Coefficient Estimates for Quasi-Subordination Classes Connected with the Combination of q-Convolution and Error Function

Abstract

We utilize quasi-subordination to analyze and introduce several new classes, and we construct a new operator by combining the error function and q-convolution. Additionally, we obtain estimates for the Fekete Szego functional and the Taylor–Maclaurin coefficients for functions in $\left|c_2\right|$ and $\left|c_3\right|$ new classes. Moreover, we discuss some applications of the operator.

1. Introduction, Definitions and Preliminaries

The error function $er\left(\zeta \right)$ defined by

$${\displaystyle er\left(\zeta \right)=\frac{2}{\sqrt{\pi }}\underset{0}{\overset{\zeta }{\int }}exp\left(-t^2\right)dt=\frac{2}{\sqrt{\pi }}\sum _{j=0}^{\infty }\frac{{(-1)}^j}{(2j+1)j!}{\zeta }^{2j+1}}$$

(1)

is a topic that has been the subject of considerable study and application during the previous few years. In [1,2,3,4], several examples of error functions are defined, and inequalities are given. The authors of [5,6] investigate the characteristics of complementary error functions which are present in almost every field of applied mathematics and mathematical physics, such as probability and statistics ([7]) and data analysis ([8]). Its inverse, introduced by Carlitz [9], which we abbreviate as inver, can be found in many branches of natural sciences and mathematics. Examples include solutions to Einstein’s scalar-field equations, concentration-dependent diffusion issues (see [10,11]), and the heat conduction problem (see [7,12]).

We let $\mathcal{A}$ stand for the class for analytical functions of the form

$$f\left(\zeta \right):=\zeta +\sum _{j=2}^{\infty }{c}_j{\zeta }^j,\zeta \in \mathbf{U}:=\{\zeta \in \mathbb{C}\mathrm{is}\mathrm{the}\mathrm{complex}\mathrm{plane}:|\zeta |<1\},$$

(2)

and we let $\mathcal{S}\subset \mathcal{A}$ consist of functions that are univalent in $\mathbf{U}$.

If function $\mathsf{{\rm Y}}\in \mathcal{A}$ is given by

$$\mathsf{{\rm Y}}\left(\zeta \right):=\zeta +\sum _{j=2}^{\infty }{b}_j{\zeta }^j,\zeta \in \mathbf{U},$$

(3)

then, the Hadamard (or convolution) product of f and $\mathsf{{\rm Y}}$ is defined by

$$(f\ast \mathsf{{\rm Y}})\left(\zeta \right):=\zeta +\sum _{j=2}^{\infty }{c}_j{b}_j{\zeta }^j,\zeta \in \mathbf{U}.$$

We let $E_r$ be a normalized analytic function which is obtained from (1) and given by

$$E_r\left(\zeta \right)=\frac{\sqrt{\pi \zeta }}{2}er\left(\sqrt{\zeta }\right)=\zeta +\sum _{j=2}^{\infty }\frac{{(-1)}^{j-1}}{(2j-1)\left(j-1\right)!}{\zeta }^j.$$

We define an analytic function as follows:

$$\mathcal{F}\left(\zeta \right)=(f\ast E_r)\left(\zeta \right)=\zeta +\sum _{j=2}^{\infty }\frac{{(-1)}^{j-1}}{(2j-1)\left(j-1\right)!}{c}_j{\zeta }^j.$$

(4)

Robertson [13] introduced the concept of quasi-subordination. For twoanalytic functions f and $\mathcal{G}$, function f is quasisubordinated to $\mathcal{G}$, written as follows:

$$f\left(\zeta \right){\prec }_t\mathcal{G}\left(\zeta \right).$$

If there exist analytic functions $\psi$ and $\omega$ with $\left|\psi \left(\zeta \right)\right|\le 1,$ $\omega \left(0\right)=0$ and $\left|\omega \left(\zeta \right)\right|<1,$ such that $f\left(\zeta \right)=\psi \left(\zeta \right)\mathcal{G}\left(\omega \right(\zeta \left)\right),$ we observe that when $\psi \left(\zeta \right)=1,$ then $f\left(\zeta \right)=\mathcal{G}\left(\omega \right(\zeta \left)\right),$ so that $f\left(\zeta \right)\prec \mathcal{G}\left(\zeta \right).$ We also notice that if $\omega \left(\zeta \right)=\zeta ,$ then $f\left(\zeta \right)=\psi \left(\zeta \right)\mathcal{G}\left(\zeta \right)$ and it said that f is majorized by $\mathcal{G}$ and written as $f\left(\zeta \right)\ll \mathcal{G}\left(\zeta \right).$ Hence, it is obvious that quasi-subordination is a generalization of subordination as well as majorization (see [1,14,15] for works related to quasi-subordination).

Domain $\mathcal{D}$ in $\mathbb{C}$ is said to be starlike with respect to point ${\zeta }_0$if the straight line segment connecting any point in $\mathcal{D}$ to ${\zeta }_0$ is contained in $\mathcal{D}$. Function $f\in \mathcal{A}$ is said to be starlike with respect to the origin (or starlike) if $\mathbf{U}$ is mapped by $f$ onto a domain starlike with respect to point zero. We let ${\mathcal{S}}^{\ast }$ denote the class of all starlike functions in $\mathcal{A}$. Analytic description of class ${\mathcal{S}}^{\ast }$ is given by

$${\mathcal{S}}^{\ast }=\left\{f\in \mathcal{A}:\Re \left\{\frac{\zeta f^{\prime }\left(\zeta \right)}{f\left(\zeta \right)}\right\}>0\left(\zeta \in \mathbf{U}\right)\right\}.$$

A special subclass of ${\mathcal{S}}^{\ast }$ is the class of starlike functions of order $\alpha \left(0\le \alpha <1\right)$ given by

$${\mathcal{S}}^{\ast }\left(\alpha \right)=\left\{f\in \mathcal{A}:\Re \left\{\frac{\zeta f^{\prime }\left(\zeta \right)}{f\left(\zeta \right)}\right\}>\alpha \left(0\le \alpha <1;\zeta \in \mathbf{U}\right)\right\}.$$

