\xi }^k,\xi \in \mathsf{\Omega }.$$

(2)

The Hadamard (or convolution) product of $\mathsf{{\rm Y}}$ and $ϝ$ is defined by

$$(\mathsf{{\rm Y}}\ast ϝ)\left(\xi \right):=\xi +\sum _{k=2}^{\infty }{a}_k{b}_k{\xi }^k,\xi \in \mathsf{\Omega }.$$

(3)

If $\mathsf{{\rm Y}}$ and $ϝ$ are analytic functions in $\mathsf{\Omega }$, we say that $\mathsf{{\rm Y}}$is subordinate to$ϝ$, written $\mathsf{{\rm Y}}\prec ϝ$, if there exists a Schwarz function $\omega$, which is analytic in $\mathsf{\Omega }$, with $\omega \left(0\right)=0$, and, $\left|\omega \left(\xi \right)\right|<1$ for all $\xi \in \mathsf{\Omega }$, such that $\mathsf{{\rm Y}}\left(\xi \right)=ϝ\left(\omega \right(\xi )$, $\xi \in \mathsf{\Omega }$. Furthermore, if the function $ϝ$ is univalent in $\mathsf{\Omega }$, then we have the following equivalence (see [1,2]):

$$\mathsf{{\rm Y}}\left(\xi \right)\prec ϝ\left(\xi \right)\iff \mathsf{{\rm Y}}\left(0\right)=ϝ\left(0\right)\mathrm{and}\mathsf{{\rm Y}}\left(\mathsf{\Omega }\right)\subset ϝ\left(\mathsf{\Omega }\right).$$

Porwal [3] studied a power series whose coefficients are probabilities of Poisson distribution as follows:

$$T_m\left(\xi \right)=\xi +\sum _{k=2}^{\infty }\frac{m^{k-1}}{\left(k-1\right)!}{e}^{-m}{\xi }^k,$$

and also, Srivastava and Porwal [4] introduced the operator $W^m:\mathsf{\Lambda }\to \mathsf{\Lambda }$ is given by

$$W^m\mathsf{{\rm Y}}\left(\xi \right)=T_m\left(\xi \right)\ast \mathsf{{\rm Y}}\left(\xi \right)=\xi +\sum _{k=2}^{\infty }\frac{m^{k-1}}{\left(k-1\right)!}{e}^{-m}{a}_k{\xi }^k.$$

El-Deeb and Lupas [5] defined a new integral operator as follows:

$${\mathcal{H}}_{\alpha ,m}^0\mathsf{{\rm Y}}\left(\xi \right)=W^m\mathsf{{\rm Y}}\left(\xi \right)=\xi +\sum _{k=2}^{\infty }\frac{m^{k-1}}{\left(k-1\right)!}{e}^{-m}{a}_k{\xi }^k,$$

$$\begin{array}{c}{\mathcal{H}}_{\alpha ,m}^1\mathsf{{\rm Y}}\left(\xi \right)=\frac{1+\alpha }{{\xi }^{\alpha }}\underset{0}{\overset{\xi }{\int }}{t}^{\alpha -1}{\mathcal{H}}_{\alpha ,m}^0\mathsf{{\rm Y}}\left(t\right)dt=\xi +\sum _{k=2}^{\infty }\left(\frac{1+\alpha }{k+\alpha }\right)\frac{m^{k-1}}{\left(k-1\right)!}{e}^{-m}{a}_k{\xi }^k,\\ ⋮\end{array}$$

$${\mathcal{H}}_{\alpha ,m}^n\mathsf{{\rm Y}}\left(\xi \right)=\frac{1+\alpha }{{\xi }^{\alpha }}\underset{0}{\overset{\xi }{\int }}{t}^{\alpha -1}{\mathcal{H}}_{\alpha ,m}^{n-1}\mathsf{{\rm Y}}\left(t\right)dt$$

$$\begin{array}{ccc}& =& \xi +\sum _{k=2}^{\infty }{\left(\frac{1+\alpha }{k+\alpha }\right)}^n\frac{m^{k-1}}{\left(k-1\right)!}{e}^{-m}{a}_k{\xi }^k\hfill \\ & =& \xi +\sum _{k=2}^{\infty }{\delta }_k{a}_k{\xi }^k,\hfill \end{array}$$

(4)

$$\left(m>0,\alpha \ge 0,n\in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}\right),$$

where ${\delta }_k$ defined by

$${\delta }_k={\left(\frac{1+\alpha }{k+\alpha }\right)}^n\frac{m^{k-1}}{\left(k-1\right)!}{e}^{-m}.$$

(5)

From (4), we obtain that

$$\xi {\left({\mathcal{H}}_{\alpha ,m}^{n}f\left(\xi \right)\right)}^{{}^{\prime }}=\left(\alpha +1\right){\mathcal{H}}_{\alpha ,m}^{n-1}f\left(\xi \right)-\alpha {\mathcal{H}}_{\alpha ,m}^{n}f\left(\xi \right),\left(\alpha >0\right).$$

(6)

Srivastava [6] made use of various operators of q-calculus and fractional q-calculus and recalling the definition and notations. The q-shifted factorial is defined for $\lambda ,q\in \mathbb{C}$ and $n\in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}$ as follows

$${(\lambda ;q)}_k=\left\{\begin{array}{c}1k=0,\hfill \\ \left(1-\lambda \right)\left(1-\lambda q\right)\dots \left(1-\lambda q^{k-1}\right)k\in \mathbb{N}.\hfill \end{array}\right.$$

By using the q-gamma function ${\mathsf{\Gamma }}_q\left(z\right),$ we get

$${\left(q^{\sigma };q\right)}_k=\frac{{\left(1-q\right)}^k{\mathsf{\Gamma }}_q\left(\sigma +k\right)}{{\mathsf{\Gamma }}_q\left(\lambda \right)},\left(k\in {\mathbb{N}}_0\right),$$

where (see [7,8,9])

$${\mathsf{\Gamma }}_q\left(\xi \right)={\left(1-q\right)}^{1-\xi }\frac{{\left(q;q\right)}_{\infty }}{{\left(q^{\xi };q\right)}_{\infty }},\left(\left|q\right|<1\right).$$

Also, we note that

$${\left(\sigma ;q\right)}_{\infty }=\prod _{k=0}^{\infty }\left(1-\sigma q^k\right),\left(\left|q\right|<1\right),$$

and, the q-gamma function ${\mathsf{\Gamma }}_q\left(\xi \right)$ is known

$${\mathsf{\Gamma }}_q(\xi +1)={\left[\xi \right]}_q{\mathsf{\Gamma }}_q\left(\xi \right),$$

where ${\left[t\right]}_q$ denotes the basic q-number defined as follows

$${\left[t\right]}_q:=\left\{\begin{array}{c}\frac{1-q^t}{1-q},t\in \mathbb{C},\\ 1+{\displaystyle \sum _{i=1}^{t-1}}{q}^i,t\in \mathbb{N}.\end{array}\right.$$

(7)

Using the definition formula (7) we have the next two products:

(i) For any non negative integer t, the q-shifted factorial is given by

$${\left[t\right]}_q!:=\left\{\begin{array}{c}1,t=0,\\ {\displaystyle \prod _{i=1}^t}{\left[i\right]}_q,t\in \mathbb{N}.\end{array}\right.$$

