Inclusion and properties neighbourhood for certain p-valent functions associated with complex order and q–p-valent Cătaş operator

Abstract

Here in our paper, we present two new subclasses of multivalent analytic functions and complex order by using q–p-valent Cătaş operator. Also we obtain coefficient estimates and consequent inclusion relationships involving the $N_{i,\delta }^{\mathbf{p},q}\left(f\right)$-neighbourhood of these classes.

Keywords:

• Coefficient estimates
• neighbourhood
• Cătaş operator
• complex order
• p-valent

1. Introduction

Let ${\mathcal{A}}_i\left(\mathbf{p}\right)$ denote the class of functions in the next formula: (1) $f\left(z\right)=z^{\mathbf{p}}+\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}{a}_k{z}^k(i,\mathbf{p}\in \mathbb{N}=\{1,2,\dots \left\}\right),$(1) which are analytic and p-valent in the open unit disk $\mathbb{U}=\{z:|z|<1\}.$ We note that ${\mathcal{A}}_1\left(\mathbf{p}\right)=\mathcal{A}\left(\mathbf{p}\right)$ (see [1,2]) and ${\mathcal{A}}_1\left(1\right)=\mathcal{A}.$ Also let $T_i\left(\mathbf{p}\right)$ denote the subclass of ${\mathcal{A}}_i\left(\mathbf{p}\right)$ which can express in the form: (2) $f\left(z\right)=z^{\mathbf{p}}-\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}{a}_k{z}^k(a_k\ge 0,i,\mathbf{p}\in \mathbb{N}).$(2) In [3,4] (see also ([5–11]) the q-derivative $D_{q,\mathbf{p}}$ of f is defined as follows: (3) $D_{q,\mathbf{p}}f\left(z\right)=\left\{\begin{array}{ll}\frac{f\left(z\right)-f\left(qz\right)}{(1-q)z}& \mathrm{f}\mathrm{o}\mathrm{r}z\ne 0,\\ f^{\mathrm{\prime }}\left(0\right)& \mathrm{f}\mathrm{o}\mathrm{r}z=0,\end{array}\right.(f\in \mathcal{A}\mathcal{(}\mathbf{p}))$(3) provided that $f^{\mathrm{\prime }}\left(0\right)$ exists. From (1(1) $f\left(z\right)=z^{\mathbf{p}}+\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}{a}_k{z}^k(i,\mathbf{p}\in \mathbb{N}=\{1,2,\dots \left\}\right),$(1) ) (with i = 1) and (3(3) $D_{q,\mathbf{p}}f\left(z\right)=\left\{\begin{array}{ll}\frac{f\left(z\right)-f\left(qz\right)}{(1-q)z}& \mathrm{f}\mathrm{o}\mathrm{r}z\ne 0,\\ f^{\mathrm{\prime }}\left(0\right)& \mathrm{f}\mathrm{o}\mathrm{r}z=0,\end{array}\right.(f\in \mathcal{A}\mathcal{(}\mathbf{p}))$(3) ), we deduce that (4) $D_{q,\mathbf{p}}f\left(z\right)={\left[\mathbf{p}\right]}_q{z}^{\mathbf{p}-1}+\sum _{k=\mathbf{p}+1}^{\mathrm{\infty }}{\left[k\right]}_q{a}_k{z}^{k-1},$(4) where $[k{]}_q$ is q-integer number k defined by (5) $[k{]}_q=\frac{1-q^k}{1-q}=1+\sum _{k=1}^{n-1}{q}^k,0<q<1,{\left[0\right]}_q=0.$(5)