Domain $\mathcal{R}$ in $\mathbb{C}$ is called convex if the line segment joining any two points in $\mathcal{R}$ lies entirely in $\mathcal{R}$. If function $f\in \mathcal{A}$ maps $\mathbf{U}$ onto a convex domain, then f is called a convex function. We let $\mathcal{K}$ denote the class of all convex functions in $\mathcal{A}$. Analytic description of class $\mathcal{K}$ is given by

$$\mathcal{K}=\left\{f\in \mathcal{A}:\Re \left\{1+\frac{\zeta f^{″}\left(\zeta \right)}{f^{\prime }\left(\zeta \right)}\right\}>0\left(\zeta \in \mathbf{U}\right)\right\}.$$

For $0\le \alpha <1,$ we let $S^{\ast }\left(\alpha \right)$ represent for the class of convex functions of order $\alpha$ in $\mathbf{U}$ such that a special subclass of $\mathcal{K}$ is the class of convex functions of order $\alpha \left(0\le \alpha <1\right)$ given by (see [16])

$$\mathcal{K}\left(\alpha \right)=\left\{f\in \mathcal{A}:\Re \left\{1+\frac{\zeta f^{″}\left(\zeta \right)}{f^{\prime }\left(\zeta \right)}\right\}>\alpha \left(0\le \alpha <1;\zeta \in \mathbf{U}\right)\right\}.$$

Throughout this paper, it is assumed that

$$\phi \left(\zeta \right)=1+d_1\zeta +d_2{\zeta }^2+\dots ,(d_1>0)$$

(5)

is an holomorphic function in the open unit disk $\mathbf{U}.$Motivated by [17], we define the following classes.

Definition 1.

Class $S_t^{\ast }\left(\phi \right)$ consists of functions $f\in \mathcal{A}$ satisfying quasi-subordination

$$\frac{\zeta f^{\prime }\left(\zeta \right)}{f\left(\zeta \right)}-1{\prec }_t\phi \left(\zeta \right)-1,\zeta \in \mathbf{U}.$$

Definition 2.

Class ${\mathcal{K}}_t\left(\phi \right)$ consists of functions $f\in \mathcal{A}$ satisfying quasi-subordination

$$\frac{\zeta f^{″}\left(\zeta \right)}{f^{\prime }\left(\zeta \right)}{\prec }_t\phi \left(\zeta \right)-1,\zeta \in \mathbf{U}.$$

Srivastava [18] made use of various operators of q-calculus and fractional q-calculus. The q-shifted factorial is defined for $\lambda ,q\in \mathbb{C}$ and $n\in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\},\mathbb{N}=\{1,2,\dots \}$ as follows:

$${(\lambda ;q)}_j=\left\{\begin{array}{ll}1& j=0,\\ \left(1-\lambda \right)\left(1-\lambda q\right)\dots \left(1-\lambda q^{j-1}\right)& j\in \mathbb{N}.\end{array}\right.$$

In addition, ${\left[j\right]}_q$ denotes a basic q-number defined as follows:

$${\left[j\right]}_q:=\frac{1-q^j}{1-q}.$$

(6)

Using definition Formula (6), we have the next two products:

(i)

For any non-negative integer j, the q-shifted factorial is given by

$${\left[j\right]}_q!:=\left\{\begin{array}{ccc}1,\hfill & \mathrm{if}\hfill & j=0,\hfill \\ {\displaystyle \prod _{n=1}^j}{\left[n\right]}_q,\hfill & \mathrm{if}\hfill & j\in \mathbb{N}.\hfill \end{array}\right.$$

(ii)

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

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

For $0<q<1$, the q-derivative operator [19] (see also [20]) for $\mathcal{F}\ast \mathsf{{\rm Y}}$ is defined by

$$\begin{array}{c}{\mathcal{D}}_q\left(\mathcal{F}\ast \mathsf{{\rm Y}}\right)\left(\zeta \right):={\mathcal{D}}_q\left(\zeta +\sum _{j=2}^{\infty }\frac{{(-1)}^{j-1}}{(2j-1)\left(j-1\right)!}{c}_j{b}_j{\zeta }^j\right)\\ =\frac{\left(\mathcal{F}\ast \mathsf{{\rm Y}}\right)\left(\zeta \right)-\left(\mathcal{F}\ast \mathsf{{\rm Y}}\right)\left(q\zeta \right)}{\zeta (1-q)}=1+\sum _{j=2}^{\infty }\frac{{(-1)}^{j-1}}{(2j-1)\left(j-1\right)!}{\left[j\right]}_q{c}_j{b}_j{\zeta }^{j-1},\zeta \in \mathbf{U},\end{array}$$

where

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

(7)

In recent years, convolution has been used to compute the error function, and the fractional q-calculus has been applied in the geometric function theory, which has a new generalization of the classical operators to define a new operator, ${\mathcal{H}}_{\mathsf{{\rm Y}}}^{\lambda ,q}$, as follows:

For $\lambda >-1$ and $0<q<1$, El-Deeb et al. [20] (see also Srivastava and El-Deeb [21]) defined linear operator ${\mathcal{H}}_{\mathsf{{\rm Y}}}^{\lambda ,q}:\mathcal{A}\to \mathcal{A}$ by

$${\mathcal{H}}_{\mathsf{{\rm Y}}}^{\lambda ,q}\mathcal{F}\left(\zeta \right)\ast {\mathcal{M}}_{q,\lambda +1}\left(\zeta \right)=\zeta {\mathcal{D}}_q\left(\mathcal{F}\ast \mathsf{{\rm Y}}\right)\left(\zeta \right),\zeta \in \mathbf{U},$$

where function ${\mathcal{M}}_{q,\lambda +1}$ is given by

$${\mathcal{M}}_{q,\lambda +1}\left(\zeta \right):=\zeta +\sum _{j=2}^{\infty }\frac{{[\lambda +1]}_{q,j-1}}{{[j-1]}_q!}{\zeta }^j,\zeta \in \mathbf{U}.$$

