Results on Univalent Functions Defined by q-Analogues of Salagean and Ruscheweh Operators

Abstract

In this paper, we define and discuss properties of various classes of analytic univalent functions by using modified q-Sigmoid functions. We make use of an idea of Salagean to introduce the q-analogue of the Salagean differential operator. In addition, we derive families of analytic univalent functions associated with new q-Salagean and q-Ruscheweh differential operators. In addition, we obtain coefficient bounds for the functions in such new subclasses of analytic functions and establish certain growth and distortion theorems. By using the concept of the (q, δ)-neighbourhood, we provide several inclusion symmetric relations for certain (q, δ)-neighbourhoods of analytic univalent functions of negative coefficients. Various q-inequalities are also discussed in more details.

1. Introduction

In recent decades, the q-derivative has experienced accelerated developments in various fields of science due to its numerous applications in mathematical analysis and physical sciences including q-difference operators, fractional and q-symmetric fractional q-calculus, optimal control, q-symmetric functions and q-integral equations, to mention but a few (see, e.g., [1,2,3,4,5,6,7,8,9]). In [10], Jackson introduces the q-difference operator and discusses some applications of the q-derivative and the q-integral (see, also, [11] for more details and concepts). In [12], Srivastava introduces a connection between the geometric function theory of the complex analysis and the theory of the q-calculus. In [13], Arif et al. describe important applications of the q-calculus concept. In [14], Ismail et al. describe starlike functions by using q-difference operators. In [15], Sokol et al. investigate a subclass of analytic functions with a Ruscheweyh q-differential operator. In [16], Kanas and Raducanu introduce q-analogues of the Ruscheweyh differential operators and establish some convolution properties of some normalized analytic functions. In [17], Darus et al. study a q-analogue of some operator by using q-hypergeometric functions. Moreover, authors of [18,19,20,21] apply properties of the q-difference operator to discuss subclasses of complex analytic functions. In this paper, we define q-analogues of the Salagean and Ruscheweyh differential operators for certain univalent functions and study some interesting properties of the obtained results.

Let $\mathcal{H}$ be the set of all analytic functions on the unite open disc $D=\{z\in \mathbb{C}:|z|<1$. Let ${\mathcal{A}}_0$ be the subset of $\mathcal{H}$ of all functions normalized by $f\left(0\right)=0$ and $f^{\prime }\left(0\right)=1$ and S be the set of univalent functions (consult, for details, [22]). For a function f in the class S, Salagean in [23] introduced a differential operator and studied some of its applications on a certain subclass of univalent functions. Ruscheweyh in [24] defined the differential operator and investigated some properties of the univalent functions. Recently, in various papers, several authors have introduced generalizations to the Salagean differential operator (see, e.g., [25]). The q-analogue of the Ruscheweyh operator was also studied in [26,27,28]. For more details on this theory, we refer to [29,30,31,32,33,34,35,36,37,38,39,40,41,42] and references cited therein.

For any real number $q,0<q<1$, the q-difference operator for a complex valued function f is defined by

$$\begin{array}{c}\hfill D_{q}f\left(z\right)=\frac{f\left(z\right)-f\left(qz\right)}{z-qz},z\in D.\end{array}$$

It is clear that $D_{q}f\to f^{\prime }$ when $q\to 1$. Let $\mathcal{U}$ denote the subclass of all normalized functions f in $\mathcal{H}$ such that

$$\begin{array}{c}\hfill f\left(z\right)=z-\sum _{k=1}^{\infty }{a}_k{z}^k,a_k\ge 0.\end{array}$$

(1)

Then, for every function f furnished by (1), we assert that

$$\begin{array}{c}\hfill D_{q}f\left(z\right)=1-\sum _{k=2}^{\infty }{\left[k\right]}_q{a}_k{z}^k,a_k\ge 0,\end{array}$$

where ${\left[k\right]}_q=\frac{1-q^n}{1-q}$, $0\le q<1$ and $n\in \mathbb{N}$. The q-real number for $a\in \mathbb{C}$ is defined by [11]

$$\begin{array}{c}\hfill {(a;q)}_n=\left\{\begin{array}{cc}1,\hfill & n=0,\hfill \\ \prod _{i=0}^{n-1}(1-a{q}^i),\hfill & n\in \mathbb{N}.\hfill \end{array}\right.\end{array}$$

(2)

Therefore, for a complex number $\alpha$, we use the following notation to denote the q-binomial coefficients [43]

$$\begin{array}{c}\hfill B_q^k\left(\alpha \right)={\left(\genfrac{}{}{0pt}{}{\alpha }{k}\right)}_q=\left\{\begin{array}{cc}1,\hfill & k=0,\hfill \\ \frac{(1-q^{\alpha })(1-q^{\alpha -1})\dots (1-q^{\alpha -k+1})}{{(q;q)}_k},\hfill & k\in \mathbb{N}.\hfill \end{array}\right.\end{array}$$

The q-analogue exponential function is defined in [43] by

$$\begin{array}{c}\hfill e_q^s=1+\sum _{k=1}^{\infty }\frac{s^k}{{\left[k\right]}_q!},\end{array}$$

where the q-factorial ${\left[k\right]}_q!$ is given by

$$\begin{array}{c}\hfill {\left[k\right]}_q!=\left\{\begin{array}{cc}1,\hfill & k=0,\hfill \\ {\left[k\right]}_q{[k-1]}_q\dots {\left[1\right]}_q,\hfill & k=1,2,\dots \hfill \end{array}\right.\end{array}$$

The modified q-Sigmoid function is defined in [43] by

$$\begin{array}{c}\hfill {\gamma }_q\left(s\right)=\frac{1}{1+e_q^s}.\end{array}$$

(3)

Alternatively, it can be expressed as

$$\begin{array}{c}\hfill {\gamma }_q\left(s\right)=1+\frac{1}{2{\left[1\right]}_q!}s+\left(\frac{1}{4{({\left[1\right]}_q!)}^2}-\frac{1}{2{\left[2\right]}_q!}\right)s^2+\left(\frac{1}{2{\left[3\right]}_q!}-\frac{1}{2{\left[1\right]}_q!{\left[2\right]}_q!}+\frac{1}{{(2{\left[1\right]}_q!)}^3}\right)s^3+\dots ,\end{array}$$

where the q-factorial ${\left[k\right]}_q!$ is given by

$$\begin{array}{c}\hfill {\left[k\right]}_q!=\left\{\begin{array}{cc}1,\hfill & k=0,\hfill \\ {\left[k\right]}_q{[k-1]}_q\dots {\left[1\right]}_q,\hfill & k=1,2,\dots \hfill \end{array}\right.\end{array}$$