(ii) For any positive number r, the q-generalized Pochhammer symbol is defined by

$${\left[r\right]}_{q,t}:=\left\{\begin{array}{ccc}1,\hfill & \mathrm{if}\hfill & t=0,\hfill \\ {\displaystyle \prod _{i=r}^{r+t-1}}{\left[i\right]}_q,\hfill & \mathrm{if}\hfill & t\in \mathbb{N}.\hfill \end{array}\right.$$

It is known in terms of the classical (Euler’s) gamma function $\mathsf{\Gamma }\left(\xi \right)$, that

$${\mathsf{\Gamma }}_q\left(\xi \right)\to \mathsf{\Gamma }\left(\xi \right)\mathrm{as}q\to 1^{-}.$$

Also, we observe that

$$\underset{q\to 1^{-}}{lim}\left\{\frac{{\left(q^{\sigma };q\right)}_k}{{\left(1-q\right)}^t}\right\}={\left(\sigma \right)}_t.$$

For $0<q<1$, the q-derivative operator for $\mathsf{{\rm Y}}$ is defined by

$$\begin{array}{c}{D}_q\mathsf{{\rm Y}}\left(\xi \right)=D_q\left[\xi +{\displaystyle \sum _{k=2}^{\infty }}{a}_k{\xi }^k\right]=\frac{\mathsf{{\rm Y}}\left(\xi \right)-\mathsf{{\rm Y}}\left(q\xi \right)}{\xi (1-q)}\\ =1+{\displaystyle \sum _{k=2}^{\infty }}{\left[k\right]}_q{a}_k{\xi }^{k-1},\xi \in \mathsf{\Omega },\end{array}$$

where

$${\left[k\right]}_q:=\frac{1-q^k}{1-q}=1+{\displaystyle \sum _{j=1}^{k-1}}{q}^j,{\left[0\right]}_q:=0.$$

(8)

Definition 1.

Let $p\left(y\right)$ and $r\left(y\right)$ be polynomials with real coefficients. The $(p,r)$-polynomials ${\mathcal{L}}_{p,r,n}\left(y\right)$ are given by the following recurrence relation (see [10,11]):

$${\mathcal{L}}_{p,r,n}\left(y\right)=p\left(y\right){\mathcal{L}}_{p,r,n-1}\left(y\right)+r\left(y\right){\mathcal{L}}_{p,r,n-2}\left(y\right)(n\in \mathbb{N}\backslash \left\{1\right\}),$$

with

$$\begin{array}{c}{\mathcal{L}}_{p,r,0}\left(y\right)=2,\hfill \\ {\mathcal{L}}_{p,r,1}\left(y\right)=p\left(y\right),\hfill \\ {\mathcal{L}}_{p,r,2}\left(y\right)=p^2\left(y\right)+2r\left(y\right),\hfill \\ {\mathcal{L}}_{p,r,3}\left(y\right)=p^3\left(y\right)+3p\left(y\right)r\left(y\right),\hfill \\ ⋮\hfill \end{array}$$

The generating function of the Lucas polynomials ${\mathcal{L}}_{p,r,n}\left(y\right)$ (see [12]) is given by:

$${\mathcal{G}}_{{\mathcal{L}}_{p,r,n}\left(y\right)}\left(\xi \right):=\sum _{n=0}^{\infty }{\mathcal{L}}_{p,r,n}\left(y\right){\xi }^n={\displaystyle \frac{2-p\left(y\right)\xi }{1-p\left(y\right)\xi -r\left(y\right){\xi }^2}}.$$

(9)

Note that for particular values of p and r, the $(p,r)-$polynomial ${\mathcal{L}}_{p,r,n}\left(y\right)$ leads to various polynomials, among those, we list few cases here (see, [12] for more details, also [13,14]):

• For $p\left(y\right)=y$ and $r\left(y\right)=1,$ we obtain the Lucas polynomials ${\mathcal{L}}_n\left(y\right)$.
• For $p\left(y\right)=2y$ and $r\left(y\right)=1,$ we attain the Pell-Lucas polynomials $Q_n\left(y\right)$.
• For $p\left(y\right)=1$ and $r\left(y\right)=2y,$ we attain the Jacobsthal-Lucas polynomials $j_n\left(y\right)$.
• For $p\left(y\right)=3y$ and $r\left(y\right)=-2,$ we attain the Fermat-Lucas polynomials $f_n\left(y\right)$.
• For $p\left(y\right)=2y$ and $r\left(y\right)=-1,$ we have the Chebyshev polynomials $T_n\left(y\right)$ of the first kind.

The Koebe one quarter theorem (see [15]) proves that the image of $\mathsf{\Omega }$ under every univalent function $\mathsf{{\rm Y}}\in \mathcal{S}$ contains a disk of radius $\frac{1}{4}.$Therefore, every function $\mathsf{{\rm Y}}\in \mathcal{S}$ has an inverse ${\mathsf{{\rm Y}}}^{-1}$ satisfied

$${\mathsf{{\rm Y}}}^{-1}\left(\mathsf{{\rm Y}}\left(\xi \right)\right)=\xi \left(\xi \in \mathsf{\Omega }\right)$$

and

$$\mathsf{{\rm Y}}\left({\mathsf{{\rm Y}}}^{-1}\left(w\right)\right)=w\left(\left|w\right|<r_0\left(f\right);r_0\left(f\right) \ge \frac{1}{4}\right),$$

where

$${\mathsf{{\rm Y}}}^{-1}\left(w\right)=w-a_2{w}^2+\left(2a_2^2-a_3\right)w^3-\left(5a_2^3-5a_2{a}_3+a_4\right)w^4+\dots .$$

(10)

A function $\mathsf{{\rm Y}}\in \mathsf{\Lambda }$ is said to be bi-univalent in $\mathsf{\Omega }$ if both $\mathsf{{\rm Y}}\left(\xi \right)$ and ${\mathsf{{\rm Y}}}^{-1}\left(\xi \right)$ are univalent in $\mathsf{\Omega }$. Let $\mathsf{\Sigma }$ denote the class of bi-univalent functions in $\mathsf{\Omega }$ given by (1). The class of analytic bi-univalent functions was first introduced by Lewin [16], where it was proved that $\left|a_2\right| < 1.51$. Brannan and Clunie [17] improved Lewin’s result to $\left|a_2\right|<\sqrt{2}$ and later Netanyahu [18] proved that $\left|a_2\right|<\frac{4}{3}.$Note that the functions ${\mathsf{{\rm Y}}}_1\left(\xi \right)={\displaystyle \frac{\xi }{1-\xi }}$, ${\mathsf{{\rm Y}}}_2\left(\xi \right)={\displaystyle \frac{1}{2}}log{\displaystyle \frac{1+\xi }{1-\xi }}$, ${\mathsf{{\rm Y}}}_3\left(\xi \right)=-log(1-\xi )$, with their corresponding inverses ${\mathsf{{\rm Y}}}_1^{-1}\left(w\right)={\displaystyle \frac{w}{1+w}}$, ${\mathsf{{\rm Y}}}_2^{-1}\left(w\right)={\displaystyle \frac{e^{2w}-1}{e^{2w}+1}}$, ${\mathsf{{\rm Y}}}_3^{-1}\left(w\right)={\displaystyle \frac{e^w-1}{e^w}}$, are elements of $\mathsf{\Sigma }$ (see [19,20,21]). For a brief history and interesting examples in the class $\mathsf{\Sigma }$(see [17]). Brannan and Taha [22] (see also [19]) introduced certain subclasses of the bi-univalent functions class $\mathsf{\Sigma }$ similar to the familiar subclasses $S^{\ast }\left(\beta \right)$ and $\mathcal{K}\left(\beta \right)$ of starlike and convex functions of order $\beta \left(0\le \beta <1\right)$, respectively (see [17]). Thus, following Brannan and Taha [22] a function $\mathsf{{\rm Y}}\in \mathsf{\Lambda }$ is said to be in the class $S_{\mathsf{\Sigma }}^{\ast }\left(\beta \right)$ of strongly bi-starlike functions of order $\beta$$\left(0<\beta \le 1\right)$ if each of the following conditions is satisfied:

$$\mathsf{{\rm Y}}\in \mathsf{\Sigma }\mathrm{and}\left|arg\left(\frac{\xi {\mathsf{{\rm Y}}}^{{}^{\prime }}\left(\xi \right)}{\mathsf{{\rm Y}}\left(\xi \right)}\right)\right|<\frac{\beta \pi }{2}\left(0<\beta \le 1;\xi \in \mathsf{\Omega }\right)$$

(11)

and

$$\left|arg\left(\frac{w{h}^{{}^{\prime }}\left(w\right)}{h\left(w\right)}\right)\right|<\frac{\beta \pi }{2}\left(0<\beta \le 1;w\in \mathsf{\Omega }\right),$$

(12)

also, a function $\mathsf{{\rm Y}}\in \mathsf{\Lambda }$ is said to be in the class ${\mathcal{K}}_{\mathsf{\Sigma }}\left(\beta \right)$ of strongly bi-convex functions of order $\beta$$\left(0<\beta \le 1\right)$ if each of the following conditions is satisfied:

$$\mathsf{{\rm Y}}\in \mathsf{\Sigma }\mathrm{and}\left|arg\left(1+\frac{\xi {\mathsf{{\rm Y}}}^{{}^{{}^{″}}}\left(\xi \right)}{{\mathsf{{\rm Y}}}^{{}^{\prime }}\left(\xi \right)}\right)\right|<\frac{\beta \pi }{2}\left(0<\beta \le 1;\xi \in \mathsf{\Omega }\right)$$

(13)

and

$$\left|arg\left(1+\frac{w{h}^{{}^{{}^{″}}}\left(w\right)}{h^{{}^{\prime }}\left(w\right)}\right)\right|<\frac{\beta \pi }{2}\left(0<\beta \le 1;w\in \mathsf{\Omega }\right),$$

(14)

where $h$is the extension of ${\mathsf{{\rm Y}}}^{-1}$ to $\mathsf{\Omega }$ is given by (10). The classes $S_{\mathsf{\Sigma }}^{\ast }\left(\beta \right)$ and ${\mathcal{K}}_{\mathsf{\Sigma }}\left(\beta \right)$ of bi-starlike functions of order $\beta$ and bi-convex functions of order $\beta$$\left(0<\beta \le 1\right),$ corresponding to the function classes $S^{\ast }\left(\beta \right)$ and $\mathcal{K}\left(\beta \right),$were also introduced analogously. For each of the function classes $S_{\mathsf{\Sigma }}^{\ast }\left(\beta \right)$ and ${\mathcal{K}}_{\mathsf{\Sigma }}\left(\beta \right),$ they found non-sharp estimates on the first two Taylor-Maclaurin coefficients $\left|a_2\right|$ and $\left|a_3\right|$ (for details, see [19,20,21,22,23,24]).

In 1916, Bieberbach [25] proved that if $\mathsf{{\rm Y}}\in \mathcal{S}$, then $\left|a_3-a_2^2\right|\le 1$ and the result is sharp for the function

$$\mathsf{{\rm Y}}\left(\xi \right)=\frac{\xi }{{(1-e^{i\alpha }\xi )}^2}=\xi +2e^{i\alpha }{\xi }^2+3e^{2i\alpha }{\xi }^3+\dots ,$$

where $\mathsf{{\rm Y}}$ is the rotation of the Koebe function defined as follows:

$$\mathsf{{\rm Y}}\left(\xi \right)=\frac{\xi }{{(1-\xi )}^2}=\frac{1}{4}{\left(\frac{1+\xi }{1-\xi }\right)}^2-\frac{1}{4}.$$

The Koebe function maps the disk $\left|\xi \right|<1$onto the plane cut along the negative real axis from $-\infty$to$-{\displaystyle \frac{1}{4}}.$

Fekete and Szegö [26] proved the inequality:

$$\left|a_3-\mu a_2^2\right|\le 1+2exp\left({\displaystyle \frac{-2\mu }{1-\mu }}\right),$$

holds for any normalized univalent function of the form (1) in the open unit disc and for $0\le \mu \le 1.$This inequality is sharp for each $\mu$(see [26]). The coefficient functional

$$a_3-\mu a_2^2=\frac{1}{6}\left({\mathsf{{\rm Y}}}^{‴}\left(0\right)-\frac{3\mu }{2}{\left\{{\mathsf{{\rm Y}}}^{″}\left(0\right)\right\}}^2\right),$$

on normalized analytic functions of the form (1) in the unit disc is important which represent various geometric quantities. Also, $a_3-a_2^2$represent $S_{\mathsf{{\rm Y}}}\left(0\right)/6,$where $S_{\mathsf{{\rm Y}}}$denote the Schwarzian derivative${({\mathsf{{\rm Y}}}^{″}/{\mathsf{{\rm Y}}}^{\prime })}^{\prime }-{({\mathsf{{\rm Y}}}^{″}/{\mathsf{{\rm Y}}}^{\prime })}^2/2$of $\mathsf{{\rm Y}}$(see [27]).

The object of the present paper is to introduce new classes of the function class $\mathsf{\Sigma }$ involving the integral operator based upon Lucas polynomial previous defined classes, and find estimates on the coefficients $\left|a_2\right|$, and $\left|a_3\right|$ for functions in these new subclasses of the function class $\mathsf{\Sigma }$.

Definition 2.

Let $\eta \ne 0$ be a complex number and $\mathsf{{\rm Y}}\in \mathsf{\Sigma }$ assumed by (1), then $\mathsf{{\rm Y}}\left(\xi \right)$ is said to be in the class ${\mathcal{M}}_{\mathsf{\Sigma }}^{q,\lambda ,\alpha }\left(\eta ,n,m,p\left(y\right),r\left(y\right)\right)$ if the following conditions are satisfied:

$$\mathsf{{\rm Y}}\in \mathsf{\Sigma }and$$

$$1+\frac{1}{\eta }\left(\frac{\lambda \xi D_q\left(D_q\left({\mathcal{H}}_{\alpha ,m}^n\mathsf{{\rm Y}}\left(\xi \right)\right)\right)+\lambda D_q\left({\mathcal{H}}_{\alpha ,m}^n\mathsf{{\rm Y}}\left(\xi \right)\right)+1-\lambda }{D_q\left({\mathcal{H}}_{\alpha ,m}^n\mathsf{{\rm Y}}\left(\xi \right)\right)}-1\right)\prec {\mathcal{G}}_{{\mathcal{L}}_{p,r,n}\left(y\right)}\left(\xi \right)-1,$$

(15)

and

$$1+\frac{1}{\eta }\left(\frac{\lambda w{D}_q\left(D_q\left({\mathcal{H}}_{\alpha ,m}^{n}h\left(w\right)\right)\right)+\lambda D_q\left({\mathcal{H}}_{\alpha ,m}^{n}h\left(w\right)\right)+1-\lambda }{D_q\left({\mathcal{H}}_{\alpha ,m}^{n}h\left(w\right)\right)}-1\right)\prec {\mathcal{G}}_{{\mathcal{L}}_{p,r,n}\left(y\right)}\left(w\right)-1,$$

(16)

with$\lambda >0,m>0,\alpha \ge 0,n\in {\mathbb{N}}_0;0<q<1;\eta \in {\mathbb{C}}^{\ast }=\mathbb{C}\setminus \left\{0\right\},$where the function $h={\mathsf{{\rm Y}}}^{-1}$ is given by (10).