A simple computation shows that

$$\begin{array}{ccc}\hfill {\mathcal{H}}_{\mathsf{{\rm Y}}}^{\lambda ,q}\mathcal{F}\left(\zeta \right)& :& =\zeta +\sum _{j=2}^{\infty }\frac{{(-1)}^{j-1}{\left[j\right]}_q!}{(2j-1)\left(j-1\right)!{[\lambda +1]}_{q,j-1}}{c}_j{b}_j{\zeta }^j,\hfill \\ & =& \zeta +\sum _{j=2}^{\infty }{\rho }_j{c}_j{\zeta }^j(\lambda >-1,0<q<1,\zeta \in \mathbf{U}),\hfill \end{array}$$

(8)

where

$${\rho }_j=\frac{{(-1)}^{j-1}{\left[j\right]}_q!}{(2j-1)\left(j-1\right)!{[\lambda +1]}_{q,j-1}}{b}_j.$$

(9)

From definition Relation (8), we can easily verify that for all $\mathcal{F}\in \mathcal{A},$

$$\begin{array}{ccc}& \left(\mathrm{i}\right)& {[\lambda +1]}_q{\mathcal{H}}_{\mathsf{{\rm Y}}}^{\lambda ,q}\mathcal{F}\left(\zeta \right)={\left[\lambda \right]}_q{\mathcal{H}}_{\mathsf{{\rm Y}}}^{\lambda +1,q}\mathcal{F}\left(\zeta \right)+q^{\lambda }\zeta {\mathcal{D}}_q\left({\mathcal{H}}_{\mathsf{{\rm Y}}}^{\lambda +1,q}\mathcal{F}\left(\zeta \right)\right),\zeta \in \mathbf{U};\hfill \\ & \left(\mathrm{ii}\right)& {\mathcal{I}}_{\mathsf{{\rm Y}}}^{\lambda }\mathcal{F}\left(\zeta \right):=\underset{q\to 1^{-}}{lim}{\mathcal{H}}_{\mathsf{{\rm Y}}}^{\lambda ,q}\mathcal{F}\left(\zeta \right)=\zeta +\sum _{j=2}^{\infty }{\displaystyle \frac{j}{{(\lambda +1)}_{j-1}}}\frac{{(-1)}^{j-1}}{(2j-1)}{a}_j{b}_j{\zeta }^j,\zeta \in \mathbf{U},\hfill \end{array}$$

(10)

the relations hold.

Remark 1.

Taking different particular cases for coefficients $b_j$, we obtain the next special cases for operator ${\mathcal{H}}_{\mathsf{{\rm Y}}}^{\lambda ,q}$:

(i) 

For $b_j=1,$ we obtain operator ${\mathcal{E}}_q^{\lambda }$ defined by (see [22])

$${\mathcal{E}}_q^{\lambda }\mathcal{F}\left(\zeta \right):=\zeta +\sum _{j=2}^{\infty }\frac{{(-1)}^{j-1}{\left[j\right]}_q!}{(2j-1)\left(j-1\right)!{[\lambda +1]}_{q,j-1}}{c}_j{\zeta }^j,(\lambda >-1,0<q<1,\zeta \in \mathbf{U});$$

(11)

(ii) 

For $b_j={\displaystyle \frac{{(-1)}^{j-1}\Gamma (\upsilon +1)}{4^{j-1}(j-1)!\Gamma (j+\upsilon )}}$, $\upsilon >0$, we obtain operator ${\mathcal{E}}_{\upsilon ,q}^{\lambda }$ defined by (see El-Deeb [23])

$$\begin{array}{ccc}\hfill {\mathcal{E}}_{\upsilon ,q}^{\lambda }\mathcal{F}\left(\zeta \right):& =& \zeta +\sum _{j=2}^{\infty }\frac{{(-1)}^{2(j-1)}\Gamma (\upsilon +1){\left[j\right]}_q!}{4^{j-1}(2j-1){\left((j-1)!\right)}^2\Gamma (j+\upsilon ){[\lambda +1]}_{q,j-1}}{c}_j{\zeta }^j\hfill \\ & =& \zeta +\sum _{j=2}^{\infty }{\xi }_j{c}_j{\zeta }^j,\hfill \end{array}$$

(12)

where $\upsilon >0,\lambda >-1,0<q<1,\zeta \in \mathbf{U}$ and

$${\xi }_j:=\frac{{(-1)}^{2\left(j-1\right)}\Gamma (\upsilon +1){\left[j\right]}_q!}{4^{j-1}(2j-1){\left((j-1)!\right)}^2\Gamma (j+\upsilon ){[\lambda +1]}_{q,j-1}};$$

(13)

(iii) 

For $b_j={\left({\displaystyle \frac{n+1}{n+j}}\right)}^{\alpha }$, $n\ge 0$, $\alpha >0$, we have operator ${\mathcal{E}}_{n,q}^{\lambda ,\alpha }$ defined by

$${\mathcal{E}}_{n,q}^{\lambda ,\alpha }\mathcal{F}\left(\zeta \right):=\zeta +\sum _{j=2}^{\infty }\frac{{(-1)}^{j-1}{\left(n+1\right)}^{\alpha }{\left[j\right]}_q!}{(2j-1){\left(n+j\right)}^{\alpha }\left(j-1\right)!{[\lambda +1]}_{q,j-1}}{c}_j{\zeta }^j,\zeta \in \mathbf{U};$$

(14)

(iv) 

For $b_j={\displaystyle \frac{{\rho }^{j-1}}{(j-1)!}}{e}^{-\rho }$, $\rho >0$, we obtain an operator defined by

$${\mathcal{E}}_q^{\lambda ,\rho }\mathcal{F}\left(\zeta \right):=\zeta +\sum _{j=2}^{\infty }\frac{{(-1)}^{j-1}{\rho }^{j-1}{\left[j\right]}_q!e^{-\rho }}{(2j-1){\left(\left(j-1\right)!\right)}^2{[\lambda +1]}_{q,j-1}}{c}_j{\zeta }^j,\zeta \in \mathbf{U}.$$

(15)

By using the idea of quasi-subordination and a new operator ${\mathcal{H}}_{\mathsf{{\rm Y}}}^{\lambda ,q}$, we define new classes ${\mathcal{S}}_{q,\eta }^{\theta ,\lambda }\left(\varphi ;\mathsf{{\rm Y}}\right)$ and ${\mathcal{C}}_{q,\eta }^{\theta ,\lambda }\left(\varphi ;\mathsf{{\rm Y}}\right)$ as follows:

Definition 3.