We observe that $\underset{q\to 1^{-}}{lim}{D}_{q,\mathbf{p}}f\left(z\right)=\underset{q\to 1^{-}}{lim}\frac{f\left(z\right)-f\left(qz\right)}{(1-q)z}=f^{\mathrm{\prime }}\left(z\right),$ for a function f which is differentiable in a given subset of $\mathbb{C}$. For $f\in T_i\left(\mathbf{p}\right),$ we introduce the q–p-valent Cătaş operator $I_{q,\mathbf{p}}^n(\varkappa ,\varrho )f\left(z\right):T_i\left(\mathbf{p}\right)\to T_i\left(\mathbf{p}\right)(\varrho ,\varkappa \ge 0,n\in {\mathbb{N}}_0=\mathbb{N}\cup \{0\},0<q<1,i,\mathbf{p}\in \mathbb{N})$ as follows: (6) $\begin{array}{rl}& I_{q,\mathbf{p}}^0(\varkappa ,\varrho )f\left(z\right)=f\left(z\right),\\ & I_{q,\mathbf{p}}^1(\varkappa ,\varrho )f\left(z\right)\\ & =(1-\varkappa )f\left(z\right)+\frac{\varkappa }{{\left[\mathbf{p}+\varrho \right]}_q{z}^{\varrho -1}}{D}_{q,\mathbf{p}}\left(z^{\varrho }f\right(z\left)\right)\\ & =z^{\mathbf{p}}-\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}\left[\frac{\begin{array}{c}{\left[\mathbf{p}+\varrho \right]}_q\\ +\varkappa \left({\left[k+\varrho \right]}_q-{\left[\mathbf{p}+\varrho \right]}_q\right)\end{array}}{{\left[\mathbf{p}+\varrho \right]}_q}\right]a_k{z}^k,\\ & I_{q,\mathbf{p}}^2(\varkappa ,\varrho )f\left(z\right)=(1-\varkappa )I_{q,\mathbf{p}}^1(\varkappa ,\varrho )f\left(z\right)\\ & +\frac{\varkappa }{{\left[\mathbf{p}+\varrho \right]}_q{z}^{\varrho -1}}{D}_{q,\mathbf{p}}\left(z^{\varrho }{I}_{q,\mathbf{p}}^1\right(\varkappa ,\varrho \left)f\right(z\left)\right)\\ & =z^{\mathbf{p}}-\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}{\left[\frac{\begin{array}{c}{\left[\mathbf{p}+\varrho \right]}_q\\ +\varkappa \left({\left[k+\varrho \right]}_q-{\left[\mathbf{p}+\varrho \right]}_q\right)\end{array}}{{\left[\mathbf{p}+\varrho \right]}_q}\right]}^2{a}_k{z}^k,\\ & ⋮\\ & I_{q,\mathbf{p}}^n(\varkappa ,\varrho )f\left(z\right)=(1-\varkappa )I_{q,\mathbf{p}}^{n-1}(\varkappa ,\varrho )f\left(z\right)\\ & +\frac{\varkappa }{{\left[\mathbf{p}+\varrho \right]}_q{z}^{\varrho -1}}{D}_{q,\mathbf{p}}\left(z^{\varrho }{I}_{q,\mathbf{p}}^{n-1}\right(\varkappa ,\varrho \left)f\right(z\left)\right)(n\in \mathbb{N}).\end{array}$(6) From (2(2) $f\left(z\right)=z^{\mathbf{p}}-\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}{a}_k{z}^k(a_k\ge 0,i,\mathbf{p}\in \mathbb{N}).$(2) ) and (6(6) $\begin{array}{rl}& I_{q,\mathbf{p}}^0(\varkappa ,\varrho )f\left(z\right)=f\left(z\right),\\ & I_{q,\mathbf{p}}^1(\varkappa ,\varrho )f\left(z\right)\\ & =(1-\varkappa )f\left(z\right)+\frac{\varkappa }{{\left[\mathbf{p}+\varrho \right]}_q{z}^{\varrho -1}}{D}_{q,\mathbf{p}}\left(z^{\varrho }f\right(z\left)\right)\\ & =z^{\mathbf{p}}-\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}\left[\frac{\begin{array}{c}{\left[\mathbf{p}+\varrho \right]}_q\\ +\varkappa \left({\left[k+\varrho \right]}_q-{\left[\mathbf{p}+\varrho \right]}_q\right)\end{array}}{{\left[\mathbf{p}+\varrho \right]}_q}\right]a_k{z}^k,\\ & I_{q,\mathbf{p}}^2(\varkappa ,\varrho )f\left(z\right)=(1-\varkappa )I_{q,\mathbf{p}}^1(\varkappa ,\varrho )f\left(z\right)\\ & +\frac{\varkappa }{{\left[\mathbf{p}+\varrho \right]}_q{z}^{\varrho -1}}{D}_{q,\mathbf{p}}\left(z^{\varrho }{I}_{q,\mathbf{p}}^1\right(\varkappa ,\varrho \left)f\right(z\left)\right)\\ & =z^{\mathbf{p}}-\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}{\left[\frac{\begin{array}{c}{\left[\mathbf{p}+\varrho \right]}_q\\ +\varkappa \left({\left[k+\varrho \right]}_q-{\left[\mathbf{p}+\varrho \right]}_q\right)\end{array}}{{\left[\mathbf{p}+\varrho \right]}_q}\right]}^2{a}_k{z}^k,\\ & ⋮\\ & I_{q,\mathbf{p}}^n(\varkappa ,\varrho )f\left(z\right)=(1-\varkappa )I_{q,\mathbf{p}}^{n-1}(\varkappa ,\varrho )f\left(z\right)\\ & +\frac{\varkappa }{{\left[\mathbf{p}+\varrho \right]}_q{z}^{\varrho -1}}{D}_{q,\mathbf{p}}\left(z^{\varrho }{I}_{q,\mathbf{p}}^{n-1}\right(\varkappa ,\varrho \left)f\right(z\left)\right)(n\in \mathbb{N}).\end{array}$(6) ), we can obtain (7) $I_{q,\mathbf{p}}^n(\varkappa ,\varrho )f\left(z\right)=z^{\mathbf{p}}-\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}{\mathrm{\Psi }}_{q,\mathbf{p}}^n(k,\varkappa ,\varrho )a_k{z}^k,$(7) where (8) $\begin{array}{rl}& {\mathrm{\Psi }}_{q,\mathbf{p}}^n(k,\varkappa ,\varrho )={\left[\frac{{\left[\mathbf{p}+\varrho \right]}_q+\varkappa \left({\left[k+\varrho \right]}_q-{\left[\mathbf{p}+\varrho \right]}_q\right)}{{\left[\mathbf{p}+\varrho \right]}_q}\right]}^n\\ & (n\in {\mathbb{N}}_0,i,\mathbf{p}\in \mathbb{N},\varrho ,\varkappa \ge 0).\end{array}$(8) We note that

