Coefficient Bounds for Symmetric Subclasses of q-Convolution-Related Analytical Functions

Abstract

By using q-convolution, we determine the coefficient bounds for certain symmetric subclasses of analytic functions of complex order, which are introduced here by means of a certain non-homogeneous Cauchy–Euler-type differential equation of order m.

1. Introduction, Definitions and Preliminaries

Assume that $\mathbb{A}$ is the class of analytic functions in the open disc $\mathsf{\Lambda }:=\{\zeta \in \mathbb{C}:|\zeta |<1\}$ of the form

$$\mathsf{{\rm Y}}\left(\zeta \right)=\zeta +\sum _{t=2}^{+\infty }{a}_t{\zeta }^t,\zeta \in \mathsf{\Lambda }.$$

(1)

If the function $h\in \mathbb{A}$ is given by

$$h\left(\zeta \right)=\zeta +\sum _{t=2}^{+\infty }{c}_t{\zeta }^t,\zeta \in \mathsf{\Lambda }.$$

(2)

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

$$(\mathsf{{\rm Y}}\ast h)\left(\zeta \right):=\zeta +\sum _{t=2}^{+\infty }{a}_t{c}_t{\zeta }^t,\zeta \in \mathsf{\Lambda }.$$

A function $\mathsf{{\rm Y}}\in \mathcal{A}$ belongs to the class $\mathcal{S}*\left(\eta \right)$ if

$$\Re \left\{1+\frac{1}{\eta }\left(\frac{\zeta {\mathsf{{\rm Y}}}^{\prime }\left(\zeta \right)}{\mathsf{{\rm Y}}\left(\zeta \right)}-1\right)\right\}>0\left(\zeta \in \mathsf{\Lambda };\eta \in \mathbb{C}*=\mathbb{C}\backslash \left\{0\right\}\right).$$

(3)

Furthermore, a function $\mathsf{{\rm Y}}\in \mathcal{A}$ be in the class $\mathcal{C}\left(\eta \right)$ if

$$\Re \left\{1+\frac{1}{\eta }\frac{\zeta {\mathsf{{\rm Y}}}^{″}\left(\zeta \right)}{{\mathsf{{\rm Y}}}^{\prime }\left(\zeta \right)}\right\}>0\left(\zeta \in \mathsf{\Lambda };\eta \in \mathbb{C}*\right).$$

(4)

The classes $\mathcal{S}*\left(\eta \right)$ and $\mathcal{C}\left(\eta \right)$ were studied by Nasr and Aouf [1,2] and Wiatrowski [3].

In a wide range of applications in the mathematical, physical, and engineering sciences, the theory of q-calculus is important. Jackson [4,5] was the first to use the q-calculus in various applications and to introduce the q-analogue of the standard derivative and integral operators; see [6,7,8,9,10]. About coefficients’ interesting results, see [11,12,13,14,15,16]. 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

$${(\lambda ;q)}_t=\left\{\begin{array}{cc} 1\hfill & t=0,\hfill \\ \left(1-\lambda \right)\left(1-\lambda q\right)\dots \left(1-\lambda q^{t-1}\right)& t\in \mathbb{N}.\hfill \end{array}\right.$$

Using the q-gamma function ${\mathsf{\Gamma }}_q\left(\zeta \right),$ we obtain

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

where

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

In addition, we note that

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

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

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

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

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

(5)

Using the definition Formula (5), we have the next two products:

(i)

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

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

(ii)

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

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

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

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

In addition, we observe that

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

where ${\left(\lambda \right)}_t$ is given by

$${\left(\lambda \right)}_t=\left\{\begin{array}{ccc}1,\hfill & \mathrm{if}\hfill & t=0,\hfill \\ \lambda \left(\lambda +1\right)\dots \left(\lambda +t-1\right),\hfill & \mathrm{if}\hfill & t\in \mathbb{N}.\hfill \end{array}\right..$$

For $0<q<1$. El-Deeb et al. [17] defined that the q-derivative operator for $\mathsf{{\rm Y}}\ast h$ is defined by

$$\begin{array}{c}{\mathcal{D}}_q\left(\mathsf{{\rm Y}}\ast h\right)\left(\zeta \right):={\mathcal{D}}_q(\zeta +{\displaystyle \sum _{t=2}^{+\infty }}{a}_t{c}_t{\zeta }^t)\\ =\frac{\left(\mathsf{{\rm Y}}\ast h\right)\left(\zeta \right)-\left(\mathsf{{\rm Y}}\ast h\right)\left(q\zeta \right)}{\zeta (1-q)}=1+{\displaystyle \sum _{t=2}^{+\infty }}{\left[t\right]}_q{a}_t{c}_t{\zeta }^{t-1},\zeta \in \mathsf{\Lambda },\end{array}$$