For $\frac{-\pi }{2}<\theta <\frac{\pi }{2},0<q<1$and $\eta \ne 0,$function $\mathcal{F}\in \mathcal{A}$ has the form of (2) and Υ is given by (3); function $f$is said to be in class $\mathcal{S}$${}_{q,\eta }^{\theta ,\lambda }\left(\varphi ;\mathsf{{\rm Y}}\right)$ if

$$\frac{1}{\eta }\left[\left(1+itan\theta \right)\left(\frac{\zeta {\mathcal{D}}_q\left({\mathcal{H}}_{\mathsf{{\rm Y}}}^{\lambda ,q}\mathcal{F}\left(\zeta \right)\right)}{{\mathcal{H}}_{\mathsf{{\rm Y}}}^{\lambda ,q}\mathcal{F}\left(\zeta \right)}\right)-itan\theta -1\right]{\prec }_t\varphi \left(\zeta \right)-1,\left(\zeta \in \mathbf{U}\right).$$

(16)

Remark 2.

(i) 

If $q\to 1^{-}$ in Definition 2, we have $\underset{q\to 1^{-}}{lim}{\mathcal{S}}_{q,\eta }^{\theta ,\lambda }\left(\varphi ;\mathsf{{\rm Y}}\right)={\mathcal{S}}_{\eta }^{\theta ,\lambda }\left(\varphi ;\mathsf{{\rm Y}}\right)$, where ${\mathcal{S}}_{\eta }^{\theta ,\lambda }\left(\varphi ;\mathsf{{\rm Y}}\right)$ represents functions $\mathcal{F}\in \mathcal{A}$ that satisfy

$$\frac{1}{\eta }\left[\left(1+itan\theta \right)\left(\frac{\zeta {\left({\mathcal{I}}_{\mathsf{{\rm Y}}}^{\lambda }\mathcal{F}\left(\zeta \right)\right)}^{{}^{\prime }}}{{\mathcal{I}}_{\mathsf{{\rm Y}}}^{\lambda }\mathcal{F}\left(\zeta \right)}\right)-itan\theta -1\right]{\prec }_t\varphi \left(\zeta \right)-1,\left(\zeta \in \mathbf{U}\right);$$

(ii) 

Fixing $b_j={\displaystyle \frac{{(-1)}^{j-1}\Gamma (\upsilon +1)}{4^{j-1}(j-1)!\Gamma (j+\upsilon )}}$, $\upsilon >0$ in Definition 2, we obtain class $\mathcal{S}$${}_{\upsilon ,q,\eta }^{\theta ,\lambda }\left(\varphi \right)$ that represents functions $\mathcal{F}\in \mathcal{A}$ that satisfy (16) for ${\mathcal{H}}_{\mathsf{{\rm Y}}}^{\lambda ,q}$ replaced with ${\mathcal{E}}_{\upsilon ,q}^{\lambda }$ (12);

(iii) 

Putting $b_j={\left({\displaystyle \frac{n+1}{n+j}}\right)}^{\alpha }$, $\alpha >0$, $n\ge 0$in Definition 2, we obtain class $\mathcal{S}$${}_{q,\eta ,n}^{\theta ,\lambda ,\alpha }\left(\varphi \right)$ that represents functions $\mathcal{F}\in \mathcal{A}$ that satisfy (16) for ${\mathcal{H}}_{\mathsf{{\rm Y}}}^{\lambda ,q}$ replaced with ${\mathcal{E}}_{n,q}^{\lambda ,\alpha }$ (14);

(iv) 

Putting $b_j={\displaystyle \frac{{\rho }^{j-1}}{(j-1)!}}{e}^{-\rho }$, $\rho >0$in Definition 2, we obtain class $\mathcal{S}$${}_{q,\eta }^{\theta ,\lambda ,\rho }\left(\varphi \right)$ that represents functions $\mathcal{F}\in \mathcal{A}$ that satisfy (16) for ${\mathcal{H}}_{\mathsf{{\rm Y}}}^{\lambda ,q}$ replaced with ${\mathcal{E}}_q^{\lambda ,\rho }$ (15).

Definition 4.

For $\frac{-\pi }{2}<\theta <\frac{\pi }{2},0<q<1$ and $\eta \ne 0,$ function $\mathcal{F}\in \mathcal{A}$ has the form of (4) and Υ is given by (3); function $\mathcal{F}$is said to be in class ${\mathcal{C}}_{q,\eta }^{\theta ,\lambda }\left(\varphi ;\mathsf{{\rm Y}}\right)$ if

$$\frac{1}{\eta }\left[\left(1+itan\theta \right)\left(\frac{{\left(\zeta {\mathcal{D}}_q\left({\mathcal{H}}_{\mathsf{{\rm Y}}}^{\lambda ,q}\mathcal{F}\left(\zeta \right)\right)\right)}^{{}^{{}^{\prime }}}}{{\mathcal{D}}_q\left({\mathcal{H}}_{\mathsf{{\rm Y}}}^{\lambda ,q}\mathcal{F}\left(\zeta \right)\right)}\right)-itan\theta -1\right]{\prec }_t\varphi \left(\zeta \right)-1,\left(\zeta \in \mathbf{U}\right).$$

Remark 3.