Remark 1.

(i) For $p\left(y\right)=y$ and $r\left(y\right)=1,$we obtain that ${\mathcal{L}}_{\mathsf{\Sigma }}^{q,\lambda ,\alpha }\left(\eta ,n,m,y\right)$defined as follows

$$\mathsf{{\rm Y}}\in \mathsf{\Sigma }and$$

$$1+\frac{1}{\eta }\left(\frac{\lambda \xi D_q\left(D_q\left({\mathcal{H}}_{\alpha ,m}^n\mathsf{{\rm Y}}\left(\xi \right)\right)\right)+\lambda D_q\left({\mathcal{H}}_{\alpha ,m}^n\mathsf{{\rm Y}}\left(\xi \right)\right)+1-\lambda }{D_q\left({\mathcal{H}}_{\alpha ,m}^n\mathsf{{\rm Y}}\left(\xi \right)\right)}-1\right)\prec {\mathcal{G}}_{{\mathcal{L}}_n\left(y\right)}\left(\xi \right)-1,$$

(17)

and

$$1+\frac{1}{\eta }\left(\frac{\lambda w{D}_q\left(D_q\left({\mathcal{H}}_{\alpha ,m}^{n}h\left(w\right)\right)\right)+\lambda D_q\left({\mathcal{H}}_{\alpha ,m}^{n}h\left(w\right)\right)+1-\lambda }{D_q\left({\mathcal{H}}_{\alpha ,m}^{n}h\left(w\right)\right)}-1\right)\prec {\mathcal{G}}_{{\mathcal{L}}_n\left(y\right)}\left(w\right)-1;$$

(18)

(ii) For $p\left(y\right)=1$ and $r\left(y\right)=2y,$we obtain that ${\mathcal{J}}_{\mathsf{\Sigma }}^{q,\lambda ,\alpha }\left(\eta ,n,m,y\right)$defined as follows

$$\mathsf{{\rm Y}}\in \mathsf{\Sigma }and$$

$$1+\frac{1}{\eta }\left(\frac{\lambda \xi D_q\left(D_q\left({\mathcal{H}}_{\alpha ,m}^n\mathsf{{\rm Y}}\left(\xi \right)\right)\right)+\lambda D_q\left({\mathcal{H}}_{\alpha ,m}^n\mathsf{{\rm Y}}\left(\xi \right)\right)+1-\lambda }{D_q\left({\mathcal{H}}_{\alpha ,m}^n\mathsf{{\rm Y}}\left(\xi \right)\right)}-1\right)\prec {\mathcal{G}}_{j_n\left(y\right)}\left(\xi \right)-1,$$

(19)

and

$$1+\frac{1}{\eta }\left(\frac{\lambda w{D}_q\left(D_q\left({\mathcal{H}}_{\alpha ,m}^{n}h\left(w\right)\right)\right)+\lambda D_q\left({\mathcal{H}}_{\alpha ,m}^{n}h\left(w\right)\right)+1-\lambda }{D_q\left({\mathcal{H}}_{\alpha ,m}^{n}h\left(w\right)\right)}-1\right)\prec {\mathcal{G}}_{j_n\left(y\right)}\left(w\right)-1;$$

(20)

(iii) For $p\left(y\right)=2y$ and $r\left(y\right)=-1,$ we obtain that ${\mathcal{T}}_{\mathsf{\Sigma }}^{q,\lambda ,\alpha }\left(\eta ,n,m,y\right)$defined as follows

$$\mathsf{{\rm Y}}\in \mathsf{\Sigma }and$$

$$1+\frac{1}{\eta }\left(\frac{\lambda \xi D_q\left(D_q\left({\mathcal{H}}_{\alpha ,m}^n\mathsf{{\rm Y}}\left(\xi \right)\right)\right)+\lambda D_q\left({\mathcal{H}}_{\alpha ,m}^n\mathsf{{\rm Y}}\left(\xi \right)\right)+1-\lambda }{D_q\left({\mathcal{H}}_{\alpha ,m}^n\mathsf{{\rm Y}}\left(\xi \right)\right)}-1\right)\prec {\mathcal{G}}_{T_n\left(y\right)}\left(\xi \right)-1,$$

(21)

and

$$1+\frac{1}{\eta }\left(\frac{\lambda w{D}_q\left(D_q\left({\mathcal{H}}_{\alpha ,m}^{n}h\left(w\right)\right)\right)+\lambda D_q\left({\mathcal{H}}_{\alpha ,m}^{n}h\left(w\right)\right)+1-\lambda }{D_q\left({\mathcal{H}}_{\alpha ,m}^{n}h\left(w\right)\right)}-1\right)\prec {\mathcal{G}}_{T_n\left(y\right)}\left(w\right)-1.$$

(22)

The following Lemma will be needed later.

Lemma 1

([27], p. 172). If $U\left(\xi \right)=\sum _{k=1}^{\infty }{u}_k{\xi }^k$ is a Schwarz function for $\xi \in \mathsf{\Omega }$, then

$$|u_1|\le 1,|u_k|\le 1-|u_1{|}^2,k\ge 1.$$

2. Coefficient Bounds for the Function Class ${\mathcal{M}}_{\mathsf{\Sigma }}^{\mathit{q},\mathit{\lambda },\mathit{\alpha }}\left(\mathit{\eta },\mathit{n},\mathit{m},\mathit{p}\left(\mathit{y}\right),\mathit{r}\left(\mathit{y}\right)\right)$

Unless otherwise mentioned, we shall assume in the reminder of this paper that $\lambda \ge 0,m>0,\alpha \ge 0,n\in {\mathbb{N}}_0;0<q<1,\eta \in {\mathbb{C}}^{\ast }$, the powers are understood as principle values.

Theorem 1.

Let the function Υ given by (1) belongs to the class${\mathcal{M}}_{\mathsf{\Sigma }}^{q,\lambda ,\alpha }\left(\eta ,n,m,p\left(y\right),r\left(y\right)\right)$, then

$$\left|a_2\right|\le$$

$$\frac{\left|\eta \right|\left|p\left(y\right)\right|\sqrt{p\left(y\right)}}{\sqrt{\left|\left(\lambda (2+q)-1\right)\left(1+q+q^2\right)\eta p^2\left(y\right){\delta }_3-\left[\eta p^2\left(y\right)+\left(2\lambda -1\right)\left(p^2\left(y\right)+2r\left(y\right)\right)\right]\left(2\lambda -1\right){(1+q)}^2{\delta }_2^2\right|}},$$

and

$$\left|a_3\right|\le \frac{\left|\eta \right|\left|p\left(y\right)\right|}{\left(\lambda (2+q)-1\right)\left(1+q+q^2\right){\delta }_3}+\frac{{\left|\eta \right|}^2{p}^2\left(y\right)}{{(1+q)}^2{\left(2\lambda -1\right)}^2{\delta }_2^2},$$

where ${\delta }_k$, $k\in \{2,3\}$, are given by (5).