Let $\vartheta >-1$ and $0<q<1$; El-Deeb et al. [17] defined the linear operator ${\mathcal{R}}_h^{\vartheta ,q}:\mathbb{A}\to \mathbb{A}$ as follows:

$${\mathcal{R}}_h^{\vartheta ,q}\mathsf{{\rm Y}}\left(\zeta \right)\ast {\mathcal{N}}_{q,\vartheta +1}\left(\zeta \right)=\zeta {\mathcal{D}}_q\left(\mathsf{{\rm Y}}\ast h\right)\left(\zeta \right),\zeta \in \mathsf{\Lambda },$$

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

$${\mathcal{N}}_{q,\vartheta +1}\left(\zeta \right):=\zeta +\sum _{t=2}^{+\infty }\frac{{[\vartheta +1]}_{q,t-1}}{{[t-1]}_q!}{\zeta }^t,\zeta \in \mathsf{\Lambda }.$$

A simple computation shows that

$${\mathcal{R}}_h^{\vartheta ,q}\mathsf{{\rm Y}}\left(\zeta \right):=\zeta +\sum _{t=2}^{+\infty }\frac{{\left[t\right]}_q!}{{[\vartheta +1]}_{q,t-1}}{a}_t{c}_t{\zeta }^t,\zeta \in \mathsf{\Lambda }(\vartheta >-1,0<q<1).$$

(6)

Remark 1 

([17]). From the definition relation (6), we can obtain that the next relations hold for all $\mathsf{{\rm Y}}\in \mathcal{A}$:

$$\begin{array}{cc}(i)& {[\vartheta +1]}_q{\mathcal{R}}_h^{\vartheta ,q}\mathsf{{\rm Y}}\left(\zeta \right)={\left[\vartheta \right]}_q{\mathcal{R}}_h^{\vartheta +1,q}\mathsf{{\rm Y}}\left(\zeta \right)+q^{\vartheta }\zeta {\mathcal{D}}_q\left({\mathcal{R}}_h^{\vartheta +1,q}\mathsf{{\rm Y}}\left(\zeta \right)\right),\zeta \in \mathsf{\Lambda };\hfill \\ (\mathit{ii})& {\mathcal{I}}_h^{\vartheta }\mathsf{{\rm Y}}\left(\zeta \right):=\underset{q\to 1^{-}}{\lim }{\mathcal{R}}_h^{\vartheta ,q}\mathsf{{\rm Y}}\left(\zeta \right)=\zeta +{\displaystyle \sum _{t=2}^{+\infty }}{\displaystyle \frac{t!}{{(\vartheta +1)}_{t-1}}}{a}_t{c}_t{\zeta }^t,\zeta \in \mathsf{\Lambda }.\hfill \end{array}$$

(7)

Remark 2 

([17]). By taking different particular cases for the coefficients $c_t,$ El-Deeb et al. [17] observed the following special cases for the operator ${\mathcal{R}}_h^{\vartheta ,q}$:

(i) 

For $c_t={\displaystyle \frac{{(-1)}^{t-1}\mathsf{\Gamma }(\rho +1)}{4^{t-1}(t-1)!\mathsf{\Gamma }(t+\rho )}}$, $\rho >0$, El-Deeb and Bulboacă [18] and El-Deeb [19] obtained the operator ${\mathcal{N}}_{\rho ,q}^{\vartheta }$ studied by:

$$\begin{array}{c}{\mathcal{N}}_{\rho ,q}^{\vartheta }\mathsf{{\rm Y}}\left(\zeta \right):=\zeta +{\displaystyle \sum _{t=2}^{+\infty }}{\displaystyle \frac{{(-1)}^{t-1}\mathsf{\Gamma }(\rho +1)}{4^{t-1}(t-1)!\mathsf{\Gamma }(t+\rho )}}·\frac{{\left[t\right]}_q!}{{[\vartheta +1]}_{q,t-1}}{a}_t{\zeta }^t\\ =\zeta +{\displaystyle \sum _{t=2}^{+\infty }}\frac{{\left[t\right]}_q!}{{[\vartheta +1]}_{q,t-1}}{\psi }_t{a}_t{\zeta }^t,\zeta \in \mathsf{\Lambda },(\rho >0,\vartheta >-1,0<q<1),\end{array}$$

(8)

where

$${\psi }_t:={\displaystyle \frac{{(-1)}^{t-1}\mathsf{\Gamma }(\rho +1)}{4^{t-1}(t-1)!\mathsf{\Gamma }(t+\rho )}};$$