1. $\underset{q\to 1^{-}}{lim}{I}_{q,\mathbf{p}}^n(\varkappa ,\varrho )f\left(z\right)=I_{\mathbf{p}}^n(\varkappa ,\varrho )f\left(z\right),I_{q,1}^n(\varkappa ,\varrho )f\left(z\right)=I_q^n(\varkappa ,\varrho )f\left(z\right),$ $\underset{q\to 1^{-}}{lim}{I}_q^n(\varkappa ,\varrho )f\left(z\right)=I^n(\varkappa ,\varrho )f\left(z\right)$ (Cătaş [12]);

2. $I_{q,\mathbf{p}}^n(1,\varrho )f\left(z\right)=I_{q,\mathbf{p}}^n\left(\varrho \right)f\left(z\right)$ $=\left\{\begin{array}{l}\begin{array}{l}f\in T_i\left(\mathbf{p}\right):I_{q,\mathbf{p}}^n\left(\varrho \right)f\left(z\right)=z^{\mathbf{p}}\\ -\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}{\left(\frac{{\left[k+\varrho \right]}_q}{{\left[\mathbf{p}+\varrho \right]}_q}\right)}^n{a}_k{z}^k,\end{array}\\ n\in {\mathbb{N}}_0,i,\mathbf{p}\in \mathbb{N},\varrho \ge 0,0<q<1\end{array}\right\};$