Proof. 

Since $\mathsf{{\rm Y}}\in {\mathcal{M}}_{\mathsf{\Sigma }}^{q,\lambda ,\alpha }\left(\eta ,n,m,p\left(y\right),r\left(y\right)\right)$. Then there exist two analytic functions R and S in $\mathsf{\Omega }$ with $R\left(0\right)=S\left(0\right)=0$, and $\left|R\right(\xi \left)\right|<1$, $\left|S\right(w\left)\right|<1$ for all $\xi ,w\in \mathsf{\Omega }$ given by

$$R\left(\xi \right)=\sum _{k=1}^{\infty }{u}_k{\xi }^k\mathrm{and}S\left(w\right)=\sum _{k=1}^{\infty }{v}_k{w}^k,\xi ,w\in \mathsf{\Omega },$$

from Lemma (1) we have

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

(23)

In view of (15) and (16), we get

$$\frac{\lambda \xi D_q\left(D_q\left({\mathcal{H}}_{\alpha ,m}^n\mathsf{{\rm Y}}\left(\xi \right)\right)\right)+\lambda D_q\left({\mathcal{H}}_{\alpha ,m}^n\mathsf{{\rm Y}}\left(\xi \right)\right)+1-\lambda }{D_q\left({\mathcal{H}}_{\alpha ,m}^n\mathsf{{\rm Y}}\left(\xi \right)\right)}-1=\eta \left({\mathcal{G}}_{{\mathcal{L}}_{p,r,n}\left(y\right)}\left(R\left(\xi \right)\right)-2\right),$$

(24)

and

$$\frac{\lambda w{D}_q\left(D_q\left({\mathcal{H}}_{\alpha ,m}^{n}h\left(w\right)\right)\right)+\lambda D_q\left({\mathcal{H}}_{\alpha ,m}^{n}h\left(w\right)\right)+1-\lambda }{D_q\left({\mathcal{H}}_{\alpha ,m}^{n}h\left(w\right)\right)}-1=\eta \left({\mathcal{G}}_{{\mathcal{L}}_{p,r,n}\left(y\right)}\left(S\left(w\right)\right)-2\right).$$

(25)

Since

$$\begin{array}{ccc}& & \frac{\lambda \xi D_q\left(D_q\left({\mathcal{H}}_{\alpha ,m}^n\mathsf{{\rm Y}}\left(\xi \right)\right)\right)+\lambda D_q\left({\mathcal{H}}_{\alpha ,m}^n\mathsf{{\rm Y}}\left(\xi \right)\right)+1-\lambda }{D_q\left({\mathcal{H}}_{\alpha ,m}^n\mathsf{{\rm Y}}\left(\xi \right)\right)}-1\hfill \\ & =& (1+q)\left(2\lambda -1\right){\delta }_2{a}_2\xi +\left[\left(\lambda (2+q)-1\right)\left(1+q+q^2\right){\delta }_3{a}_3-\left(2\lambda -1\right){(1+q)}^2{\delta }_2^2{a}_2^2\right]{\xi }^2+\cdots ,\hfill \end{array}$$

$$\begin{array}{ccc}& & \frac{\lambda w{D}_q\left(D_q\left({\mathcal{H}}_{\alpha ,m}^{n}h\left(w\right)\right)\right)+\lambda D_q\left({\mathcal{H}}_{\alpha ,m}^{n}h\left(w\right)\right)+1-\lambda }{D_q\left({\mathcal{H}}_{\alpha ,m}^{n}h\left(w\right)\right)}-1\hfill \\ & =& -(1+q)\left(2\lambda -1\right){\delta }_2{a}_{2}w\hfill \\ & & +\left[\left(\lambda (2+q)-1\right)\left(1+q+q^2\right){\delta }_3\left(2a_2^2-a_3\right)-\left(2\alpha -1\right){(1+q)}^2{\delta }_2^2{a}_2^2\right]w^2+\cdots ,\hfill \end{array}$$

and

$$\begin{array}{c}\eta \left({\mathcal{G}}_{{\mathcal{L}}_{p,r,n}\left(y\right)}\left(R\left(\xi \right)\right)-2\right)=\\ \eta {\mathcal{L}}_{p,r,1}\left(y\right)u_1\xi +\left({\mathcal{L}}_{p,r,1}\left(y\right)u_2+{\mathcal{L}}_{p,r,2}\left(y\right)u_1^2\right)\eta {\xi }^2+\cdots ,\end{array}$$

$$\eta \left({\mathcal{G}}_{{\mathcal{L}}_{p,r,n}\left(y\right)}\left(S\left(w\right)\right)-2\right)=\eta {\mathcal{L}}_{p,r,1}\left(y\right)v_{1}w+\left({\mathcal{L}}_{p,r,1}\left(y\right)v_2+{\mathcal{L}}_{p,r,2}\left(y\right)v_1^2\right)\eta w^2+\cdots .$$

Next, equating the corresponding coefficients of $\xi$ and w in (24) and(25), we get

$$(1+q)\left(2\lambda -1\right){\delta }_2{a}_2=\eta {\mathcal{L}}_{p,r,1}\left(y\right)u_1,$$

(26)

$$\left(\lambda (2+q)-1\right)\left(1+q+q^2\right){\delta }_3{a}_3-\left(2\lambda -1\right){(1+q)}^2{\delta }_2^2{a}_2^2=\eta {\mathcal{L}}_{p,r,1}\left(y\right)u_2+\eta {\mathcal{L}}_{p,r,2}\left(y\right)u_1^2$$

(27)

$$-(1+q)\left(2\lambda -1\right){\delta }_2{a}_2=\eta {\mathcal{L}}_{p,r,1}\left(y\right)v_1,$$

(28)

$$\left(\lambda (2+q)-1\right)\left(1+q+q^2\right){\delta }_3\left(2a_2^2-a_3\right)-\left(2\lambda -1\right){(1+q)}^2{\delta }_2^2{a}_2^2=\eta {\mathcal{L}}_{p,r,1}\left(y\right)v_2+\eta {\mathcal{L}}_{p,r,2}\left(y\right)v_1^2.$$

(29)

From (26) and (28), we have

$$u_1=-v_1$$

(30)

By squaring (26) and (28), then adding the new relations we get

$$2{(1+q)}^2{\left(2\lambda -1\right)}^2{a}_2^2{\delta }_2^2={\eta }^2{\mathcal{L}}_{p,r,1}^2\left(y\right)\left(u_1^2+v_1^2\right).$$

(31)

If we add (27) and (29) we obtain

$$\begin{array}{c}2\left[\left(\lambda (2+q)-1\right)\left(1+q+q^2\right){\delta }_3-\left(2\lambda -1\right){(1+q)}^2{\delta }_2^2\right]a_2^2=\\ \eta {\mathcal{L}}_{p,r,1}\left(y\right)\left(u_2+v_2\right)+\eta {\mathcal{L}}_{p,r,2}\left(y\right)\left(u_1^2+v_1^2\right).\end{array}$$