(9)

(ii) 

For $c_t={\left({\displaystyle \frac{m+1}{m+t}}\right)}^{\alpha }$, $\alpha >0$, $m\ge 0$, El-Deeb and Bulboacă [20] and Srivastava and El-Deeb [21] obtained the operator ${\mathcal{N}}_{m,1,q}^{\vartheta ,\alpha }=:{\mathcal{M}}_{m,q}^{\vartheta ,\alpha }$ studied by:

$${\mathcal{M}}_{m,q}^{\vartheta ,\alpha }\mathsf{{\rm Y}}\left(\zeta \right):=\zeta +\sum _{t=2}^{+\infty }{\left(\frac{m+1}{m+t}\right)}^{\alpha }·\frac{{\left[t\right]}_q!}{{[\vartheta +1]}_{q,t-1}}{a}_t{\zeta }^t,\zeta \in \mathsf{\Lambda };$$

(10)

(iii) 

For $c_t={\displaystyle \frac{n^{t-1}}{(t-1)!}}{e}^{-n}$, $n>0$, El-Deeb et al. [17] obtained the q-analogue of Poisson operator defined by:

$${\mathcal{I}}_q^{\vartheta ,n}\mathsf{{\rm Y}}\left(\zeta \right):=\zeta +\sum _{t=2}^{+\infty }{\displaystyle \frac{n^{t-1}}{(t-1)!}}{e}^{-n}·\frac{{\left[t\right]}_q!}{{[\vartheta +1]}_{q,t-1}}{a}_t{\zeta }^t,\zeta \in \mathsf{\Lambda };$$

(11)

(iv) 

For $c_t={\left[{\displaystyle \frac{1+\ell +\lambda (t-1)}{1+\ell }}\right]}^n$, $n\in \mathbb{Z}$, $\ell \ge 0$, $\lambda \ge 0$, El-Deeb et al. [17] obtained the q-analogue of Prajapat operator defined by

$${\mathcal{J}}_{q,\ell ,\lambda }^{\vartheta ,n}\mathsf{{\rm Y}}\left(\zeta \right):=\zeta +\sum _{t=2}^{+\infty }{\left[{\displaystyle \frac{1+\ell +\lambda (t-1)}{1+\ell }}\right]}^n·\frac{{\left[t\right]}_q!}{{[\vartheta +1]}_{q,t-1}}{a}_t{\zeta }^t,\zeta \in \mathsf{\Lambda }.$$

(12)

In this paper, we define the following subclasses ${\mathcal{SC}}_h^{\vartheta ,q}\left(\eta ,\gamma ,\beta \right)$ and ${\mathcal{N}}_h^{\vartheta ,q}\left(\eta ,\gamma ,\beta ,m,\mu \right)(\eta \in \mathbb{C}*,0\le \gamma \le 1,0\le \beta <1,\vartheta >-1,0<q<1,m\in \mathbb{N}*=\mathbb{N}\backslash \left\{1\right\}=\{2,3,4,\dots \},\mu \in \mathbb{R}\backslash \left(-\infty ,-1\right])$ as follows:

Definition 1.

For a function Υ has the form (1) and h is defined by (2), the function Υ belongs to the class ${\mathcal{SC}}_h^{\vartheta ,q}\left(\eta ,\gamma ,\beta \right)$ if

$$\Re \left\{1+\frac{1}{\eta }\left[\frac{\zeta {\left[\left(1-\gamma \right){\mathcal{R}}_h^{\vartheta ,q}\mathsf{{\rm Y}}\left(\zeta \right)+\gamma \zeta {\left({\mathcal{R}}_h^{\vartheta ,q}\mathsf{{\rm Y}}\left(\zeta \right)\right)}^{\prime }\right]}^{\prime }}{\left(1-\gamma \right){\mathcal{R}}_h^{\vartheta ,q}\mathsf{{\rm Y}}\left(\zeta \right)+\gamma \zeta {\left({\mathcal{R}}_h^{\vartheta ,q}\mathsf{{\rm Y}}\left(\zeta \right)\right)}^{\prime }}-1\right]\right\}>\beta$$

$$\left(\eta \in \mathbb{C}*;0\le \gamma \le 1;0\le \beta <1;\vartheta >-1,0<q<1;\zeta \in \mathsf{\Lambda }\right).$$

(13)

Remark 3.