(i) 

Putting $q\to 1^{-}$ in Definition 3, we obtain $\underset{q\to 1^{-}}{lim}{\mathcal{C}}_{q,\eta }^{\theta ,\lambda }\left(\varphi ;\mathsf{{\rm Y}}\right)={\mathcal{C}}_{\eta }^{\theta ,\lambda }\left(\varphi ;\mathsf{{\rm Y}}\right)$, where ${\mathcal{C}}_{\eta }^{\theta ,\lambda }\left(\varphi ;\mathsf{{\rm Y}}\right)$ represents functions $\mathcal{F}\in \mathcal{A}$ that satisfy

$$\frac{1}{\eta }\left[\left(1+itan\theta \right)\left(\frac{\zeta {\left({\mathcal{I}}_{\mathsf{{\rm Y}}}^{\lambda }\mathcal{F}\left(\zeta \right)\right)}^{{}^{{}^{\prime \prime }}}}{{\left({\mathcal{I}}_{\mathsf{{\rm Y}}}^{\lambda }\mathcal{F}\left(\zeta \right)\right)}^{{}^{\prime }}}\right)\right]{\prec }_t\varphi \left(\zeta \right)-1,\left(\zeta \in \mathbf{U}\right);$$

(ii) 

Putting $b_j={\displaystyle \frac{{(-1)}^{j-1}\Gamma (\upsilon +1)}{4^{j-1}(j-1)!\Gamma (j+\upsilon )}}$, $\upsilon >0$ in Definition 3, we obtain class ${\mathcal{C}}_{\upsilon ,q,\eta }^{\theta ,\lambda }\left(\varphi \right)$ that represents functions $\mathcal{F}\in \mathcal{A}$ that satisfy (16) for ${\mathcal{H}}_{\mathsf{{\rm Y}}}^{\lambda ,q}$ replaced with ${\mathcal{E}}_{\upsilon ,q}^{\lambda }$ (12);

(iii) 

Putting $b_j={\left({\displaystyle \frac{n+1}{n+j}}\right)}^{\alpha }$, $\alpha >0$, $n\ge 0$ in Definition 3, we obtain class ${\mathcal{C}}_{q,\eta ,n}^{\theta ,\lambda ,\alpha }\left(\varphi \right)$ that represents functions $\mathcal{F}\in \mathcal{A}$ that satisfy (16) for ${\mathcal{H}}_{\mathsf{{\rm Y}}}^{\lambda ,q}$ replaced with ${\mathcal{E}}_{n,q}^{\lambda ,\alpha }$ (14);

(iv) 

Putting $b_j={\displaystyle \frac{{\rho }^{j-1}}{(j-1)!}}{e}^{-\rho }$, $\rho >0$ in Definition 3, we obtain class ${\mathcal{C}}_{q,\eta }^{\theta ,\lambda ,\rho }\left(\varphi \right)$ that represents functions $\mathcal{F}\in \mathcal{A}$ that satisfy (16) for ${\mathcal{H}}_{\mathsf{{\rm Y}}}^{\lambda ,q}$ replaced with ${\mathcal{E}}_q^{\lambda ,\rho }$ (15).

The following lemma is needed to prove our main results.

Lemma 1.

[24] If $\vartheta \in \mathsf{\Omega },$ then

$$\left|w_2-\vartheta w_1^2\right|\le \left\{\begin{array}{c}-\vartheta if\vartheta <-1,\\ 1if-1\le \vartheta \le 1,\\ \vartheta if\vartheta >1.\end{array}\right.$$

When $\vartheta <-1$ or $\vartheta >1,$the equality holds if and only if $w\left(\zeta \right)=\zeta$ or one of its rotations. If $-1<\vartheta <1,$ then equality holds if $w\left(\zeta \right)={\zeta }^2$ or one of its rotations. Equality holds for $\vartheta =$$-1$ if and only if

$$w\left(\zeta \right)=\zeta \frac{\mu +\zeta }{1+\mu \zeta }\left(0\le \mu \le 1\right),$$

or one of its rotations, while for $\vartheta =1$ the equality holds if and only if

$$w\left(\zeta \right)=-\zeta \frac{\mu +\zeta }{1+\mu \zeta }\left(0\le \mu \le 1\right),$$

or one of its rotations. Although the above is sharp, it can be improved in the case when $-1<t<1$,

$$\begin{array}{ccc}\hfill \left|w_2-\vartheta w_1^2\right|+\left(1+\vartheta \right){\left|w_1\right|}^2& \le & 1\left(-1<\vartheta \le 0\right),\hfill \\ \hfill \left|w_2-\vartheta w_1^2\right|+\left(1-\vartheta \right){\left|w_1\right|}^2& \le & 1\left(0<\vartheta <1\right).\hfill \end{array}$$

We now focus on the expansion of the idea of subordination. The claim of quasi-subordination is the main distinction. Therefore, we must strengthen Theorems 1 and 2 in the following ways to obtain estimates for the Fekete Szego functional and for the function coefficients, $\left|c_2\right|$ and $\left|c_3\right|$, of the classes, ${\mathcal{S}}_{q,\eta }^{\theta ,\lambda }\left(\varphi ;\mathsf{{\rm Y}}\right)$ and ${\mathcal{C}}_{q,\eta }^{\theta ,\lambda }\left(\varphi ;\mathsf{{\rm Y}}\right)$.

2. Main Results

Theorem 1.

We let $\frac{-\pi }{2}<\theta <\frac{\pi }{2},\lambda >-1,0<q<1$and $\eta \ne 0.$ If function $\mathcal{F}$ given by (4) belongs to class ${\mathcal{S}}_{q,\eta }^{\theta ,\lambda }\left(\varphi ;\mathsf{{\rm Y}}\right)$, then

$$\left|c_2\right|\le \frac{3\eta r_1}{\left(1+itan\theta \right)\left(1-{\left[2\right]}_q\right){\rho }_2},$$

(17)

$$\left|c_3\right|\le \frac{10\eta }{\left(1+itan\theta \right)\left({\left[3\right]}_q-1\right){\rho }_3}\left(r_1+max\left\{r_1,\left|\frac{\eta r_1^2}{\left(1+itan\theta \right)\left(1-{\left[2\right]}_q\right){\rho }_2}\right|+\left|r_2\right|\right\}\right),$$

(18)

and for any complex number ϑ,

$$\left|c_3-\vartheta c_2^2\right|\le \frac{10\eta }{\left(1+itan\theta \right)\left({\left[3\right]}_q-1\right){\rho }_3}\left(r_1+max\left\{r_1,\left|\frac{\left[10\left(1-{\left[2\right]}_q\right){\rho }_2+9\vartheta \eta \left({\left[3\right]}_q-1\right){\rho }_3\right]\eta r_1^2}{10\left(1+itan\theta \right){\left(1-{\left[2\right]}_q\right)}^2{\rho }_2^2}\right|+\left|r_2\right|\right\}\right),$$

(19)

where ${\rho }_j$ for $j=2,3$are given by (9).

Proof. 