By keeping track of the definition of the q-Sigmoid function ${\gamma }_q\left(s\right)$, we define

$$\begin{array}{c}\hfill f_{{\gamma }_q}\left(z\right)=z-\sum _{k=2}^{\infty }{\gamma }_q\left(s\right)a_k{z}^k,a_k\ge 0.\end{array}$$

(4)

Hence, we introduce the $(q,\lambda )$-Salagean differential operator for the function $f_{{\gamma }_q}\left(z\right)$ to be

$$\begin{array}{ccc}\hfill D_{q,\lambda }^0{f}_{{\gamma }_q}\left(z\right)& =& f_{{\gamma }_q}\left(z\right);\hfill \\ \hfill D_{q,\lambda }{f}_{{\gamma }_q}\left(z\right)& =& {\gamma }_q\left(s\right)\left((1-\lambda )f_{{\gamma }_q}\left(z\right)+\lambda z{D}_q{f}_{{\gamma }_q}\left(z\right)\right);\hfill \\ & ⋮& \\ \hfill D_{q,\lambda }^n{f}_{{\gamma }_q}\left(z\right)& =& D_{q,\lambda }\left(D_{q,\lambda }^{n-1}{f}_{{\gamma }_q}\left(z\right)\right),\hfill \end{array}$$

where $n\in {\mathbb{N}}_0=\mathbb{N}\bigcup \left\{0\right\}$.

If f is satisfies (4), then we have

$$\begin{array}{c}\hfill D_{q,\lambda }^n{f}_{{\gamma }_q}\left(z\right)={\gamma }_q^n\left(s\right)z-\sum _{n=k}^{\infty }{\gamma }_q^{n+1}\left(s\right){\left[(1-\lambda )+\lambda {\left[k\right]}_q\right]}^n{a}_k{z}^k.\end{array}$$

(5)

From the preceding definition, we observe that if $s=0$ and $q\to 1^{-}$, then we have

$$\begin{array}{c}\hfill D_{\lambda }^{n}f\left(z\right)=z-\sum _{k=2}^{\infty }{\left[1+(k-1)\lambda \right]}^n{a}_k{z}^k,\end{array}$$

where the operator $D_{\lambda }f$ is the generalized Salagean differential operator defined by [25]. Let f be given by (1), then the q-analogue involving a modified q-Sigmoid function of the Ruscheweyh operator is defined by

$$\begin{array}{c}\hfill R_q^n{f}_{{\gamma }_q}\left(z\right)=z-\sum _{n=2}^{\infty }{\gamma }_q\left(s\right)B_q^k\left(n\right)a_k{z}^k,\end{array}$$

(6)

where $B_q^k\left(n\right)$ has a significance of (2).

Definition 1.

Let $f\in \mathcal{U}$ and $0\le q\le 1$. Then, we define $S_{{\gamma }_q}^{n,R}\left(\alpha \right)$ to be the subclass of $\mathcal{U}$ defined as

$$\begin{array}{c}\hfill S_{{\gamma }_q}^{n,R}\left(\alpha \right)=\left\{f\in \mathcal{U}:Re\left(\frac{z{D}_q{R}_q^n{f}_{{\gamma }_q}\left(z\right)}{R_q^n{f}_{{\gamma }_q}\left(z\right)}\right)>\alpha ,z\in D\right\}.\end{array}$$

Definition 2.

Let $f\in \mathcal{U}$ and $0\le q\le 1$. Then, by $C_{{\gamma }_q}^{n,R}\left(\alpha \right)$ we denote the subclass of $\mathcal{U}$ such that

$$\begin{array}{c}\hfill C_{{\gamma }_q}^{n,R}\left(\alpha \right)=\left\{f\in \mathcal{U}:Re\left(1+\frac{z{D}_q\left(z{D}_q{R}_q^n{f}_{{\gamma }_q}\left(z\right)\right)}{z{D}_q{R}_q^n{f}_{{\gamma }_q}\left(z\right)}\right)>\alpha ,z\in D\right\}.\end{array}$$

In [44,45], the authors raised a definition of the $(n,\delta )$-neighbourhood of a function f in $\mathcal{U}$. In [46], they introduced the $(\delta ,q)$-neighbourhood of a function $f\in \mathcal{U}$ in the form

$$\begin{array}{c}\hfill N_{\delta ,q}\left(f\right)=\left\{g\in {\mathcal{U}}_p:g\left(z\right)=z-\sum _{k=n}^{\infty }{b}_k{z}^{k}and\sum _{k=n}^{\infty }{\left[k\right]}_q|a_k-b_k|\le \delta \right\},(\delta >0).\end{array}$$

(7)

Therefore, it can be inferred from (7) that if $h\left(z\right)=z,$ then

$$\begin{array}{c}\hfill N_{\delta ,q}\left(h\right)=\left\{g\in \mathcal{U}:g\left(z\right)=z-\sum _{k=n}^{\infty }{b}_k{z}^{k}and\sum _{k=n}^{\infty }{\left[k\right]}_q\left|b_k\right|\le \delta \right\},(\delta >0).\end{array}$$

(8)

However, the aim of the present paper is to discuss various characteristics of an analytic univalent function in the class $T_{{\gamma }_q}^{n,R}(\mu ,\alpha )$ of those functions f possessing the inequality

$$\begin{array}{c}\hfill Re\left\{\frac{z{D}_q{R}_q^n{f}_{{\gamma }_q}\left(z\right)+\mu z{D}_q\left(z{D}_q{R}_q^n{f}_{{\gamma }_q}\left(z\right)\right)}{(1-\mu )R_q^n{f}_{{\gamma }_q}\left(z\right)+\mu z{D}_q{R}_q^n{f}_{{\gamma }_q}\left(z\right)}\right\}>\alpha ,(z\in D,0\le \mu \le 10\le \alpha <1).\end{array}$$

(9)

Clearly, in terms of the simpler classes $S_{{\gamma }_q}^{n,R}\left(\alpha \right)$ and $C_{{\gamma }_q}^{n,R}\left(\alpha \right)$, we, respectively, have

$$\begin{array}{c}\hfill T_{{\gamma }_q}^{n,R}(0,\alpha )=S_{{\gamma }_q,\lambda }^{n,R}\left(\alpha \right)and{T}_{{\gamma }_q}^{n,R}(1,\alpha )=C_{{\gamma }_q}^{n,R}\left(\alpha \right).\end{array}$$