(i) 

For $q\to 1^{-},$ we obtain that $\underset{q\to 1^{-}}{\lim }{\mathcal{SC}}_h^{\vartheta ,q}\left(\eta ,\gamma ,\beta \right)=:{\mathcal{G}}_h^{\vartheta }\left(\eta ,\gamma ,\beta \right)$, where ${\mathcal{G}}_h^{\vartheta }\left(\eta ,\gamma ,\beta \right)$ represents the functions $\mathsf{{\rm Y}}\in \mathbb{A}$ that satisfies (13) for ${\mathcal{R}}_h^{\vartheta ,q}$ replaced with ${\mathcal{I}}_h^{\vartheta }$ (7).

(ii) 

For $c_t={\displaystyle \frac{{(-1)}^{t-1}\mathsf{\Gamma }(\rho +1)}{4^{t-1}(t-1)!\mathsf{\Gamma }(t+\rho )}}$, $\rho >0$, we obtain the subclass ${\mathcal{B}}_{\rho }^{\vartheta ,q}\left(\eta ,\gamma ,\beta \right)$, that represents the functions $\mathsf{{\rm Y}}\in \mathbb{A}$ that satisfies (13) for ${\mathcal{R}}_h^{\vartheta ,q}$ replaced with ${\mathcal{N}}_{\rho ,q}^{\vartheta }$ (8).

(iii) 

For $c_t={\left({\displaystyle \frac{m+1}{m+t}}\right)}^{\alpha }$, $\alpha >0$, $m\ge 0$, we obtain the class ${\mathcal{M}}_{m,\alpha }^{\vartheta ,q}\left(\eta ,\gamma ,\beta \right)$, that represents the functions $\mathsf{{\rm Y}}\in \mathbb{A}$ that satisfies (13) for ${\mathcal{R}}_h^{\vartheta ,q}$ replaced with ${\mathcal{M}}_{m,q}^{\vartheta ,\alpha }$ (10).

(iv) 

For $c_t={\displaystyle \frac{n^{t-1}}{(t-1)!}}{e}^{-n}$, $n>0$, we obtain the class ${\mathcal{I}}_t^{\vartheta ,q}\left(\eta ,\gamma ,\beta \right)$, that represents the functions $\mathsf{{\rm Y}}\in \mathbb{A}$ that satisfies (13) for ${\mathcal{R}}_h^{\vartheta ,q}$ replaced with ${\mathcal{I}}_q^{\vartheta ,t}$ (11).

(v) 

For $c_t={\left[{\displaystyle \frac{1+\ell +\lambda (t-1)}{1+\ell }}\right]}^n$, $n\in \mathbb{Z}$, $\ell \ge 0$, $\lambda \ge 0$, we obtain the class ${\mathcal{J}}_{n,\ell ,\lambda }^{\vartheta ,q}\left(\eta ,\gamma ,\beta \right)$, that represents the functions $\mathsf{{\rm Y}}\in \mathbb{A}$ that satisfies (13) for ${\mathcal{R}}_h^{\vartheta ,q}$ replaced with ${\mathcal{J}}_{q,\ell ,\lambda }^{\vartheta ,n}$ (12).

The following lemma must be used in to show our study results:

Definition 2.

A function $\mathsf{{\rm Y}}\in \mathbb{A}$ belongs to the class ${\mathcal{N}}_h^{\vartheta ,q}\left(\eta ,\gamma ,\beta ,m,\mu \right)$ if it satisfies the following non-homogeneous Cauchy–Euler type differential equation of order m:

$${\zeta }^m\frac{d^{m}w}{d{\zeta }^m}+\left(\genfrac{}{}{0pt}{}{m}{1}\right)\left(\mu +m-1\right){\zeta }^{m-1}\frac{d^{m-1}w}{d{\zeta }^{m-1}}+\cdots +\left(\genfrac{}{}{0pt}{}{m}{m}\right)w{\displaystyle \prod _{j=0}^{m-1}}\left(\mu +j\right)=g\left(\zeta \right){\displaystyle \prod _{j=0}^{m-1}}\left(\mu +j+1\right)$$

$$\begin{array}{c}\left(w=\mathsf{{\rm Y}}\left(\zeta \right);g\left(\zeta \right)\in {\mathcal{SC}}_h^{\vartheta ,q}\left(\eta ,\gamma ,\beta \right);\eta \in \mathbb{C}*,0\le \gamma \le 1,0\le \beta <1;\vartheta >-1;0<q<1;\right.\hfill \\ \left.m\in \mathbb{N}*;\mu \in \mathbb{R}\backslash \left(-\infty ,-1\right]\right).\hfill \end{array}$$

Remark 4.