We can rewrite (31) as

$$u_1^2+v_1^2=\frac{2{(1+q)}^2{\left(2\lambda -1\right)}^2}{{\eta }^2{\mathcal{L}}_{p,r,1}^2\left(y\right)}{\delta }_2^2{a}_2^2.$$

From above equation, we get

$$\begin{array}{c}2\left[\left(\lambda (2+q)-1\right)\left(1+q+q^2\right)\eta {\mathcal{L}}_{p,r,1}^2\left(y\right){\delta }_3-\right.\\ \left.\left[\eta {\mathcal{L}}_{p,r,1}^2\left(y\right)+\left(2\lambda -1\right){\mathcal{L}}_{p,r,2}\left(y\right)\right]\left(2\lambda -1\right){(1+q)}^2{\delta }_2^2\right]a_2^2\\ ={\eta }^2{\mathcal{L}}_{p,r,1}^3\left(y\right)\left(u_2+v_2\right),\end{array}$$

it follows that

$$\begin{array}{c}{a}_2^2=\\ \frac{{\eta }^2{\mathcal{L}}_{p,r,1}^3\left(y\right)\left(u_2+v_2\right)}{2\left[\left(\lambda (2+q)-1\right)\left(1+q+q^2\right)\eta {\mathcal{L}}_{p,r,1}^2\left(y\right){\delta }_3-\left[\eta {\mathcal{L}}_{p,r,1}^2\left(y\right)+\left(2\lambda -1\right){\mathcal{L}}_{p,r,2}\left(y\right)\right]\left(2\lambda -1\right){(1+q)}^2{\delta }_2^2\right]},\end{array}$$

(32)

Then taking the absolute value to the above equation and from (17) and (23), we obtain

$$\begin{array}{c}\left|a_2\right|\le \\ \frac{\left|\eta \right|\left|p\left(y\right)\right|\sqrt{p\left(y\right)}}{\sqrt{\left|\left(\lambda (2+q)-1\right)\left(1+q+q^2\right)\eta p^2\left(y\right){\delta }_3-\left[\eta p^2\left(y\right)+\left(2\lambda -1\right)\left(p^2\left(y\right)+2r\left(y\right)\right)\right]\left(2\lambda -1\right){(1+q)}^2{\delta }_2^2\right|}},\end{array}$$

which gives the bound for $\left|a_2\right|$ as we asserted in our theorem.

To find the bound for $\left|a_3\right|$. Using (27) from (29), we have

$$2\left(\lambda (2+q)-1\right)\left(1+q+q^2\right){\delta }_3\left(a_3-a_2^2\right)=\eta \left[{\mathcal{L}}_{p,r,1}\left(y\right)\left(u_2-v_2\right)+{\mathcal{L}}_{p,r,2}\left(y\right)\left(u_1^2-v_1^2\right)\right].$$

(33)

Form (30)–(33), we obtain

$$a_3=\frac{\eta {\mathcal{L}}_{p,r,1}\left(y\right)\left(u_2-v_2\right)}{2\left(\lambda (2+q)-1\right)\left(1+q+q^2\right){\delta }_3}+\frac{{\eta }^2{\mathcal{L}}_{p,r,1}^2\left(y\right)\left(u_1^2+v_1^2\right)}{2{(1+q)}^2{\left(2\lambda -1\right)}^2{\delta }_2^2}.$$

(34)

Using (17) and (23), we get

$$\left|a_3\right|\le \frac{\left|\eta \right|\left|p\left(y\right)\right|}{\left(\lambda (2+q)-1\right)\left(1+q+q^2\right){\delta }_3}+\frac{{\left|\eta \right|}^2{p}^2\left(y\right)}{{(1+q)}^2{\left(2\lambda -1\right)}^2{\delta }_2^2}.$$

□

In view of Theorem (1) we obtain the following results.

Putting $p\left(y\right)=y$ and $r\left(y\right)=1,$ we get the following corollary:

Corollary 1.

Let the function Υ given by (1) belongs to the class${\mathcal{L}}_{\mathsf{\Sigma }}^{q,\lambda ,\alpha }\left(\eta ,n,m,y\right)$, then

$$\left|a_2\right|\le \frac{\left|\eta \right|\left|y\right|\sqrt{y}}{\sqrt{\left|\left(\lambda (2+q)-1\right)\left(1+q+q^2\right)\eta y^2{\delta }_3-\left[\eta y^2+\left(2\lambda -1\right)\left(y^2+2\right)\right]\left(2\lambda -1\right){(1+q)}^2{\delta }_2^2\right|}},$$

and

$$\left|a_3\right|\le \frac{\left|\eta \right|\left|y\right|}{\left(\lambda (2+q)-1\right)\left(1+q+q^2\right){\delta }_3}+\frac{{\left|\eta \right|}^2{y}^2}{{(1+q)}^2{\left(2\lambda -1\right)}^2{\delta }_2^2},$$

where ${\delta }_k$, $k\in \{2,3\}$, are given by (5).

Putting $p\left(y\right)=1$ and $r\left(y\right)=2y,$ we get the following corollary:

Corollary 2.

Let the function Υ given by (1) belongs to the class${\mathcal{J}}_{\mathsf{\Sigma }}^{q,\lambda ,\alpha }\left(\eta ,n,m,y\right)$, then

$$\left|a_2\right|\le \frac{\left|\eta \right|}{\sqrt{\left|\left(\lambda (2+q)-1\right)\left(1+q+q^2\right)\eta {\delta }_3-\left[\eta +\left(2\lambda -1\right)\left(1+4y\right)\right]\left(2\lambda -1\right){(1+q)}^2{\delta }_2^2\right|}},$$

and

$$\left|a_3\right|\le \frac{\left|\eta \right|}{\left(\lambda (2+q)-1\right)\left(1+q+q^2\right){\delta }_3}+\frac{{\left|\eta \right|}^2}{{(1+q)}^2{\left(2\lambda -1\right)}^2{\delta }_2^2},$$

where ${\delta }_k$, $k\in \{2,3\}$, are given by (5).

Putting $p\left(y\right)=2y$ and $r\left(y\right)=-1,$ we get the following corollary:

Corollary 3.

Let the function Υ given by (1) belongs to the class${\mathcal{T}}_{\mathsf{\Sigma }}^{q,\lambda ,\alpha }\left(\eta ,n,m,y\right)$, then

$$\left|a_2\right|\le \frac{\left|\eta \right|\left|2y\right|\sqrt{2y}}{\sqrt{\left|4\left(\lambda (2+q)-1\right)\left(1+q+q^2\right)\eta y^2{\delta }_3-\left[4\eta y^2+\left(2\lambda -1\right)\left(4y^2-2\right)\right]\left(2\lambda -1\right){(1+q)}^2{\delta }_2^2\right|}},$$

and

$$\left|a_3\right|\le \frac{\left|\eta \right|\left|2y\right|}{\left(\lambda (2+q)-1\right)\left(1+q+q^2\right){\delta }_3}+\frac{4y^2{\left|\eta \right|}^2}{{(1+q)}^2{\left(2\lambda -1\right)}^2{\delta }_2^2},$$

where ${\delta }_k$, $k\in \{2,3\}$, are given by (5).

Putting $q\to 1^{-}$ in Theorem (1), we get the following Example:

Example 1.