We let $f\in {\mathcal{S}}_{q,\eta }^{\theta ,\lambda }\left(\varphi ;\mathsf{{\rm Y}}\right)$ and there exist analytic functions $\psi$ and $\omega$ such that $\left|\phi \left(\zeta \right)\right|\le 1,$ $\varpi \left(0\right)=0$ and $\left|\varpi \left(\zeta \right)\right|<1,$ then

$$\frac{1}{\eta }\left[\left(1+itan\theta \right)\left(\frac{\zeta {\left({\mathcal{H}}_{\mathsf{{\rm Y}}}^{\lambda ,q}\mathcal{F}\left(\zeta \right)\right)}^{{}^{\prime }}}{{\mathcal{H}}_{\mathsf{{\rm Y}}}^{\lambda ,q}\mathcal{F}\left(\zeta \right)}\right)-itan\theta -1\right]=\psi \left(\zeta \right)\left(\varphi \left(\omega \left(\zeta \right)\right)-1\right).$$

(20)

Since

$$\frac{\zeta {\left({\mathcal{H}}_{\mathsf{{\rm Y}}}^{\lambda ,q}\mathcal{F}\left(\zeta \right)\right)}^{\prime }}{{\mathcal{H}}_{\mathsf{{\rm Y}}}^{\lambda ,q}\mathcal{F}\left(\zeta \right)}=1+\frac{1-{\left[2\right]}_q}{3}{\rho }_2{c}_2\zeta +\left(\frac{{\left[3\right]}_q-1}{10}{\rho }_3{c}_3+\frac{1-{\left[2\right]}_q}{9}{\rho }_2^2{c}_2^2\right){\zeta }^2+\dots ,$$

(21)

and

$$\psi \left(\zeta \right)\left(\varphi \left(\omega \left(\zeta \right)\right)-1\right)=r_1{d}_0{\omega }_1\zeta +\left(r_1{d}_1{\omega }_1+d_0\left(r_1{\omega }_2+r_2{\omega }_1^2\right)\right){\zeta }^2+\dots .$$

(22)

From (20), (21) and (22), we obtain

$$c_2=\frac{3\eta r_1{d}_0{\omega }_1}{\left(1+itan\theta \right)\left(1-{\left[2\right]}_q\right){\rho }_2},$$

$$c_3=\frac{10\eta }{\left(1+itan\theta \right)\left(1-{\left[3\right]}_q\right){\rho }_3}\left[r_1{d}_1{\omega }_1+r_1{d}_0{\omega }_2+d_0\left(r_2-\frac{\eta r_1^2{d}_0}{\left(1+i tan\theta \right)\left(1-{\left[2\right]}_q\right){\rho }_2}\right){\omega }_1^2\right],$$

and

$$\left|c_3-\vartheta c_2^2\right|\le \frac{10\eta }{\left(1+i tan\theta \right)\left({\left[3\right]}_q-1\right){\rho }_3}\left[{}^{\begin{array}{c}\end{array}}\left|r_1{d}_1{\omega }_1\right|+\right.$$

$$\left|r_1{d}_0\left\{{\omega }_2-\left(\frac{\eta r_1{d}_0}{\left(1+i tan\theta \right)\left(1-{\left[2\right]}_q\right){\rho }_2}+\frac{9\vartheta \eta r_1{d}_0\left({\left[3\right]}_q-1\right){\rho }_3}{10\left(1+i tan\theta \right){\left(1-{\left[2\right]}_q\right)}^2{\rho }_2^2}-\frac{r_2}{r_1}\right){\omega }_1^2\right\}\right|,$$

where $\psi$ is analytic in $\mathbf{U}.$ By using inequalities $\left|d_n\right|\le 1$ and $\left|{\omega }_1\right|\le 1,$ we obtain

$$\left|c_2\right|\le \frac{3\eta r_1}{\left(1+i tan\theta \right)\left(1-{\left[2\right]}_q\right){\rho }_2},$$

$$\left|c_3-\vartheta c_2^2\right|\le \frac{10\eta r_1}{\left(1+i tan\theta \right)\left({\left[3\right]}_q-1\right){\rho }_3}\left(1+\left|{\omega }_2-\left(-\frac{r_2}{r_1}-\left[\frac{\eta r_1}{\left(1+i tan\theta \right)\left(1-{\left[2\right]}_q\right){\rho }_2}+\right.\right.\right.\right.$$

$$\left.\left.\left.\left.\frac{9\vartheta {\eta }^2{r}_1\left({\left[3\right]}_q-1\right){\rho }_3}{10\left(1+i tan\theta \right){\left(1-{\left[2\right]}_q\right)}^2{\rho }_2^2}\right]r_1\right){\omega }_1^2\right|\right).$$

Applying Lemma 1 to

$$\left|{\omega }_2-\left(-\frac{r_2}{r_1}-\left[\frac{\eta r_1}{\left(1+i tan\theta \right)\left(1-{\left[2\right]}_q\right){\rho }_2}+\frac{9\vartheta {\eta }^2{r}_1\left({\left[3\right]}_q-1\right){\rho }_3}{10\left(1+i tan\theta \right){\left(1-{\left[2\right]}_q\right)}^2{\rho }_2^2}\right]r_1\right){\omega }_1^2\right|,$$

we obtain

$$\left|c_3-\vartheta c_2^2\right|\le \frac{10\eta }{\left(1+i tan\theta \right)\left({\left[3\right]}_q-1\right){\rho }_3}\left(r_1+max\left\{r_1,\left|\frac{\eta r_1^2\left[10\left(1-{\left[2\right]}_q\right){\rho }_2+9\vartheta \eta \left({\left[3\right]}_q-1\right){\rho }_3\right]}{10\left(1+i tan\theta \right){\left(1-{\left[2\right]}_q\right)}^2{\rho }_2^2}\right|+\left|r_2\right|\right\}\right).$$

Putting $\vartheta =0,$ we obtain

$$\left|c_3\right|\le \frac{10\eta }{\left(1+i tan\theta \right)\left({\left[3\right]}_q-1\right){\rho }_3}\left(r_1+max\left\{r_1,\left|\frac{\eta r_1^2}{\left(1+i tan\theta \right)\left(1-{\left[2\right]}_q\right){\rho }_2}\right|+\left|r_2\right|\right\}\right).$$

□

Putting $b_j={\displaystyle \frac{{(-1)}^{j-1}\Gamma (\upsilon +1)}{4^{j-1}(j-1)!\Gamma (j+\upsilon )}}$, $\upsilon >0$ in Theorem 1, we obtain the following example:

Example 1.