3. $I_{q,\mathbf{p}}^n(\varkappa ,0)f\left(z\right)=D_{q,\mathbf{p}}^n\left(\varkappa \right)f\left(z\right)$ $=\left\{\begin{array}{l}\begin{array}{l}f\in T_i\left(\mathbf{p}\right):D_{q,\mathbf{p}}^n\left(\varkappa \right)f\left(z\right)=z^{\mathbf{p}}\\ -\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}{\left[\frac{{\left[\mathbf{p}\right]}_q+\varkappa \left({\left[k\right]}_q-{\left[\mathbf{p}\right]}_q\right)}{{\left[\mathbf{p}\right]}_q}\right]}^n{a}_k{z}^k,\end{array}\\ n\in {\mathbb{N}}_0,i,\mathbf{p}\in \mathbb{N},\varkappa \ge 0,0<q<1\end{array}\right\};$

4. $I_{q,\mathbf{p}}^n(1,0)f\left(z\right)=D_{q,\mathbf{p}}^{n}f\left(z\right)$ $=\left\{\begin{array}{l}f\in T_i\left(\mathbf{p}\right):D_{q,\mathbf{p}}^{n}f\left(z\right)\\ =z^{\mathbf{p}}-\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}{\left(\frac{{\left[k\right]}_q}{{\left[\mathbf{p}\right]}_q}\right)}^n{a}_k{z}^k,\\ n\in {\mathbb{N}}_0,i,\mathbf{p}\in \mathbb{N},0<q<1\end{array}\right\};$ where $\underset{q\to 1^{-}}{lim}{D}_{q,\mathbf{p}}^{n}f\left(z\right)=D_{\mathbf{p}}^{n}f\left(z\right)$ (see [13–15]);

5. $I_{q,1}^n(1,\varrho )f\left(z\right)=I_q^n\left(\varrho \right)f\left(z\right),\underset{q\to 1^{-}}{lim}{I}_q^n\left(\varrho \right)f\left(z\right)=I^n\left(\varrho \right)f\left(z\right)$ (see [16,17]);

6. $I_{q,1}^n(\varkappa ,0)f\left(z\right)=D_{q,\varkappa }^{n}f\left(z\right),\underset{q\to 1^{-}}{lim}{D}_{q,\varkappa }^{n}f\left(z\right)=D_{\varkappa }^{n}f\left(z\right)$ (see [18–20]);

7. $I_{q,1}^n(1,0)f\left(z\right)=D_q^{n}f\left(z\right)$ (see [21]), $\underset{q\to 1^{-}}{lim}{D}_q^{n}f\left(z\right)=D^{n}f\left(z\right)$ (see Sălăgean [22]) see also [23,24].

Now by using the operator $I_{q,\mathbf{p}}^n(\varkappa ,\varrho )f\left(z\right)$ we defined the classes $S_q^n(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β})$ and $K_q^n(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β})$ as follows:

Definition 1

Let $f\left(z\right)\in T_i\left(\mathbf{p}\right).$ Then $f\in S_q^n(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β})$ if it satisfies the following inequality: (9) $\begin{array}{rl}& \left|\frac{1}{b}\left[\frac{\begin{array}{c}(1-\sigma )z{D}_{q,\mathbf{p}}\left(I_{q,\mathbf{p}}^n\right(\varkappa ,\varrho \left)f\right(z\left)\right)\\ +\sigma z{D}_{q,\mathbf{p}}\left(z{D}_{q,\mathbf{p}}\right(I_{q,\mathbf{p}}^n(\varkappa ,\varrho )f\left(z\right)\left)\right)\end{array}}{\begin{array}{c}(1-\sigma )I_{q,\mathbf{p}}^n(\varkappa ,\varrho )f\left(z\right)\\ +\sigma z{D}_{q,\mathbf{p}}\left(I_{q,\mathbf{p}}^n\right(\varkappa ,\varrho \left)f\right(z\left)\right)\end{array}}-{\left[\mathbf{p}\right]}_q\right]\right|<\text{β}\\ & (b\in {\mathbb{C}}^{\ast }=\mathbb{C}\mathrm{\setminus }\{0\},n\in {\mathbb{N}}_0,\mathbf{p},i\in \mathbb{N},0<q<1,\varrho ,\\ & \varkappa \ge 0,0\le \sigma \le 1,0<\text{β}\le 1).\end{array}$(9)

We note that:

1. $\underset{q\to 1^{-}}{lim}{S}_q^n(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β})=S^n(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β})$ $\begin{array}{rl}& \left\{f\in T_i\left(\mathbf{p}\right):\left|\frac{1}{b}\left[\frac{\begin{array}{c}z\left(I_{\mathbf{p}}^n\right(\varkappa ,\varrho \left)f\right(z){)}^{\mathrm{\prime }}\\ +\sigma z^2\left(I_{\mathbf{p}}^n\right(\varkappa ,\varrho \left)f\right(z){)}^{\mathrm{\prime }\mathrm{\prime }}\end{array}}{\begin{array}{c}(1-\sigma )I_{\mathbf{p}}^n(\varkappa ,\varrho )f\left(z\right)\\ +\sigma z\left(I_{\mathbf{p}}^n\right(\varkappa ,\varrho \left)f\right(z){)}^{\mathrm{\prime }}\end{array}}-\mathbf{p}\right]\right|<\text{β}\right.\\ & (b\in {\mathbb{C}}^{\ast },n\in {\mathbb{N}}_0,\mathbf{p},i\in \mathbb{N},\varrho ,\varkappa \ge 0,\\ & \left.0\le \sigma \le 1,0<\text{β}\le 1)\phantom{\frac{1}{b}}\phantom{\frac{\begin{array}{c}z\left(I_{\mathbf{p}}^n\right(\varkappa ,\varrho \left)f\right(z){)}^{\mathrm{\prime }}\\ +\sigma z^2\left(I_{\mathbf{p}}^n\right(\varkappa ,\varrho \left)f\right(z){)}^{\mathrm{\prime }\mathrm{\prime }}\end{array}}{\begin{array}{c}(1-\sigma )I_{\mathbf{p}}^n(\varkappa ,\varrho )f\left(z\right)\\ +\sigma z\left(I_{\mathbf{p}}^n\right(\varkappa ,\varrho \left)f\right(z){)}^{\mathrm{\prime }}\end{array}}}\right\};\end{array}$

2. $S_q^n(i,\mathbf{p},1,\varrho ,\sigma ,b,\text{β})=S_q^n(i,\mathbf{p},\varrho ,\sigma ,b,\text{β})$ $\begin{array}{rl}& \left\{f\in T_i\left(\mathbf{p}\right):\left|\frac{1}{b}\left[\frac{\begin{array}{c}(1-\sigma )z{D}_{q,\mathbf{p}}\left(I_{q,\mathbf{p}}^n\right(\varrho \left)f\right(z\left)\right)\\ +\sigma z{D}_{q,\mathbf{p}}\left(z{D}_{q,\mathbf{p}}\right(I_{q,\mathbf{p}}^n\left(\varrho \right)f\left(z\right)\left)\right)\end{array}}{\begin{array}{c}(1-\sigma )I_{q,\mathbf{p}}^n\left(\varrho \right)f\left(z\right)\\ +\sigma z{D}_{q,\mathbf{p}}\left(I_{q,\mathbf{p}}^n\right(\varrho \left)f\right(z\left)\right)\end{array}}\right.\right.\right.\\ & \left.-\left.\phantom{\frac{\begin{array}{c}(1-\sigma )z{D}_{q,\mathbf{p}}\left(I_{q,\mathbf{p}}^n\right(\varrho \left)f\right(z\left)\right)\\ +\sigma z{D}_{q,\mathbf{p}}\left(z{D}_{q,\mathbf{p}}\right(I_{q,\mathbf{p}}^n\left(\varrho \right)f\left(z\right)\left)\right)\end{array}}{\begin{array}{c}(1-\sigma )I_{q,\mathbf{p}}^n\left(\varrho \right)f\left(z\right)\\ +\sigma z{D}_{q,\mathbf{p}}\left(I_{q,\mathbf{p}}^n\right(\varrho \left)f\right(z\left)\right)\end{array}}}{\left[\mathbf{p}\right]}_q\right]\right|<\text{β}(b\in {\mathbb{C}}^{\ast },n\in {\mathbb{N}}_0,\mathbf{p},i\in \mathbb{N},\\ & \left.0<q<1,\varrho \ge 0,0\le \sigma \le 1,0<\text{β}\le 1)\phantom{\frac{\begin{array}{c}z\left(I_{\mathbf{p}}^n\right(\varkappa ,\varrho \left)f\right(z){)}^{\mathrm{\prime }}\\ +\sigma z^2\left(I_{\mathbf{p}}^n\right(\varkappa ,\varrho \left)f\right(z){)}^{\mathrm{\prime }\mathrm{\prime }}\end{array}}{\begin{array}{c}(1-\sigma )I_{\mathbf{p}}^n(\varkappa ,\varrho )f\left(z\right)\\ +\sigma z\left(I_{\mathbf{p}}^n\right(\varkappa ,\varrho \left)f\right(z){)}^{\mathrm{\prime }}\end{array}}}\right\};\end{array}$