For further demonstration, we denote by ${\mathcal{K}}_{{\gamma }_q}^{n,R}(\mu ,\alpha )$ the subclass of $\mathcal{U}$ of all functions f possessing the inequality

$$\begin{array}{c}\hfill Re\left\{R_q^n{f}_{{\gamma }_q}\left(z\right)+\mu z{D}_q{R}_q^n{f}_{{\gamma }_q}\left(z\right)\right\}>\alpha .\end{array}$$

Now, we combine the q-analogue of the generalized Salagean differential operator involving the modified q-Sigmoid function defined by (5) and the Ruscheweyh operator involving the modified q-Sigmoid function expressed in (6) to obtain a new operator as follows:

$$\begin{array}{c}\hfill {\mathsf{\Psi }}_{q,\lambda }^n{f}_{{\gamma }_q}\left(z\right)=\beta D_{{\gamma }_q,\lambda }^n{f}_{{\gamma }_q}\left(z\right)+(1-\beta )R_q^n{f}_{{\gamma }_q}\left(z\right),0\le \lambda \le 1,0\le \beta \le 1,z\in U.\end{array}$$

Thus, we write

$$\begin{array}{c}\hfill {\mathsf{\Psi }}_{q,\lambda }^n{f}_{{\gamma }_q}\left(z\right)=\left(\beta {\gamma }_q^n\left(s\right)-\beta +1\right)z-\sum _{k=2}^{\infty }{\gamma }_q\left(s\right)\left[\beta {\gamma }_q^n\left(s\right){\left[(1-\lambda )+\lambda {\left[k\right]}_q\right]}^n+(1-\beta )B_q^k\left(n\right)\right]a_k{z}^k.\end{array}$$

Definition 3.

Let $f\in \mathcal{U}$, $0\le q\le 1$ and $0\le \lambda \le 1$. Then, we define the subclass $S_{{\gamma }_q,\lambda }^{n,\mathsf{\Psi }}\left(\alpha \right)$ of the class $\mathcal{U}$ by

$$\begin{array}{c}\hfill S_{{\gamma }_q,\lambda }^{n,\mathsf{\Psi }}\left(\alpha \right)=\left\{f\in \mathcal{U}:Re\left(\frac{z{D}_q{\mathsf{\Psi }}_{q,\lambda }^n{f}_{{\gamma }_q}\left(z\right)}{{\mathsf{\Psi }}_{q,\lambda }^n{f}_{{\gamma }_q}\left(z\right)}\right)>\alpha ,z\in D\right\}.\end{array}$$

Definition 4.

Let $f\in \mathcal{U}$, $0\le q\le 1$ and $0\le \lambda \le 1$. Then, we define the subclass $C_{{\gamma }_q,\lambda }^{n,\mathsf{\Psi }}\left(\alpha \right)$ of the class $\mathcal{U}$ by

$$\begin{array}{c}\hfill C_{{\gamma }_q,\lambda }^{n,\mathsf{\Psi }}\left(\alpha \right)=\left\{f\in \mathcal{U}:Re\left(1+\frac{z{D}_q\left(z{D}_q{\mathsf{\Psi }}_{q,\lambda }^n{f}_{{\gamma }_q}\left(z\right)\right)}{z{D}_q{\mathsf{\Psi }}_{q,\lambda }^n{f}_{{\gamma }_q}\left(z\right)}\right)>\alpha ,z\in D\right\}.\end{array}$$

Finally, let $T_{{\gamma }_q,\lambda }^{n,\mathsf{\Psi }}(\mu ,\alpha )$ denote the subclass of $\mathcal{U}$ of functions such that the following inequality holds

$$\begin{array}{c}\hfill Re\left\{\frac{z{D}_q{\mathsf{\Psi }}_{q,\lambda }^n{f}_{{\gamma }_q}\left(z\right)+\mu z{D}_q\left(z{D}_q{\mathsf{\Psi }}_{q,\lambda }^n{f}_{{\gamma }_q}\left(z\right)\right)}{(1-\mu ){\mathsf{\Psi }}_{q,\lambda }^n{f}_{{\gamma }_q}\left(z\right)+\mu z{D}_q{\mathsf{\Psi }}_{q,\lambda }^n{f}_{{\gamma }_q}\left(z\right)}\right\}>\alpha ,(z\in D,0\le \mu \le 10\le \alpha <1).\end{array}$$

(10)

Then, in terms of the simpler classes $S_{{\gamma }_q,\lambda }^{n,\mathsf{\Psi }}\left(\alpha \right)$ and $C_{{\gamma }_q,\lambda }^{n,\mathsf{\Psi }}\left(\alpha \right)$, we, respectively, have

$$\begin{array}{c}\hfill T_{{\gamma }_q,\lambda }^{n,\mathsf{\Psi }}(0,\alpha )=S_{{\gamma }_q,\lambda }^{n,\mathsf{\Psi }}\left(\alpha \right)and{T}_{{\gamma }_q,\lambda }^{n,\mathsf{\Psi }}(1,\alpha )=C_{{\gamma }_q,\lambda }^{n,\mathsf{\Psi }}\left(\alpha \right).\end{array}$$

Furthermore, let ${\mathcal{K}}_{{\gamma }_q,\lambda }^{n,\mathsf{\Psi }}(\mu ,\alpha )$ denote the subclass of $\mathcal{U}$ of functions f that satisfy the inequality

$$\begin{array}{c}\hfill Re\left\{{\mathsf{\Psi }}_{q,\lambda }^n{f}_{{\gamma }_q}\left(z\right)+\mu z{D}_q{\mathsf{\Psi }}_{q,\lambda }^n{f}_{{\gamma }_q}\left(z\right)\right\}>\alpha .\end{array}$$

One part of deriving a set of coefficient bounds for each of such function classes is to establish several inclusion relationships associated with the $(q,\delta )$-neighbourhoods of analytic univalent functions of negative missing coefficients in the same subclasses.

2. A Set of Coefficient q-Inequalities

In this section, we establish the following result, which gives a coefficient inequality for functions in the subclass $T_{{\gamma }_q,\lambda }^{n,\mathsf{\Psi }}(\mu ,\alpha )$.

Theorem 1.

Let $n\in \mathbb{N}\bigcup \left\{0\right\}$ be a fixed number. A function $f\in \mathcal{U}$ is in the class $T_{{\gamma }_q,\lambda }^{n,\mathsf{\Psi }}(\mu ,\alpha )$ if and only if

$$\begin{array}{c}\hfill \sum _{k=2}^{\infty }{\varphi }_k\left[\alpha \mu -\alpha +(1-\alpha \mu ){\left[k\right]}_q+\mu {\left[k\right]}_q^2\right]a_k,\le (1+\mu -\alpha )(\beta {\gamma }_q^n\left(s\right)-\beta +1)\end{array}$$

(11)

$$\begin{array}{c}\hfill (0\le q<1,0\le \beta \le 1,0\le \mu <1,0\le \alpha <1,z\in D),\end{array}$$

where

$$\begin{array}{c}\hfill {\varphi }_k={\gamma }_q\left(s\right)\left[\beta {\gamma }_q^n\left(s\right){\left[(1-\lambda )+\lambda {\left[k\right]}_q\right]}^n+(1-\beta )B_q^k\left(n\right)\right],k=2,3,\dots .\end{array}$$

(12)

This result is sharp.