Let the function Υ given by (1) belongs to the class$\underset{q\to 1^{-}}{lim}{\mathcal{M}}_{\mathsf{\Sigma }}^{q,\lambda ,\alpha }\left(\eta ,n,m,p\left(y\right),r\left(y\right)\right)$, then

$$\left|a_2\right|\le$$

$${\displaystyle \frac{\left|\eta \right|\left|p\left(y\right)\right|\sqrt{p\left(y\right)}}{\sqrt{\left|3\eta \left(3\lambda -1\right)p^2\left(y\right){\delta }_3-4\left[\eta p^2\left(y\right)+\left(2\lambda -1\right)\left(p^2\left(y\right)+2r\left(y\right)\right)\right]\left(2\lambda -1\right){\delta }_2^2\right|}}},$$

and

$$\left|a_3\right|\le \frac{\left|\eta \right|\left|p\left(y\right)\right|}{3\left(3\lambda -1\right){\delta }_3}+\frac{{\left|\eta \right|}^2{p}^2\left(y\right)}{4{\left(2\lambda -1\right)}^2{\delta }_2^2},$$

where ${\delta }_k$, $k\in \{2,3\}$, are given by (5).

3. Fekete-Szegö Problem for the Function Class ${\mathcal{M}}_{\mathsf{\Sigma }}^{\mathit{q},\mathit{\lambda },\mathit{\alpha }}\left(\mathit{\eta },\mathit{n},\mathit{m},\mathit{p}\left(\mathit{y}\right),\mathit{r}\left(\mathit{y}\right)\right)$

Theorem 2.

If Υ assumed (1) belongs to the class ${\mathcal{M}}_{\mathsf{\Sigma }}^{q,\lambda ,\alpha }\left(\eta ,n,m,p\left(y\right),r\left(y\right)\right)$, and $\eta \in {\mathbb{C}}^{\ast }$, then

$$\left|a_3-\mu a_2^2\right|\le \left|\eta \right|\left|p\left(y\right)\right|\left(|\mathcal{M}+\mathcal{N}|+|\mathcal{M}-\mathcal{N}|\right),$$

(35)

where

$$\mathcal{M}=\frac{\left(1-\mu \right)\eta p^2\left(y\right)}{2\left[\left(\lambda (2+q)-1\right)\left(1+q+q^2\right)\eta p^2\left(y\right){\delta }_3-\left[\eta p^2\left(y\right)+\left(2\lambda -1\right)(p^2\left(y\right)+2r\left(y\right))\right]\left(2\lambda -1\right){(1+q)}^2{\delta }_2^2\right]},$$

(36)

and

$$\mathcal{N}=\frac{1}{2\left(\lambda (2+q)-1\right)\left(1+q+q^2\right){\delta }_3},$$

(37)

where $\mu \in \mathbb{C}$, and ${\delta }_k$, $k\in \{2,3\}$, are given by (5).

Proof. 

If $\mathsf{{\rm Y}}\in {\mathcal{M}}_{\mathsf{\Sigma }}^{q,\lambda ,\alpha }\left(\eta ,n,m,p\left(y\right),r\left(y\right)\right)$. As in the proof of Theorem (1), from (30) and (33), we have

$$a_3-a_2^2=\frac{\eta {\mathcal{L}}_{p,r,1}\left(y\right)\left(u_2-v_2\right)}{2\left(\lambda (2+q)-1\right)\left(1+q+q^2\right){\delta }_3},$$

(38)

and multiplying (32) by $(1-\mu )$ we get

$$\left(1-\mu \right)a_2^2=\frac{\left(1-\mu \right){\eta }^2{\mathcal{L}}_{p,r,1}^3\left(y\right)\left(u_2+v_2\right)}{2\left[\left(\lambda (2+q)-1\right)\left(1+q+q^2\right)\eta {\mathcal{L}}_{p,r,1}^2\left(y\right){\delta }_3-\left[\eta {\mathcal{L}}_{p,r,1}^2\left(y\right)+\left(2\lambda -1\right){\mathcal{L}}_{p,r,2}\left(y\right)\right]\left(2\lambda -1\right){(1+q)}^2{\delta }_2^2\right]}.$$

(39)

Adding (38) and (39) leads to

$$a_3-\mu a_2^2=\eta p\left(y\right)\left[\left(\mathcal{M}+\mathcal{N}\right)u_2+\left(\mathcal{M}-\mathcal{N}\right)v_2\right],$$

(40)

where $\mathcal{M}$ and $\mathcal{N}$ are given by (36) and (37), and taking the absolute value of (40), from (23) we obtain the inequality (35). The proof is complete. □

Remark 2.

A simple computation shows that the inequality $\left|\mathcal{M}\right|\le \mathcal{N}$ is equivalent to

$$\left|\mu -1\right|\le \left|1-\frac{\left[\eta p^2\left(y\right)+\left(2\lambda -1\right)(p^2\left(y\right)+r\left(y\right))\right]\left(2\lambda -1\right){(1+q)}^2{\delta }_2^2}{\eta p^2\left(y\right)\left(\lambda (2+q)-1\right)\left(1+q+q^2\right){\delta }_3}\right|,$$

therefore, from Theorem (2) we get the next result:

If the function Υ given by (1) belongs to the class ${\mathcal{M}}_{\mathsf{\Sigma }}^{q,\lambda ,\alpha }\left(\eta ,n,m,p\left(y\right),r\left(y\right)\right)$, and $\eta \in {\mathbb{C}}^{\ast }$, then

$$\left|a_3-\mu a_2^2\right|\le \frac{\eta p\left(y\right)}{\left(\lambda (2+q)-1\right)\left(1+q+q^2\right){\delta }_3},$$

where $\mu \in \mathbb{C}$, with

$$\left|\mu -1\right|\le \left|1-\frac{\left[\eta p^2\left(y\right)+\left(2\lambda -1\right)(p^2\left(y\right)+r\left(y\right))\right]\left(2\lambda -1\right){(1+q)}^2{\delta }_2^2}{\eta p^2\left(y\right)\left(\lambda (2+q)-1\right)\left(1+q+q^2\right){\delta }_3}\right|,$$

and ${\delta }_k$, $k\in \{2,3\}$, are given by (5).

We conclude our result with the following consequence of Theorem (2)

Putting$p\left(y\right)=y$ and $r\left(y\right)=1$, we obtain the following corollary:

Corollary 4.

If Υ assumed (1) belongs to the class ${\mathcal{L}}_{\mathsf{\Sigma }}^{q,\lambda ,\alpha }\left(\eta ,n,m,y\right)$, and $\eta \in {\mathbb{C}}^{\ast }$, then

$$\left|a_3-\mu a_2^2\right|\le \left|\eta \right|\left|y\right|\left(|{\mathcal{M}}^{\ast }+{\mathcal{N}}^{\ast }|+|{\mathcal{M}}^{\ast }-{\mathcal{N}}^{\ast }|\right),$$

where

$${\mathcal{M}}^{\ast }=\frac{\left(1-\mu \right)\eta y^2}{2\left[\left(\lambda (2+q)-1\right)\left(1+q+q^2\right)\eta y^2{\delta }_3-\left[\eta y^2+\left(2\lambda -1\right)(y^2+2)\right]\left(2\lambda -1\right){(1+q)}^2{\delta }_2^2\right]},$$

and

$${\mathcal{N}}^{\ast }=\frac{1}{2\left(\lambda (2+q)-1\right)\left(1+q+q^2\right){\delta }_3},$$

where $\mu \in \mathbb{C}$, and ${\delta }_k$, $k\in \{2,3\}$, are given by (5).