3. $S_q^n(i,1,\varkappa ,\varrho ,\sigma ,b,\text{β})=S_q^n(i,\varkappa ,\varrho ,\sigma ,b,\text{β})$ $\begin{array}{rl}& \left\{f\in T_i\left(1\right):\left|\frac{1}{b}\left[\frac{\begin{array}{c}(1-\sigma )z{D}_q\left(I_q^n\right(\varkappa ,\varrho \left)f\right(z\left)\right)+\sigma z{D}_q\\ \left(z{D}_q\right(I_q^n(\varkappa ,\varrho )f\left(z\right)\left)\right)\end{array}}{\begin{array}{c}(1-\sigma )I_q^n(\varkappa ,\varrho )f\left(z\right)\\ +\sigma z{D}_q\left(I_q^n\right(\varkappa ,\varrho \left)f\right(z\left)\right)\end{array}}\right.\right.\right.\\ & \left.\left.-1\phantom{\frac{\begin{array}{c}(1-\sigma )z{D}_q\left(I_q^n\right(\varkappa ,\varrho \left)f\right(z\left)\right)\\ +\sigma z{D}_q(z{D}_q\\ \left(I_q^n\right(\varkappa ,\varrho \left)f\right(z\left)\right))\end{array}}{\begin{array}{c}(1-\sigma )I_q^n(\varkappa ,\varrho )f\left(z\right)\\ +\sigma z{D}_q\left(I_q^n\right(\varkappa ,\varrho \left)f\right(z\left)\right)\end{array}}}\right]\right|<\text{β}(b\in {\mathbb{C}}^{\ast },i\in \mathbb{N},n\in {\mathbb{N}}_0,0<q<1,\\ & \left.\varrho ,\varkappa \ge 0,0\le \sigma \le 1,0<\text{β}\le 1)\phantom{\frac{\begin{array}{c}z\left(I_{\mathbf{p}}^n\right(\varkappa ,\varrho \left)f\right(z){)}^{\mathrm{\prime }}\\ +\sigma z^2\left(I_{\mathbf{p}}^n\right(\varkappa ,\varrho \left)f\right(z){)}^{\mathrm{\prime }\mathrm{\prime }}\end{array}}{\begin{array}{c}(1-\sigma )I_{\mathbf{p}}^n(\varkappa ,\varrho )f\left(z\right)\\ +\sigma z\left(I_{\mathbf{p}}^n\right(\varkappa ,\varrho \left)f\right(z){)}^{\mathrm{\prime }}\end{array}}}\right\};\end{array}$