(i) 

Putting $q\to 1^{-},$ we obtain that $\underset{q\to 1^{-}}{\lim }{\mathcal{N}}_h^{\vartheta ,q}\left(\eta ,\gamma ,\beta ,m,\mu \right)=:{\mathcal{T}}_h^{\vartheta }\left(\eta ,\gamma ,\beta ,m,\mu \right)$, where ${\mathcal{T}}_h^{\vartheta }\left(\eta ,\gamma ,\beta ,m,\mu \right)$ represents the functions $\mathsf{{\rm Y}}\in \mathbb{A}$ that satisfies (13) for ${\mathcal{R}}_h^{\vartheta ,q}$ replaced with ${\mathcal{I}}_h^{\lambda }$ (7).

(ii) 

Putting $c_t={\displaystyle \frac{{(-1)}^{t-1}\mathsf{\Gamma }(\rho +1)}{4^{t-1}(t-1)!\mathsf{\Gamma }(t+\rho )}}$, $\rho >0$, we get the subclass ${\mathcal{P}}_{\rho }^{\vartheta ,q}\left(\eta ,\gamma ,\beta ,m,\mu \right)$, that represents the functions $\mathsf{{\rm Y}}\in \mathbb{A}$ that satisfies (13) for ${\mathcal{R}}_h^{\vartheta ,q}$ replaced with ${\mathcal{N}}_{\rho ,q}^{\vartheta }$ (8).

(iii) 

Putting $c_t={\left({\displaystyle \frac{m+1}{m+t}}\right)}^{\alpha }$, $\alpha >0$, $m\ge 0$, we have the class ${\mathcal{R}}_{m,\alpha }^{\vartheta ,q}\left(\eta ,\gamma ,\beta ,m,\mu \right)$, that represents the functions $\mathsf{{\rm Y}}\in \mathbb{A}$ that satisfies (13) for ${\mathcal{R}}_h^{\vartheta ,q}$ replaced with ${\mathcal{M}}_{m,q}^{\vartheta ,\alpha }$ (10).

(iv) 

Putting $c_t={\displaystyle \frac{n^{t-1}}{(t-1)!}}{e}^{-n}$, $n>0$, we get the class ${\mathcal{D}}_n^{\vartheta ,q}\left(\eta ,\gamma ,\beta ,m,\mu \right)$, that represents the functions $\mathsf{{\rm Y}}\in \mathbb{A}$ that satisfies (13) for ${\mathcal{R}}_h^{\vartheta ,q}$ replaced with ${\mathcal{I}}_q^{\vartheta ,n}$ (11).

(v) 

Putting $c_t={\left[{\displaystyle \frac{1+\ell +\lambda (t-1)}{1+\ell }}\right]}^n$, $n\in \mathbb{Z}$, $\ell \ge 0$, $\lambda \ge 0$, we have the class ${\mathcal{J}}_{n,\ell ,\lambda }^{\vartheta ,q}\left(\eta ,\gamma ,\beta ,m,\mu \right)$, that represents the functions $\mathsf{{\rm Y}}\in \mathbb{A}$ that satisfies (13) for ${\mathcal{R}}_h^{\vartheta ,q}$ replaced with ${\mathcal{J}}_{q,\ell ,\lambda }^{\vartheta ,n}$ (12).

The main object of the present investigation is to derive some coefficient bounds for functions in the subclasses ${\mathcal{SC}}_h^{\vartheta ,q}\left(\eta ,\gamma ,\beta \right)$ and ${\mathcal{N}}_h^{\vartheta ,q}\left(\eta ,\gamma ,\beta ,m,\mu \right)$ of $\mathbb{A}.$

2. Coefficient Estimates for the Function Class ${\mathcal{SC}}_h^{\vartheta ,q}\left(\eta ,\gamma ,\beta \right)$

Unless otherwise mentioned, we assume throughout this paper that:

$\eta \in \mathbb{C}*,0\le \gamma \le 1,0\le \beta <1;m\in \mathbb{N}*;\mu \in \mathbb{R}\backslash \left(-\infty ,-1\right],$$\vartheta >-1;0<q<1,\zeta \in \mathsf{\Lambda }.$

Theorem 1.

Assume that the function Υ given by (1) belongs to the class ${\mathcal{SC}}_h^{\vartheta ,q}\left(\eta ,\gamma ,\beta \right)$, then

$$\left|a_t\right|\le \frac{{[\vartheta +1]}_{q,t-1}{\displaystyle \prod _{i=0}^{t-2}}[i+2(1-\beta )|\eta \left|\right]}{(t-1)![1+\gamma (t-1)]{\left[t\right]}_q!c_t}\left(t\in \mathbb{N}*\right).$$

(14)

Proof. 