Proof. 

Assume that $f\in T_{{\gamma }_q,\lambda }^{n,\mathsf{\Psi }}(\mu ,\alpha )$. Then, we have

$$\begin{array}{c}\hfill Re\left\{\frac{(1+\mu )(\beta {\gamma }_q^n\left(s\right)-\beta +1)-{\sum }_{k=2}^{\infty }{\varphi }_k\left[{\left[k\right]}_q+\mu {\left[k\right]}_q^2\right]a_k{z}^{k-1}}{\beta {\gamma }_q^n\left(s\right)-\beta +1-{\sum }_{k=2}^{\infty }{\varphi }_k\left[1-\mu +\mu {\left[k\right]}_q\right]a_k{z}^{k-1}}\right\}>\alpha ,\end{array}$$

$$\begin{array}{c}\hfill (0\le q<1,0\le \beta \le 1,0\le \mu <1,0\le \alpha <1,z\in D).\end{array}$$

Choose z to be real and let $z\to 1^{-}$. Then, we obtain

$$\begin{array}{c}\hfill \frac{(1+\mu )(\beta {\gamma }_q^n\left(s\right)-\beta +1)-{\sum }_{k=2}^{\infty }{\varphi }_k\left[{\left[k\right]}_q+\mu {\left[k\right]}_q^2\right]a_k}{{\gamma }_q^n\left(s\right)-\beta +1-{\sum }_{k=2}^{\infty }{\varphi }_k\left[1-\mu +\mu {\left[k\right]}_q\right]a_k}>\alpha ,\end{array}$$

$$\begin{array}{c}\hfill (0\le q<1,0\le \beta \le 1,0\le \mu <1,0\le \alpha <1,z\in D).\end{array}$$

Or, alternatively, we write

$$\begin{array}{c}\hfill (1+\mu )(\beta {\gamma }_q^n\left(s\right)-\beta +1)-\sum _{k=2}^{\infty }{\varphi }_k\left[{\left[k\right]}_q+\mu {\left[k\right]}_q^2\right]a_k\ge \alpha \left({\gamma }_q^n\left(s\right)-\beta +1\right)-\sum _{k=2}^{\infty }\alpha {\varphi }_k\left[1-\mu +\mu {\left[k\right]}_q\right]a_k,\end{array}$$

$$\begin{array}{c}\hfill (0\le q<1,0\le \beta \le 1,0\le \mu <1,0\le \alpha <1,z\in D).\end{array}$$

Thus, it follows

$$\begin{array}{c}\hfill -\sum _{k=2}^{\infty }{\varphi }_k\left[\alpha \mu -\alpha +(1-\alpha \mu ){\left[k\right]}_q+\mu {\left[k\right]}_q^2\right]a_k\ge (\alpha -1-\mu )(\beta {\gamma }_q^n\left(s\right)-\beta +1)\end{array}$$

$$\begin{array}{c}\hfill (0\le q<1,0\le \beta \le 1,0\le \mu <1,0\le \alpha <1,z\in D),\end{array}$$

which is precisely the assertion (11) of Theorem 1.

Conversely, assume that the inequality (11) truly holds and let

$$\begin{array}{c}\hfill z\in \partial D=\{z\in \mathbb{C}:|z|=1\}.\end{array}$$

Then, from the assertion (10) and the assumption that the inequality (11) holds, we find that

$$\begin{array}{ccc}& & \left|\frac{z{D}_q{\mathsf{\Psi }}_{q,\lambda }^n{f}_{{\gamma }_q}\left(z\right)+\mu z{D}_q\left(z{D}_q{\mathsf{\Psi }}_{q,\lambda }^n{f}_{{\gamma }_q}\left(z\right)\right)}{(1-\mu ){\mathsf{\Psi }}_{q,\lambda }^n{f}_{{\gamma }_q}\left(z\right)+\mu z{D}_q{\mathsf{\Psi }}_{q,\lambda }^n{f}_{{\gamma }_q}\left(z\right)}-(1+\mu -\alpha )\right|\hfill \\ & =& \left|\frac{\alpha (\beta {\gamma }_q^n\left(s\right)-\beta +1)-{\sum }_{k=2}^{\infty }{\varphi }_k\left[\mu {\left[k\right]}_q^2+\left(1-\mu \left(1+\mu -\alpha \right)\right){\left[k\right]}_q-\left(1-\mu \right)\left(1+\mu -\alpha \right)\right]a_k{z}^{k-1}}{\beta {\gamma }_q^n\left(s\right)-\beta +1-{\sum }_{k=2}^{\infty }{\varphi }_k\left[1-\mu +\mu {\left[k\right]}_q\right]a_k{z}^{k-1}}\right|\hfill \\ & \le & \frac{\alpha (\beta {\gamma }_q^n\left(s\right)-\beta +1)+{\sum }_{k=2}^{\infty }{\varphi }_k\left[\mu {\left[k\right]}_q^2+\left(1-\mu \left(1+\mu -\alpha \right)\right){\left[k\right]}_q-\left(1-\mu \right)\left(1+\mu -\alpha \right)\right]a_k{\left|z\right|}^{k-1}}{\beta {\gamma }_q^n\left(s\right)-\beta +1-{\sum }_{k=2}^{\infty }{\varphi }_k\left[1-\mu +\mu {\left[k\right]}_q\right]a_k{\left|z\right|}^{k-1}}\hfill \\ & =& \frac{\alpha (\beta {\gamma }_q^n\left(s\right)-\beta +1)+{\sum }_{k=2}^{\infty }{\varphi }_k\left[\mu {\left[k\right]}_q^2+{\left[k\right]}_q\right]a_k{\left|z\right|}^{k-1}-(1+\mu -\alpha ){\sum }_{k=2}^{\infty }{\varphi }_k\left[\mu {\left[k\right]}_q+(1-\mu )\right]a_k{\left|z\right|}^{k-1}}{\beta {\gamma }_q^n\left(s\right)-\beta +1-{\sum }_{k=2}^{\infty }{\varphi }_k\left[1-\mu +\mu {\left[k\right]}_q\right]a_k{\left|z\right|}^{k-1}}\hfill \\ & \le & 1+\mu -2\alpha .\hfill \end{array}$$

Thus, the maximum modulus theorem reveals $f\in T_{q,\lambda }^{n,\mathsf{\Psi }}(\mu ,\alpha )$. Finally, it is clear that the assertion (11) of Theorem 1 is sharp, where the extremal function is given by

$$\begin{array}{c}\hfill f\left(z\right)=z-\frac{(1+\mu -\alpha )(\beta {\gamma }_q^n\left(s\right)-\beta +1)}{{\varphi }_k\left[\alpha \mu -\alpha +(1-\alpha \mu ){\left[k\right]}_q+\mu {\left[k\right]}_q^2\right]}{z}^k\end{array}$$

$$\begin{array}{c}\hfill (0\le q<1,0\le \beta \le 1,0\le \mu <1,0\le \alpha <1,z\in D).\end{array}$$

Hence, we have the required result. □

Putting $\beta =0$ in Theorem 1, one may derive the following corollary.