We let $\frac{-\pi }{2}<\theta <\frac{\pi }{2},\lambda >-1,0<q<1$and $\eta \ne 0.$ If function $\mathcal{F}$ given by (4) belongs to class $\mathcal{S}$${}_{\upsilon ,q,\eta }^{\theta ,\lambda }\left(\varphi \right)$, then

$$\left|c_2\right|\le \frac{36\eta r_1(1+\upsilon ){\left[\lambda +1\right]}_q}{\left(1+i tan\theta \right)\left(1-{\left[2\right]}_q\right){\left[2\right]}_q!},$$

$$\left|c_3\right|\le \frac{3200\eta (1+\upsilon )(2+\upsilon ){\left[\lambda +1\right]}_{q,2}}{\left(1+i tan\theta \right)\left({\left[3\right]}_q-1\right){\left[3\right]}_q!}\left(r_1+max\left\{r_1,\left|\frac{12\eta r_1^2(1+\upsilon ){\left[\lambda +1\right]}_q}{\left(1+i tan\theta \right)\left(1-{\left[2\right]}_q\right){\left[2\right]}_q!}\right|+\left|r_2\right|\right\}\right),$$

and for any complex number ϑ,

$$\left|c_3-\vartheta c_2^2\right|\le \frac{3200\eta (1+\upsilon )(2+\upsilon ){\left[\lambda +1\right]}_{q,2}}{\left(1+i tan\theta \right)\left({\left[3\right]}_q-1\right){\left[3\right]}_q!}\left({}^{\begin{array}{c}\end{array}}{r}_1+max\left\{{}^{\begin{array}{c}\end{array}}{r}_1,\left|\frac{12(1+\upsilon ){\left[\lambda +1\right]}_q\eta r_1^2}{\left(1+i tan\theta \right)\left(1-{\left[2\right]}_q\right){\left[2\right]}_q!}+\right.\right.\right.$$

$$\left.\left.\left.\frac{81\vartheta {\eta }^2{r}_1^2(1+\upsilon )\left({\left[3\right]}_q-1\right){\left[3\right]}_q{\left[\lambda +1\right]}_q^2}{200(2+\upsilon )\left(1+i tan\theta \right){\left(1-{\left[2\right]}_q\right)}^2{\left[2\right]}_q!{\left[\lambda +1\right]}_{q,2}}\right|+\left|r_2\right|\right\}\right).$$

Putting ${\left({\displaystyle \frac{n+1}{n+j}}\right)}^{\alpha }$, $\alpha >0$, $n\ge 0$ in Theorem 1, we obtain the following example:

Example 2.

We let $\frac{-\pi }{2}<\theta <\frac{\pi }{2},\lambda >-1,0<q<1$and $\eta \ne 0.$ If function $\mathcal{F}$ given by (4) belongs to class $\mathcal{S}$${}_{q,\eta ,n}^{\theta ,\lambda ,\alpha }\left(\varphi \right)$, then

$$\left|c_2\right|\le \frac{9\eta r_1{\left(n+2\right)}^{\alpha }{\left[\lambda +1\right]}_q}{\left(1+i tan\theta \right)\left(1-{\left[2\right]}_q\right){\left(n+1\right)}^{\alpha }{\left[2\right]}_q!},$$

$$\left|c_3\right|\le \frac{100\eta {\left(n+3\right)}^{\alpha }{\left[\lambda +1\right]}_{q,2}}{\left(1+i tan\theta \right)\left({\left[3\right]}_q-1\right){\left(n+1\right)}^{\alpha }{\left[3\right]}_q!}\left(r_1+max\left\{r_1,\left|\frac{3\eta r_1^2{\left(n+2\right)}^{\alpha }{\left[\lambda +1\right]}_q}{\left(1+i tan\theta \right)\left(1-{\left[2\right]}_q\right){\left(n+1\right)}^{\alpha }{\left[2\right]}_q!}\right|+\left|r_2\right|\right\}\right),$$

and for any complex number ϑ,

$$\left|c_3-\vartheta c_2^2\right|\le \frac{100\eta {\left(n+3\right)}^{\alpha }{\left[\lambda +1\right]}_{q,2}}{\left(1+i tan\theta \right)\left({\left[3\right]}_q-1\right){\left(n+1\right)}^{\alpha }{\left[3\right]}_q!}\left({}^{\begin{array}{c}\end{array}}{r}_1+max\left\{{}^{\begin{array}{c}\end{array}}{r}_1,\left|\frac{3\eta r_1^2{\left(n+2\right)}^{\alpha }{\left[\lambda +1\right]}_q}{\left(1+i tan\theta \right)\left(1-{\left[2\right]}_q\right){\left(n+1\right)}^{\alpha }{\left[2\right]}_q!}+\right.\right.\right.$$

$$\left.\left.\left.\frac{81\vartheta {\eta }^2{r}_1^2\left({\left[3\right]}_q-1\right){\left[3\right]}_q{\left(n+2\right)}^{2\alpha }{\left[\lambda +1\right]}_q^2}{100\left(1+i tan\theta \right){\left(1-{\left[2\right]}_q\right)}^2{\left(n+3\right)}^{\alpha }{\left[\lambda +1\right]}_{q,2}{\left(n+1\right)}^{\alpha }{\left[2\right]}_q!}\right|+\left|r_2\right|\right\}\right).$$

Putting $b_j={\displaystyle \frac{{\rho }^{j-1}}{(j-1)!}}{e}^{-\rho }$, $\rho >0$ in Theorem 1, we obtain the following example:

Example 3.