The function $\mathsf{{\rm Y}}\in \mathbb{A}$ be given by (1)and let the function $\mathcal{F}\left(\zeta \right)$ be defined by

$$\mathcal{F}\left(\zeta \right)=\left(1-\gamma \right){\mathcal{R}}_h^{\vartheta ,q}\mathsf{{\rm Y}}\left(\zeta \right)+\gamma \zeta {\left({\mathcal{R}}_h^{\vartheta ,q}\mathsf{{\rm Y}}\left(\zeta \right)\right)}^{\prime }.$$

Then from (13) and the definition of the function $\mathcal{F}\left(\zeta \right)$ above, it is easily seen that

$$\Re \left\{1+\frac{1}{\eta }\left(\frac{\zeta {\mathcal{F}}^{\prime }\left(\zeta \right)}{\mathcal{F}\left(\zeta \right)}-1\right)\right\}>\beta$$

with

$$\mathcal{F}\left(\zeta \right)=\zeta +\sum _{t=2}^{+\infty }{\mathsf{\Theta }}_t{\zeta }^t\left({\mathsf{\Theta }}_t=\frac{{\left[t\right]}_q!}{{[\vartheta +1]}_{q,t-1}}\left[1+\gamma (t-1)\right]a_t{c}_t;t\in \mathbb{N}*\right).$$

Thus, by setting

$$\frac{1+\frac{1}{\eta }\left(\frac{\zeta {\mathcal{F}}^{\prime }\left(\zeta \right)}{\mathcal{F}\left(\zeta \right)}-1\right)-\beta }{1-\beta }=g\left(\zeta \right)$$

or, equivalently,

$$\zeta {\mathcal{F}}^{\prime }\left(\zeta \right)=\left[1+\eta \left(1-\beta \right)\left(g\left(\zeta \right)-1\right)\right]\mathcal{F}\left(\zeta \right),$$

(15)

we get

$$g\left(\zeta \right)=1+d_1\zeta +d_2{\zeta }^2+\dots \dots .$$

(16)

Since $\Re \left\{g\left(\zeta \right)\right\}>0,$ we conclude that $\left|d_t\right|\le 2\left(t\in \mathbb{N}\right)$ (see [14]).

We get from (15) and (16) that

$$(t-1){\mathsf{\Theta }}_t=\eta \left(1-\beta \right)\left[d_1{\mathsf{\Theta }}_{t-1}+d_2{\mathsf{\Theta }}_{t-2}+\cdots +d_{t-1}\right].$$

For $t=2,3,4$, we have

$${\mathsf{\Theta }}_2=\eta \left(1-\beta \right)d_1\Rightarrow \left|{\mathsf{\Theta }}_2\right|\le 2\left(1-\beta \right)\left|\eta \right|,$$

$$2{\mathsf{\Theta }}_3=\eta \left(1-\beta \right)\left(d_1{\mathsf{\Theta }}_2+d_2\right)\Rightarrow \left|{\mathsf{\Theta }}_3\right|\le \frac{2\left(1-\beta \right)\left|\eta \right|\left[1+2\left(1-\beta \right)\left|\eta \right|\right]}{2!},$$

and

$$3{\mathsf{\Theta }}_4=\eta \left(1-\beta \right)\left(d_1{\mathsf{\Theta }}_3+d_2{\mathsf{\Theta }}_2+d_3\right)\Rightarrow \left|{\mathsf{\Theta }}_4\right|\le \frac{2\left(1-\beta \right)\left|\eta \right|\left[1+2\left(1-\beta \right)\right]\left|\eta \right|\left[2+2\left(1-\beta \right)\left|\eta \right|\right]}{3!},$$

respectively. Using the principle of mathematical induction, we obtain

$$\left|{\mathsf{\Theta }}_t\right|\le \frac{{\displaystyle \prod _{i=0}^{t-2}}\left[i+2(1-\beta )\left|\eta \right|\right]}{(t-1)!}\left(t\in \mathbb{N}*\right).$$

(17)

Using the relationship between the functions $\mathsf{{\rm Y}}\left(\zeta \right)$ and $\mathcal{F}\left(\zeta \right)$, we get

$${\mathsf{\Theta }}_t=\frac{{\left[t\right]}_q!}{{[\vartheta +1]}_{q,t-1}}\left[1+\gamma (t-1)\right]a_t{c}_t\left(t\in \mathbb{N}*\right),$$

(18)

and then we get

$$\left|a_t\right|\le \frac{{[\vartheta +1]}_{q,t-1}{\displaystyle \prod _{i=0}^{t-2}}\left[i+2(1-\beta )\left|\eta \right|\right]}{(t-1)!\left[1+\gamma (t-1)\right]{\left[t\right]}_q!c_t}\left(t\in \mathbb{N}*\right).$$

This completes the proof of Theorem 1. □

Putting $q\to 1^{-}$ in Theorem 1, we obtain the following corollary:

Corollary 1.