Corollary 1.

Let $n\in \mathbb{N}\bigcup \left\{0\right\}$ be a fixed number. A function $f\in \mathcal{U}$ is in the class $T_{q,\lambda }^{n,R}(\mu ,\alpha )$ if and only if

$$\begin{array}{c}\hfill \sum _{k=2}^{\infty }{B}_q^k\left(n\right)\left[\alpha \mu -\alpha +(1-\alpha \mu ){\left[k\right]}_q+\mu {\left[k\right]}_q^2\right]a_k\le \frac{1+\mu -\alpha }{{\gamma }_q\left(s\right)}\end{array}$$

$$\begin{array}{c}\hfill (0\le q<1,,0\le \mu <1,0\le \alpha <1,z\in D).\end{array}$$

This result is indeed sharp for a function f in the form

$$\begin{array}{c}\hfill f\left(z\right)=z-\frac{1+\mu -\alpha }{{\gamma }_q\left(s\right)\left(B_q^k\left(n\right)\left[\alpha \mu -\alpha +(1-\alpha \mu ){\left[k\right]}_q+\mu {\left[k\right]}_q^2\right]\right)}{z}^k,(0\le q<1,0\le \mu <1,0\le \alpha <1,z\in D).\end{array}$$

Similarly, we can prove the following theorem.

Theorem 2.

Let $n\in \mathbb{N}\bigcup \left\{0\right\}$ be a fixed number. A function $f\in \mathcal{U}$ is a member in the class ${\mathcal{K}}_{q,\lambda }^{n\mathsf{\Psi }}(\mu ,\alpha )$ if and only if

$$\begin{array}{c}\hfill \sum _{k=2}^{\infty }{\varphi }_k(1+\mu {\left[k\right]}_q)a_k\le (1+\mu )(\beta {\gamma }_q^n\left(s\right)-\beta +1)-\alpha ,\end{array}$$

(13)

$$\begin{array}{c}\hfill (0\le q<1,0\le \beta \le 1,0\le \mu <1,0\le \alpha <1,z\in D),\end{array}$$

where ${\varphi }_k$ is given by (12).

This result is sharp for a function f given by

$$\begin{array}{c}\hfill f\left(z\right)=z-\frac{(1+\mu )(\beta {\gamma }_q^n\left(s\right)-\beta +1)-\alpha }{{\varphi }_k(1+\mu {\left[k\right]}_q)}{z}^k,(0\le q<1,0\le \beta \le 1,0\le \mu <1,0\le \alpha <1,z\in D).\end{array}$$

Putting $\beta =0$ in Theorem 2, we derive a corollary as follows.

Corollary 2.

Let $n\in \mathbb{N}\bigcup \left\{0\right\}$ be a fixed number. A function $f\in \mathcal{U}$ is in the class ${\mathcal{N}}_q^{n,R}(\mu ,\alpha )$ if and only if

$$\begin{array}{c}\hfill \sum _{k=2}^{\infty }{B}_q^k\left(n\right)a_k\le \frac{1+\mu -\alpha }{{\gamma }_q\left(s\right)},0\le q<1,0\le \mu <1,0\le \alpha <1,z\in D.\end{array}$$

$$\begin{array}{c}\hfill (0\le q<1,,0\le \mu <1,0\le \alpha <1,z\in D).\end{array}$$

This result is indeed sharp for a function f in the form

$$\begin{array}{c}\hfill f\left(z\right)=z-\frac{1+\mu -\alpha }{{\gamma }_q\left(s\right)B_q^k\left(n\right)a_k},(0\le q<1,,0\le \mu <1,0\le \alpha <1,z\in D).\end{array}$$

Theorem 3.

Let the function f be given by (1) and ${\varphi }_k$ be given by (12). If f is in the class $T_{q,\lambda }^{n,\mathsf{\Psi }}(\mu ,\alpha )$, then we have

$$\begin{array}{c}\hfill \sum _{k=2}^{\infty }{\left[k\right]}_q{a}_k\le \frac{(1+\mu -\alpha )(\beta {\gamma }_q^n\left(s\right)-\beta +1)}{{\varphi }_k\left[1-\alpha \mu +\mu {\left[k\right]}_q\right]}\left(1+\frac{\alpha \mu -\alpha }{\alpha \mu -\alpha +(1-\alpha \mu ){\left[k\right]}_q+\mu {\left[k\right]}_q^2}\right).\end{array}$$

(14)

Proof. 

By using the assertion (11) in Theorem 1, we obtain

$$\begin{array}{ccc}\hfill {\varphi }_k\left(\alpha \mu -\alpha +(1-\alpha \mu {\left[k\right]}_q+\mu {\left[k\right]}_q^2\right)\sum _{k=2}^{\infty }{a}_k{z}^k& \le & \sum _{k=2}^{\infty }{\varphi }_k\left(\alpha \mu -\alpha +(1-\alpha \mu ){\left[k\right]}_q+\mu {\left[k\right]}_q^2\right)a_k{z}^k\hfill \\ & \le & (1+\mu -\alpha )(\beta {\gamma }_q^n\left(s\right)-\beta +1),\hfill \end{array}$$

which immediately yields

$$\begin{array}{c}\hfill \sum _{k=2}^{\infty }{a}_k{z}^k\le \frac{(1+\mu -\alpha )(\beta {\gamma }_q^n\left(s\right)-\beta +1)}{{\varphi }_k\left(\alpha \mu -\alpha +(1-\alpha \mu ){\left[k\right]}_q+\mu {\left[k\right]}_q^2\right)}.\end{array}$$

(15)