If we put $p\left(y\right)=1$and$r\left(y\right)=2y,$ we obtain the following result.

Corollary 5.

If Υ assumed (1) belongs to the class ${\mathcal{J}}_{\mathsf{\Sigma }}^{q,\lambda ,\alpha }\left(\eta ,n,m,y\right)$, and $\eta \in {\mathbb{C}}^{\ast }$, then

$$\left|a_3-\mu a_2^2\right|\le \left|\eta \right|\left(|{\mathcal{M}}^{\ast \ast }+{\mathcal{N}}^{\ast \ast }|+|{\mathcal{M}}^{\ast \ast }-{\mathcal{N}}^{\ast \ast }|\right),$$

where

$${\mathcal{M}}^{\ast \ast }=\frac{\left(1-\mu \right)\eta }{2\left[\left(\lambda (2+q)-1\right)\left(1+q+q^2\right)\eta {\delta }_3-\left[\eta +\left(2\lambda -1\right)(1+4y)\right]\left(2\lambda -1\right){(1+q)}^2{\delta }_2^2\right]},$$

and

$${\mathcal{N}}^{\ast \ast }=\frac{1}{2\left(\lambda (2+q)-1\right)\left(1+q+q^2\right){\delta }_3},$$

where $\mu \in \mathbb{C}$, and ${\delta }_k$, $k\in \{2,3\}$, are given by (5).

Considering $p\left(y\right)=2y$ and $r\left(y\right)=-1,$ we get the following corollary.

Corollary 6.

If Υ assumed (1) belongs to the class ${\mathcal{T}}_{\mathsf{\Sigma }}^{q,\lambda ,\alpha }\left(\eta ,n,m,y\right)$, and $\eta \in {\mathbb{C}}^{\ast }$, then

$$\left|a_3-\mu a_2^2\right|\le \left|\eta \right|\left|2y\right|\left(|{\mathcal{M}}^{\ast \ast \ast }+{\mathcal{N}}^{\ast \ast \ast }|+|{\mathcal{M}}^{\ast \ast \ast }-{\mathcal{N}}^{\ast \ast \ast }|\right),$$

where

$${\mathcal{M}}^{\ast \ast \ast }=\frac{2\left(1-\mu \right)\eta y^2}{4\left(\lambda (2+q)-1\right)\left(1+q+q^2\right)\eta y^2{\delta }_3-\left[4\eta y^2+\left(2\lambda -1\right)(4y^2-2)\right]\left(2\lambda -1\right){(1+q)}^2{\delta }_2^2},$$

and

$${\mathcal{N}}^{\ast \ast \ast }=\frac{1}{2\left(\lambda (2+q)-1\right)\left(1+q+q^2\right){\delta }_3},$$

where $\mu \in \mathbb{C}$, and ${\delta }_k$, $k\in \{2,3\}$, are given by (5).

Now the following examples are presented here to illustrate our results.

For $\eta =1$ and $\lambda =1.$ Therefore, from Theorems (1) and (2).

Example 2.

Let the function Υ given by (1) belongs to the class ${\mathcal{M}}_{\mathsf{\Sigma }}^{q,\lambda ,1}\left(1,n,m,p\left(y\right),r\left(y\right)\right)$, then

$$\left|a_2\right|\le \frac{\left|\eta \right|\left|p\left(y\right)\right|\sqrt{p\left(y\right)}}{\sqrt{\left|\left(q+1\right)\left(1+q+q^2\right)p^2\left(y\right){\delta }_3-2\left(p^2\left(y\right)+r\left(y\right)\right){(1+q)}^2{\delta }_2^2\right|}},$$

$$\left|a_3\right|\le \frac{\left|p\right(y\left)\right|}{(1+q)\left(1+q+q^2\right){\delta }_3}+\frac{p^2\left(y\right)}{{(1+q)}^2{\delta }_2^2},$$

and

$$\left|a_3-\mu a_2^2\right|\le \left|p\left(y\right)\right|\left(|K+L|+|K-L|\right),$$

with

$$K={\displaystyle \frac{\left(1-\mu \right)p^2\left(y\right)}{2\left[(1+q)\left(1+q+q^2\right)p^2\left(y\right){\delta }_3-2(p^2\left(y\right)+r\left(y\right)){(1+q)}^2{\delta }_2^2\right]}},$$

and

$$L=\frac{1}{2\left(\lambda (2+q)-1\right)\left(1+q+q^2\right){\delta }_3},$$

where $\mu \in \mathbb{C}$, and ${\delta }_k$, $k\in \{2,3\}$, are given by (5).

Putting $q\to 1^{-}$ in Theorem (2), we get the following Example:

Example 3.

If Υ assumed (1) belongs to the class $\underset{q\to 1^{-}}{lim}{\mathcal{M}}_{\mathsf{\Sigma }}^{q,\lambda ,\alpha }\left(\eta ,n,m,p\left(y\right),r\left(y\right)\right)$, and $\eta \in {\mathbb{C}}^{\ast }$, then

$$\left|a_3-\mu a_2^2\right|\le \left|\eta \right|\left|p\left(y\right)\right|\left(|\mathcal{P}+\mathcal{R}|+|\mathcal{P}-\mathcal{R}|\right),$$

where

$$\mathcal{P}={\displaystyle \frac{\left(1-\mu \right)\eta p^2\left(y\right)}{2\left[3\left(3\lambda -1\right)\eta p^2\left(y\right){\delta }_3-4\left[\eta p^2\left(y\right)+\left(2\lambda -1\right)(p^2\left(y\right)+2r\left(y\right))\right]\left(2\lambda -1\right){\delta }_2^2\right]}},$$

and

$$\mathcal{R}=\frac{1}{6\left(3\lambda -1\right){\delta }_3},$$

where $\mu \in \mathbb{C}$, and ${\delta }_k$, $k\in \{2,3\}$, are given by (5).

Remark 3.

We mention that all the above estimations for the coefficients $\left|a_2\right|$, $\left|a_3\right|$, and Fekete-Szegö problem for the function class ${\mathcal{M}}_{\mathsf{\Sigma }}^{q,\lambda ,\alpha }\left(\eta ,n,m,p\left(y\right),r\left(y\right)\right)$are not sharp. To find the sharp upper bounds for the above functionals remains an interesting open problem, as well as those for $\left|a_n\right|$, $n\ge 4$.

4. Conclusions

The original results presented in this paper refer to the study on the new defined subclass of analytic functions ${\mathcal{M}}_{\mathsf{\Sigma }}^{q,\lambda ,\alpha }\left(\eta ,n,m,p\left(y\right),r\left(y\right)\right)$. We mainly get upper bounds of the initial Taylors coefficients for functions in this subclass are found. Furthermore, we find the Fekete-Szegö inequalities for function in these classes. Several consequences of the results are also pointed out as examples. For future investigations, the class could be transformed considering aspects of fuzzy differential subordination and superordination and quantum calculus. It can be studied the coefficient inequalities, partial sums of functions for this subclass and second Hankel for this class, the functions in the class can be evaluated regarding distortion properties and also starlikeness and convexity theorems can be obtained providing radii for close-to-convexity, starlikeness and convexity. The study of functions from this subclass could provide interesting results given their symmetry properties.