We let $\frac{-\pi }{2}<\theta <\frac{\pi }{2},\lambda >-1,0<q<1$and $\eta \ne 0.$ If function $\mathcal{F}$ given by (4) belongs to class $\mathcal{S}$${}_{q,\eta }^{\theta ,\lambda ,\rho }\left(\varphi \right)$, then

$$\left|c_2\right|\le \frac{9\eta r_1{\left[\lambda +1\right]}_q}{\rho \left(1+i tan\theta \right)\left(1-{\left[2\right]}_q\right){\left[2\right]}_q!e^{-\rho }},$$

$$\left|c_3\right|\le \frac{200\eta {\left[\lambda +1\right]}_{q,2}}{{\rho }^2\left(1+i tan\theta \right)\left({\left[3\right]}_q-1\right){\left[3\right]}_q!e^{-\rho }}\left(r_1+max\left\{r_1,\left|\frac{3\eta r_1^2{\left[\lambda +1\right]}_q}{\rho \left(1+i tan\theta \right)\left(1-{\left[2\right]}_q\right){\left[2\right]}_q!e^{-\rho }}\right|+\left|r_2\right|\right\}\right),$$

and for any complex number ϑ,

$$\begin{gathered} \left|c_3-\vartheta c_2^2\right|\le \frac{200\eta {\left[\lambda +1\right]}_{q,2}}{{\rho }^2\left(1+i tan\theta \right)\left({\left[3\right]}_q-1\right){\left[3\right]}_q!e^{-\rho }}\left({}^{\begin{array}{c}\end{array}}{r}_1+max\left\{{}^{\begin{array}{c}\end{array}}{r}_1,\left|\frac{3\eta r_1^2{\left[\lambda +1\right]}_q}{\rho \left(1+i tan\theta \right)\left(1-{\left[2\right]}_q\right){\left[2\right]}_q!e^{-\rho }}+\right.\right.\right. \\ \left.\left.\left.\frac{81\vartheta {\eta }^2{r}_1^2\left({\left[3\right]}_q-1\right){\left[3\right]}_q{\left[\lambda +1\right]}_q^2}{200\left(1+i tan\theta \right){\left(1-{\left[2\right]}_q\right)}^2{\left[\lambda +1\right]}_{q,2}{\left[2\right]}_q!e^{-\rho }}\right|+\left|r_2\right|\right\}\right). \end{gathered}$$

Analysis similar to that in the proof of the previous Theorem shows the following:

Theorem 2.

We let $\frac{-\pi }{2}<\theta <\frac{\pi }{2},\lambda >-1,0<q<1$and $\eta \ne 0.$ If function $\mathcal{F}$ given by (4) belongs to class ${\mathcal{C}}_{q,\eta }^{\theta ,\lambda }\left(\varphi ;\mathsf{{\rm Y}}\right)$, then

$$\left|c_2\right|\le \frac{3\eta r_1}{\left(1+i tan\theta \right){\left[2\right]}_q{\rho }_2},$$

(23)

$$\left|c_3\right|\le \frac{5\eta }{\left(1+i tan\theta \right){\left[3\right]}_q{\rho }_3}\left(r_1+max\left\{r_1,\left|\frac{\eta r_1^2}{\left(1+i tan\theta \right){\rho }_2}\right|+\left|r_2\right|\right\}\right),$$

(24)

and for any complex number ϑ,

$$\left|c_3-\vartheta c_2^2\right|\le \frac{5\eta }{\left(1+i tan\theta \right){\left[3\right]}_q{\rho }_3}\left(r_1+max\left\{r_1,\left|\frac{\left[{\left(1+i tan\theta \right)}^2{\left[2\right]}_q^2{\rho }_2^2+9\vartheta \eta \right]\eta r_1^2}{{\left(1+i tan\theta \right)}^2{\left[2\right]}_q^2{\rho }_2^2}\right|+\left|r_2\right|\right\}\right).$$

(25)

Putting $b_j={\displaystyle \frac{{(-1)}^{j-1}\Gamma (\upsilon +1)}{4^{j-1}(j-1)!\Gamma (j+\upsilon )}}$, $\upsilon >0$ in Theorem 2, we obtain the following example:

Example 4.

We let $\frac{-\pi }{2}<\theta <\frac{\pi }{2},\lambda >-1,0<q<1$and $\eta \ne 0.$ If function $\mathcal{F}$ given by (4) belongs to class ${\mathcal{C}}_{\upsilon ,q,\eta }^{\theta ,\lambda }\left(\varphi \right)$, then

$$\left|c_2\right|\le \frac{36\eta r_1(1+\upsilon ){\left[\lambda +1\right]}_q}{\left(1+i tan\theta \right){\left[2\right]}_q{\left[2\right]}_q!},$$

$$\left|c_3\right|\le \frac{1600\eta (1+\upsilon )(2+\upsilon ){\left[\lambda +1\right]}_{q,2}}{\left(1+i tan\theta \right){\left[3\right]}_q{\left[3\right]}_q!}\left(r_1+max\left\{r_1,\left|\frac{12\eta r_1^2(1+\upsilon ){\left[\lambda +1\right]}_q}{\left(1+i tan\theta \right){\left[2\right]}_q!}\right|+\left|r_2\right|\right\}\right),$$

and for any complex number ϑ

$$\left|c_3-\vartheta c_2^2\right|\le \frac{1600\eta (1+\upsilon )(2+\upsilon ){\left[\lambda +1\right]}_{q,2}}{\left(1+i tan\theta \right){\left[3\right]}_q{\left[3\right]}_q!}\left(r_1+max\left\{r_1,\left|\eta r_1^2+\frac{1296\vartheta {\eta }^2{r}_1^2{(1+\upsilon )}^2{\left[\lambda +1\right]}_q^2}{{\left(1+i tan\theta \right)}^2{\left[2\right]}_q^2{\left({\left[2\right]}_q!\right)}^2}\right|+\left|r_2\right|\right\}\right).$$

3. Concluding Remarks and Observations

Using quasi-subordination, we were able to introduce some new subclasses of the class of analytic functions in the open unit disk $\mathbf{U}$ by employing the q-derivative operator and error function in the current study. In addition to other features and outcomes, we estimated the initial Taylor–Maclaurin coefficients $|c_2|$ and $\left|c_3\right|$ for functions that fall into the function classes considered in this work. We also obtained the Fekete–Szego functional. Furthermore, as mentioned in Theorem 1, we chose to deduce several examples of our primary claims.