Furthermore, by using the inequality (11), we have

$$\begin{array}{ccc}& & {\varphi }_k(\alpha \mu -\alpha )\sum _{k=2}^{\infty }{a}_k+{\varphi }_k\left(1-\alpha \mu +\mu {\left[k\right]}_q\right)\sum _{k=1}^{\infty }{\left[k\right]}_q{a}_k\hfill \\ & & \le \sum _{k=2}^{\infty }{\varphi }_k\left(\alpha -\alpha \mu +(1-\alpha \mu ){\left[k\right]}_q+\mu {\left[k\right]}_q^2\right)a_k\le (1+\mu -\alpha )(\beta {\gamma }_q^n\left(s\right)-\beta +1).\hfill \end{array}$$

It follows from the inequality (15) that

$$\begin{array}{ccc}\hfill {\varphi }_k\left(1-\alpha \mu +\mu {\left[k\right]}_q\right)\sum _{k=1}^{\infty }{\left[k\right]}_q{a}_k& \le & (\alpha -\alpha \mu )\frac{(1+\mu -\alpha )(\beta {\gamma }_q^n\left(s\right)-\beta +1)}{\alpha \mu -\alpha +(1-\alpha \mu ){\left[k\right]}_q+\mu {\left[k\right]}_q^2}\hfill \\ & +& (1+\mu -\alpha )(\beta {\gamma }_q^n\left(s\right)-\beta +1).\hfill \end{array}$$

Hence, we derived the desired inequality as presented in (14). □

Putting $\beta =0$ in Theorem 3, we state without proof a corollary as follows.

Corollary 3.

Let f be a function furnished by (1). If f is in the class $T_{q,\lambda }^{n,R}(\mu ,\alpha )$, then we have

$$\begin{array}{c}\hfill \sum _{k=2}^{\infty }{\left[k\right]}_q{a}_k=\frac{1+\mu -\alpha }{{\gamma }_q\left(s\right)B_q^k\left(n\right)\left[1-\alpha \mu +\mu {\left[k\right]}_q\right]}\left(1+\frac{\alpha \mu -\alpha }{\alpha \mu -\alpha +(1-\alpha \mu ){\left[k\right]}_q+\mu {\left[k\right]}_q^2}\right).\end{array}$$

(16)

Similarly, by invoking inequality (13) in Theorem 2, we prove a new theorem as follows.

Theorem 4.

Let f be a function given by (1) and ${\varphi }_k$ be given by (12). If f is in the class ${\mathcal{N}}_{q,\lambda }^{n,\mathsf{\Psi }}(\mu ,\alpha )$, then we have

$$\begin{array}{c}\hfill \sum _{k=2}^{\infty }{a}_k\le \frac{(1+\mu )(\beta {\gamma }_q^n\left(s\right)-\beta +1)-\alpha }{{\varphi }_k\left(1+\mu {\left[k\right]}_q\right)}\end{array}$$

and

$$\begin{array}{c}\hfill \sum _{k=2}^{\infty }{\left[k\right]}_q{a}_k=\frac{(1+\mu )(\beta {\gamma }_q^n\left(s\right)-\beta +1)}{\mu {\varphi }_k}\left(1-\frac{1}{1+\mu {\left[k\right]}_q}\right)+\frac{\alpha }{\mu {\varphi }_k(1+\mu {\left[k\right]}_q)}.\end{array}$$

(17)

Putting $\beta =0$ in Theorem 4, we derive a new corollary as follows.

Corollary 4.

Let the function f be given by (1). If f is in the class ${\mathcal{N}}_{q,\lambda }^{n,R}(\mu ,\alpha )$, then we have

$$\begin{array}{c}\hfill \sum _{k=2}^{\infty }{\left[k\right]}_q{a}_k=\frac{1+\mu }{\mu {\gamma }_q\left(s\right)B_q^k\left(n\right)}\left(1-\frac{1}{1+\mu {\left[k\right]}_q}\right)+\frac{\alpha }{{\mu }_q\left(s\right)B_q^k\left(n\right)(1+\mu {\left[k\right]}_q)}.\end{array}$$

(18)

In the following, we have the following growth and distortion theorem for the defined subclasses of univalent functions.

Theorem 5.

Let a function f be given by (1). If $f\in T_{{\gamma }_q,\lambda }^{n,\mathsf{\Psi }}(\mu ,\alpha )$, then we have

$$\begin{array}{c}\hfill r-r^2\frac{(1+\mu -\alpha )(\beta {\gamma }_q^n\left(s\right)-\beta +1)}{{\varphi }_k\left(\alpha \mu -\alpha +\left(1-\alpha \mu \right){\left[k\right]}_q+\mu {\left[k\right]}_q^2\right)}\le \left|f\left(z\right)\right|\le r+r^2\frac{(1+\mu -\alpha )(\beta {\gamma }_q^n\left(s\right)-\beta +1)}{{\varphi }_k\left(\alpha \mu -\alpha +\left(1-\alpha \mu \right){\left[k\right]}_q+\mu {\left[k\right]}_q^2\right)},\end{array}$$

(19)

where ${\varphi }_k$ is defined in assertion (12).

Proof. 

Since $f\in T_{q,\lambda }^{n,\mathsf{\Psi }}(\mu ,\alpha )$, let $z\to 1^{-}$ in the above inequality to have

$$\begin{array}{c}\hfill \sum _{k=2}^{\infty }{a}_k\le \frac{(1+\mu -\alpha )(\beta {\gamma }_q^n\left(s\right)-\beta +1)}{{\varphi }_k\left(\alpha \mu -\alpha +(1-\alpha \mu {\left[k\right]}_q+\mu {\left[k\right]}_q^2\right)}.\end{array}$$

(20)

In addition, since the function f is given by (1), we obtain

$$\begin{array}{c}\hfill \left|z\right|-{\left|z\right|}^2\sum _{k=2}^{\infty }{a}_k{\left|z\right|}^{k-2}\le \left|f\left(z\right)\right|\le \left|z\right|+{\left|z\right|}^2\sum _{k=2}^{\infty }{a}_k{\left|z\right|}^{k-2}.\end{array}$$

Thus, it gives

$$\begin{array}{c}\hfill \left|z\right|-{\left|z\right|}^2\sum _{k=2}^{\infty }{a}_k\le \left|f\left(z\right)\right|\le \left|z\right|+{\left|z\right|}^2\sum _{k=2}^{\infty }{a}_k.\end{array}$$

Hence, by using the inequality (20) we obtain the desired inequality of (19). □

Putting $\beta =0$ in Theorem 5, we derive the following corollary.

Corollary 5.