If the function Υ given by (1) belongs to the class ${\mathcal{G}}_h^{\vartheta }\left(\eta ,\gamma ,\beta \right)$, then

$$\left|a_t\right|\le \frac{{\left(\vartheta +1\right)}_{t-1}{\displaystyle \prod _{i=0}^{t-2}}\left[i+2(1-\beta )\left|\eta \right|\right]}{t{\left[(t-1)!\right]}^2\left[1+\gamma (t-1)\right]c_t}\left(t\in \mathbb{N}*\right).$$

(19)

Taking $c_t={\displaystyle \frac{{(-1)}^{t-1}\mathsf{\Gamma }(\rho +1)}{4^{t-1}(t-1)!\mathsf{\Gamma }(t+\rho )}}$, $\rho >0$ in Theorem 1, we obtain the following special case:

Example 1.

If the function Υ given by (1) belongs to the class ${\mathcal{B}}_{\rho }^{\vartheta ,q}\left(\eta ,\gamma ,\beta \right)$, then

$$\left|a_t\right|\le \frac{4^{t-1}\mathsf{\Gamma }(t+\rho ){[\vartheta +1]}_{q,t-1}{\displaystyle \prod _{i=0}^{t-2}}\left[i+2(1-\beta )\left|\eta \right|\right]}{{(-1)}^{t-1}\mathsf{\Gamma }(\rho +1)\left[1+\gamma (t-1)\right]{\left[t\right]}_q!}\left(t\in \mathbb{N}*\right).$$

Considering $c_t={\left({\displaystyle \frac{m+1}{m+t}}\right)}^{\alpha }$, $\alpha >0$, $m\ge 0$ in Theorem 1, we obtain the following result:

Example 2.

If the function Υ given by (1) belongs to the class ${\mathcal{M}}_{m,\alpha }^{\vartheta ,q}\left(\eta ,\gamma ,\beta \right)$, then

$$\left|a_t\right|\le \frac{{\left(m+t\right)}^{\alpha }{[\vartheta +1]}_{q,t-1}{\displaystyle \prod _{i=0}^{t-2}}\left[i+2(1-\beta )\left|\eta \right|\right]}{(t-1)!\left[1+\gamma (t-1)\right]{\left[t\right]}_q!{\left(m+1\right)}^{\alpha }}\left(t\in \mathbb{N}*\right).$$

Putting $c_t={\displaystyle \frac{n^{t-1}}{(t-1)!}}{e}^{-n}$, $n>0$ in Theorem 1, we obtain the following special case:

Example 3.

If the function Υ given by (1) belongs to the class ${\mathcal{I}}_n^{\vartheta ,q}\left(\eta ,\gamma ,\beta \right)$, then

$$\left|a_t\right|\le \frac{{[\vartheta +1]}_{q,t-1}{\displaystyle \prod _{i=0}^{t-2}}\left[i+2(1-\beta )\left|\eta \right|\right]}{n^{t-1}\left[1+\gamma (t-1)\right]{\left[t\right]}_q!e^{-n}}\left(t\in \mathbb{N}*\right).$$

Putting $c_t={\left[{\displaystyle \frac{1+\ell +\lambda (t-1)}{1+\ell }}\right]}^n$, $n\in \mathbb{Z}$, $\ell \ge 0$, $\lambda \ge 0$ in Theorem 1, we obtain the following special case:

Example 4.

If the function Υ given by (1) belongs to the class ${\mathcal{J}}_{n,\ell ,\lambda }^{\vartheta ,q}\left(\eta ,\gamma ,\beta ,m,\mu \right)$, then

$$\left|a_t\right|\le \frac{{\left(1+\ell \right)}^n{[\vartheta +1]}_{q,t-1}{\displaystyle \prod _{i=0}^{t-2}}\left[i+2(1-\beta )\left|\eta \right|\right]}{(t-1)!\left[1+\gamma (t-1)\right]{\left[t\right]}_q!{\left(1+\ell +\lambda (t-1)\right)}^n}\left(t\in \mathbb{N}*\right).$$

Putting $c_t=1$ and $\vartheta =1$ in Corollary 1, we obtain the following special case:

Example 5.