Let f be a function given by (1). If $f\in T_{q,\lambda }^{n,R}(\mu ,\alpha )$, then we have

$$\begin{array}{c}\hfill r-r^2\frac{(1+\mu -\alpha )}{B_q^k\left(n\right){\gamma }_q\left(s\right)\left(\alpha \mu -\alpha +\left(1-\alpha \mu \right){\left[k\right]}_q+\mu {\left[k\right]}_q^2\right)}\le \left|f\left(z\right)\right|\\ \hfill \le r+r^2\frac{(1+\mu -\alpha )}{B_q^k\left(n\right){\gamma }_q\left(s\right)\left(\alpha \mu -\alpha +\left(1-\alpha \mu \right){\left[k\right]}_q+\mu {\left[k\right]}_q^2\right)}.\end{array}$$

By following a proof similar to the proof of Theorem 5, we prove the following theorem.

Theorem 6.

Let f be a function given by (1). If $f\in T_{{\gamma }_q,\lambda }^{n,\mathsf{\Psi }}(\mu ,\alpha )$, then we have

$$\begin{array}{c}\hfill |D_{q}f\left(z\right)|\le 1+r\frac{(1+\mu )(\beta {\gamma }_q^n\left(s\right)-\beta +1)}{\mu {\varphi }_k}\left(1-\frac{1}{1+\mu {\left[k\right]}_q}\right)+\frac{\alpha }{\mu {\varphi }_k(1+\mu {\left[k\right]}_q)},\\ \hfill |D_{q}f\left(z\right)|\ge 1-r\frac{(1+\mu )(\beta {\gamma }_q^n\left(s\right)-\beta +1)}{\mu {\varphi }_k}\left(1-\frac{1}{1+\mu {\left[k\right]}_q}\right)+\frac{\alpha }{\mu {\varphi }_k(1+\mu {\left[k\right]}_q)}.\end{array}$$

where ${\varphi }_k$ is defined in assertion (12).

Putting $\beta =0$ in the Theorem 5, we derive the following corollary.

Corollary 6.

Let the function f be given by (1). If $f\left(z\right)\in T_{{\gamma }_q,\lambda }^{n,R}(\mu ,\alpha )$, then we have

$$\begin{array}{c}\hfill |D_{q}f\left(z\right)|\le 1+r\frac{1+\mu -\alpha }{{\gamma }_q\left(s\right)B_q^k\left(n\right)\left[1-\alpha \mu +\mu {\left[k\right]}_q\right]}\left(1+\frac{\alpha \mu -\alpha }{\alpha \mu -\alpha +(1-\alpha \mu ){\left[k\right]}_q+\mu {\left[k\right]}_q^2}\right),\\ \hfill |D_{q}f\left(z\right)|\ge 1-r\frac{1+\mu -\alpha }{{\gamma }_q\left(s\right)B_q^k\left(n\right)\left[1-\alpha \mu +\mu {\left[k\right]}_q\right]}\left(1+\frac{\alpha \mu -\alpha }{\alpha \mu -\alpha +(1-\alpha \mu ){\left[k\right]}_q+\mu {\left[k\right]}_q^2}\right).\end{array}$$

3. Inclusion Relations Involving the $(\mathit{n},\mathit{\delta })$-Neighborhoods

In this section, we derive inclusion relations associated with the $(n,\delta )$-neighbourhoods and properties for each of the following subclasses of univalent functions with negative coefficients as follows.

Theorem 7.

Let f be given by (1), ${\varphi }_k$ be given by (12) and $h\left(z\right)=z$ for each $z\in D$. If f is in the class $T_{q,\lambda }^{n,\mathsf{\Psi }}(\mu ,\alpha )$, then we have

$$\begin{array}{c}\hfill T_{q,\lambda }^{n,\mathsf{\Psi }}(\mu ,\alpha )\subseteq N_{\delta ,q}\left(h\right),\end{array}$$

(21)

where

$$\begin{array}{c}\hfill \delta :=\frac{(1+\mu -\alpha )(\beta {\gamma }_q^n\left(s\right)-\beta +1)}{{\varphi }_k\left[1-\alpha \mu +\mu {\left[k\right]}_q\right]}\left(1+\frac{\alpha \mu -\alpha }{\alpha \mu -\alpha +(1-\alpha \mu ){\left[k\right]}_q+\mu {\left[k\right]}_q^2}\right).\end{array}$$

Proof. 

By using the definition of $N_{\delta ,q}\left(h\right)$ in (8), which is given when g is replaced by f, and using inequality (14) in Theorem 3, we derive the assertion presented (22). □

Similarly, by using the inequalities (16)–(18) and the definition of $N_{\delta ,q}\left(h\right)$ presented in (8), we derive the three following theorems.

Theorem 8.

Let f be a function be given by (1) and $h\left(z\right)=z$ for each $z\in D$. If f belongs to the class $T_{q,\lambda }^{n,R}(\mu ,\alpha )$, then we have

$$\begin{array}{c}\hfill T_{q,\lambda }^{n,R}(\mu ,\alpha )\subseteq N_{\delta ,q}\left(h\right),\end{array}$$

where

$$\begin{array}{c}\hfill \delta :=\frac{1+\mu -\alpha }{{\gamma }_q\left(s\right)B_q^k\left(n\right)\left[1-\alpha \mu +\mu {\left[k\right]}_q\right]}\left(1+\frac{\alpha \mu -\alpha }{\alpha \mu -\alpha +(1-\alpha \mu ){\left[k\right]}_q+\mu {\left[k\right]}_q^2}\right).\end{array}$$

Theorem 9.

Let f be a function given by (1), ${\varphi }_k$ be given by (12) and $h\left(z\right)=z$ for each $z\in D$. If f falls in the class ${\mathcal{N}}_{q,\lambda }^{n,\mathsf{\Psi }}(\mu ,\alpha )$, then we have

$$\begin{array}{c}\hfill {\mathcal{N}}_{q,\lambda }^{n,\mathsf{\Psi }}(\mu ,\alpha )\subseteq N_{\delta ,q}\left(h\right),\end{array}$$

(22)

where

$$\begin{array}{c}\hfill \delta :=\frac{(1+\mu )(\beta {\gamma }_q^n\left(s\right)-\beta +1)}{\mu {\varphi }_k}\left(1-\frac{1}{1+\mu {\left[k\right]}_q}\right)+\frac{\alpha }{\mu {\varphi }_k(1+\mu {\left[k\right]}_q)}.\end{array}$$

Theorem 10.

Let the function f be given by (1) and $h\left(z\right)=z$ for each $z\in D$. If f is in the class ${\mathcal{N}}_{q,\lambda }^{n,R}(\mu ,\alpha )$, then we have

$$\begin{array}{c}\hfill {\mathcal{N}}_{q,\lambda }^{n,R}(\mu ,\alpha )\subseteq N_{\delta ,q}\left(h\right),\end{array}$$

where

$$\begin{array}{c}\hfill \delta :=\frac{1+\mu }{\mu {\gamma }_q\left(s\right)B_q^k\left(n\right)}\left(1-\frac{1}{1+\mu {\left[k\right]}_q}\right)+\frac{\alpha }{{\mu }_q\left(s\right)B_q^k\left(n\right)(1+\mu {\left[k\right]}_q)}.\end{array}$$

Definition 5.

The function $f\in \mathcal{U}$ is said to belong to the class $S_{q,\lambda }^{n,\mathsf{\Psi }}(\mu ,\zeta )$ if there exists a function $g\in {\mathcal{N}}_{q,\lambda }^{n,\mathsf{\Psi }}(\mu ,\alpha ))$ such that

$$\begin{array}{c}\hfill \left|\frac{f\left(z\right)}{g\left(z\right)}-1\right|\le \eta ,(0<\eta \le 1),\end{array}$$

where

$$\begin{array}{c}\hfill g\left(z\right)=z+\sum _{k=2}^{\infty }{b}_n{z}^n,(z\in D).\end{array}$$

(23)

Theorem 11.

Let the function f be given by (23), ${\varphi }_k$ be given by (12) and $g\in {\mathcal{N}}_{q,\lambda }^{n,\mathsf{\Psi }}(\mu ,\alpha ))$. If

$$\zeta \le 1-\frac{2\delta {\varphi }_k(1+\mu {\left[k\right]}_q)}{{\left[2\right]}_q\left({\varphi }_k\left(1+\mu {\left[k\right]}_q\right)-(1+\mu )(\beta {\gamma }_q^n\left(s\right)-\beta +1)\right)},$$

then we have

$$\begin{array}{c}\hfill N_{\delta ,q}\left(h\right)\subseteq S_{q,\lambda }^{n,\mathsf{\Psi }}(\mu ,\zeta ).\end{array}$$

Proof. 

Suppose that $f\in N_{\delta ,q}\left(g\right)$, then in view of the relation (7), we have

$$\begin{array}{c}\hfill \sum _{k=2}^{\infty }{\left[k\right]}_q|a_k-b_k|\le \delta .\end{array}$$

(24)

Since $\left\{{\left[n\right]}_q\right\}$ is a non-decreasing sequence, we obtain

$$\begin{array}{c}\hfill \sum _{k=2}^{\infty }{\left[2\right]}_q|a_k-b_k|\le \sum _{k=2}^{\infty }{\left[k\right]}_q|a_k-b_k|.\end{array}$$

This implies that

$$\begin{array}{c}\hfill {\left[2\right]}_q\sum _{k=2}^{\infty }|a_k-b_k|\le \sum _{k=2}^{\infty }{\left[k\right]}_q|a_k-b_k|,\end{array}$$

which, in view of the inequality (24), reveals

$$\begin{array}{c}\hfill \sum _{k=2}^{\infty }|a_k-b_k|\le \frac{\delta }{{\left[2\right]}_q},(0\le q<1,\delta \ge 0).\end{array}$$

Therefore, for a function g in the class $S_{q,\lambda }^{n,\mathsf{\Psi }}(\mu ,\zeta )$, expressed by (23), by using the inequality (17), we obtain

$$\begin{array}{c}\hfill \sum _{k=2}^{\infty }\left|b_k\right|\le \frac{(1+\mu )(\beta {\gamma }_q^n\left(s\right)-\beta +1)-\alpha }{{\varphi }_k(1+\mu {\left[k\right]}_q)}.\end{array}$$

By applying the equations (1) and (23) and the fact that $\left|z\right|<1$, we get

$$\begin{array}{}\hfill \mathrm{(25)}& \left|\frac{f\left(z\right)}{g\left(z\right)}-1\right|& =\left|\frac{{\sum }_{k=2}^{\infty }(a_k-b_k)z^{k-1}}{1+{\sum }_{k=2}^{\infty }{b}_n{z}^{n-1}}\right|\le \frac{{\sum }_{k=2}^{\infty }|a_k-b_k|}{1-{\sum }_{k=2}^{\infty }{b}_k}\hfill \mathrm{(26)}& & \le \frac{\delta }{{\left[2\right]}_q}\left(\frac{1}{1-{\sum }_{k=2}^{\infty }\left|b_k\right|}\right)\hfill \mathrm{(27)}& & \le \frac{\delta }{{\left[2\right]}_q}\left[\frac{{\varphi }_k\left(1+\mu {\left[k\right]}_q\right)}{{\varphi }_k\left(1+\mu {\left[k\right]}_q\right)-(1+\mu )(\beta {\gamma }_q^n\left(s\right)-\beta +1)}\right].\hfill \end{array}$$

If we set

$$\zeta \le 1-\frac{2\delta {\varphi }_k(1+\mu {\left[k\right]}_q)}{{\left[2\right]}_q\left({\varphi }_k\left(1+\mu {\left[k\right]}_q\right)-(1+\mu )(\beta {\gamma }_q^n\left(s\right)-\beta +1)\right)},$$

then, in view of the definition (5) and the inequality (25), we obtain that $f\in S_{q,\lambda }^{n,\mathsf{\Psi }}(\mu ,\zeta )$. Hence, the proof of the theorem is completed. □

Putting $\beta =0$ in Theorem 11 leads to the following theorem.

Theorem 12.

Let the function f be given by (23) and $g\in {\mathcal{N}}_{q,\lambda }^{n,R}(\mu ,\alpha ))$. If

$$\zeta \le 1-\frac{2\delta {\gamma }_q\left(s\right)B_q^k\left(n\right)(1+\mu {\left[k\right]}_q)}{{\left[2\right]}_q\left({\gamma }_q\left(s\right)B_q^k\left(n\right)\left(1+\mu {\left[k\right]}_q\right)-(1+\mu )\right)},$$

then we have

$$\begin{array}{c}\hfill N_{\delta ,q}\left(h\right)\subseteq S_{q,\lambda }^{n,R}(\mu ,\zeta ).\end{array}$$

4. Conclusions

In the present work, certain analytic functions and new q-analogues of the Salagean differential operator are, respectively, obtained by using a modified q-Sigmoid function and a recent idea of Salagean. In addition, certain classes of analytic univalent functions associated with new q-Salagean differential operators and q-Ruscheweh operators are obtained. Moreover, coefficient bounds for functions in the mentioned subclasses and the growth and distortion theorems are established. Following the concept of $(q,\delta )$-neighbourhoods of analytic univalent functions, several inclusion relations for the $(q,\delta )$-neighbourhood of these functions are discussed.