If the function Υ given by (1) belongs to the class ${\mathcal{G}}_{\frac{\zeta }{1-\zeta }}^1\left(\eta ,\gamma ,\beta \right)$, then

$$\left|a_t\right|\le \frac{{\displaystyle \prod _{i=0}^{t-2}}\left[i+2(1-\beta )\left|\eta \right|\right]}{(t-1)!\left[1+\gamma (t-1)\right]}\left(t\in \mathbb{N}*\right).$$

3. Coefficient Estimates for the Function Class ${\mathcal{N}}_h^{\vartheta ,q}\left(\eta ,\gamma ,\beta ,m,\mu \right)$

Our main coefficient bounds for function in the class ${\mathcal{N}}_h^{\vartheta ,q}\left(\eta ,\gamma ,\beta ,m,\mu \right)$ are given by Theorem 2 below.

Theorem 2.

If the function Υ given by (1) belongs to the class ${\mathcal{N}}_h^{\vartheta ,q}\left(\eta ,\gamma ,\beta ,m,\mu \right)$, then

$$\left|a_t\right|\le \frac{{[\vartheta +1]}_{q,t-1}{\displaystyle \prod _{i=0}^{t-2}}\left[i+2(1-\beta )\left|\eta \right|\right]{\displaystyle \prod _{i=0}^{m-1}}\left(\mu +i+1\right)}{(t-1)!\left[1+\gamma (t-1)\right]{\left[t\right]}_q!{\displaystyle \prod _{i=0}^{m-1}}\left(\mu +i+t\right)c_t}\left(t\in \mathbb{N}*\right).$$

(20)

Proof. 

Let the function $\mathsf{{\rm Y}}\in \mathbb{A}$ be given by (1)and let the function $g$ define as follows

$$g\left(\zeta \right)=\zeta +\sum _{t=2}^{+\infty }{d}_t{\zeta }^t\in {\mathcal{SC}}_h^{\vartheta ,q}\left(\eta ,\gamma ,\beta \right),$$

(21)

so that

$$a_t=\frac{{\displaystyle \prod _{i=0}^{m-1}}\left(\mu +i+1\right)}{{\displaystyle \prod _{i=0}^{m-1}}\left(\mu +i+1\right)}{d}_t\left(t,m\in \mathbb{N}*;\mu \in \mathbb{R}\backslash \left(-\infty ,-1\right]\right).$$

(22)

$$\left|a_t\right|\le \frac{{[\vartheta +1]}_{q,t-1}{\displaystyle \prod _{r=0}^{t-2}}\left[r+2(1-\beta )\left|\eta \right|\right]{\displaystyle \prod _{i=0}^{m-1}}\left(\mu +i+1\right)}{(t-1)!\left[1+\gamma (t-1)\right]{\left[t\right]}_q!{\displaystyle \prod _{i=0}^{m-1}}\left(\mu +i+t\right)c_t}\left(j\in \mathbb{N}*\right).$$

Thus, by using Theorem 1, we readily complete the proof of Theorem 2. □

Putting $q\to 1^{-}$ in Theorem 1, we obtain the following corollary:

Corollary 2.

If the function Υ given by (1) belongs to the class ${\mathcal{T}}_h^{\vartheta }\left(\eta ,\gamma ,\beta ,m,\mu \right)$, then

$$\left|a_t\right|\le \frac{{\left(\vartheta +1\right)}_{t-1}{\displaystyle \prod _{r=0}^{t-2}}\left[r+2(1-\beta )\left|\eta \right|\right]{\displaystyle \prod _{i=0}^{m-1}}\left(\mu +i+1\right)}{t{\left[(t-1)!\right]}^2\left[1+\gamma (t-1)\right]{\displaystyle \prod _{i=0}^{m-1}}\left(\mu +i+t\right)c_t}\left(t\in \mathbb{N}*\right).$$

Putting $c_t=1$ and $\vartheta =1$ in Corollary 2, we obtain the following example:

Example 6.

If the function Υ given by (1) belongs to the class ${\mathcal{T}}_{\frac{\zeta }{1-\zeta }}^1\left(\eta ,\gamma ,\beta ,m,\mu \right)$, then

$$\left|a_j\right|\le \frac{{\displaystyle \prod _{r=0}^{t-2}}\left[r+2(1-\beta )\left|\eta \right|\right]{\displaystyle \prod _{i=0}^{m-1}}\left(\mu +i+1\right)}{(t-1)!\left[1+\gamma (t-1)\right]{\displaystyle \prod _{i=0}^{m-1}}\left(\mu +i+t\right)}\left(t\in \mathbb{N}*\right).$$

4. Conclusions

We investigated certain subclasses of analytic functions of complex order combined with the linear q-convolution operator. For the functions in this new class, we obtained the coefficient bounds and introduced here by means of a certain non-homogeneous Cauchy–Euler-type differential equation of order m. There was also consideration of several interesting corollaries and applications of the results by suitably fixing the parameters, as illustrated in Remark 1.