4. $S_q^n(i,\mathbf{p},\varkappa ,0,\sigma ,b,\text{β})=S_q^n(i,\mathbf{p},\varkappa ,\sigma ,b,\text{β})$ $\begin{array}{rl}& \left\{f\in T_i\left(\mathbf{p}\right):\left|\frac{1}{b}\left[\frac{\begin{array}{c}(1-\sigma )z{D}_{q,\mathbf{p}}\\ \left(D_{q,\mathbf{p}}^n\right(\varkappa \left)f\right(z\left)\right)\\ +\sigma z{D}_{q,\mathbf{p}}(z{D}_{q,\mathbf{p}}\\ \left(D_{q,\mathbf{p}}^n\right(\varkappa \left)f\right(z\left)\right))\end{array}}{\begin{array}{c}(1-\sigma )D_{q,\mathbf{p}}^n\left(\varkappa \right)f\left(z\right)\\ +\sigma z{D}_{q,\mathbf{p}}\left(D_{q,\mathbf{p}}^n\right(\varkappa \left)f\right(z\left)\right)\end{array}}-{\left[\mathbf{p}\right]}_q\right]\right|<\text{β}\right.\\ & (b\in {\mathbb{C}}^{\ast },n\in {\mathbb{N}}_0,\mathbf{p},i\in \mathbb{N},0<q<1,\varkappa \ge 0,\\ & \left.0\le \sigma \le 1,0<\text{β}\le 1)\phantom{\frac{\begin{array}{c}z\left(I_{\mathbf{p}}^n\right(\varkappa ,\varrho \left)f\right(z){)}^{\mathrm{\prime }}\\ +\sigma z^2\left(I_{\mathbf{p}}^n\right(\varkappa ,\varrho \left)f\right(z){)}^{\mathrm{\prime }\mathrm{\prime }}\end{array}}{\begin{array}{c}(1-\sigma )I_{\mathbf{p}}^n(\varkappa ,\varrho )f\left(z\right)\\ +\sigma z\left(I_{\mathbf{p}}^n\right(\varkappa ,\varrho \left)f\right(z){)}^{\mathrm{\prime }}\end{array}}}\right\};\end{array}$

5. $S_q^n(i,\mathbf{p},1,0,\sigma ,b,\text{β})=S_q^n(i,\mathbf{p},\sigma ,b,\text{β})$ $\begin{array}{rl}& \left\{f\in T_i\left(\mathbf{p}\right):\left|\frac{1}{b}\left[\frac{\begin{array}{c}(1-\sigma )z{D}_{q,\mathbf{p}}\\ \left(D_{q,\mathbf{p}}^{n}f\right(z\left)\right)\\ +\sigma z{D}_{q,\mathbf{p}}(z{D}_{q,\mathbf{p}}\\ \left(D_{q,\mathbf{p}}^{n}f\right(z\left)\right))\end{array}}{\begin{array}{c}(1-\sigma )D_{q,\mathbf{p}}^{n}f\left(z\right)\\ +\sigma z{D}_{q,\mathbf{p}}\\ \left(D_{q,\mathbf{p}}^{n}f\right(z\left)\right)\end{array}}-{\left[\mathbf{p}\right]}_q\right]\right|<\text{β}\right.\\ & (b\in {\mathbb{C}}^{\ast },n\in {\mathbb{N}}_0,\mathbf{p},i\in \mathbb{N},\\ & \left.0<q<1,0\le \sigma \le 1,0<\text{β}\le 1)\phantom{\frac{\begin{array}{c}z\left(I_{\mathbf{p}}^n\right(\varkappa ,\varrho \left)f\right(z){)}^{\mathrm{\prime }}\\ +\sigma z^2\left(I_{\mathbf{p}}^n\right(\varkappa ,\varrho \left)f\right(z){)}^{\mathrm{\prime }\mathrm{\prime }}\end{array}}{\begin{array}{c}(1-\sigma )I_{\mathbf{p}}^n(\varkappa ,\varrho )f\left(z\right)\\ +\sigma z\left(I_{\mathbf{p}}^n\right(\varkappa ,\varrho \left)f\right(z){)}^{\mathrm{\prime }}\end{array}}}\right\}.\end{array}$

Definition 2

Let $f\in T_i\left(\mathbf{p}\right).$ Then $f\in K_q^n(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β})$ if it satisfies the next inequality (10) $\begin{array}{rl}& \left|\frac{1}{b}\left[(1-\sigma )\frac{I_{q,\mathbf{p}}^n(\varkappa ,\varrho )f\left(z\right)}{z^{\mathbf{p}}}\right.\right.\\ & \left.+\left.\sigma \frac{D_{q,\mathbf{p}}\left(I_{q,\mathbf{p}}^n\right(\varkappa ,\varrho \left)f\right(z\left)\right)}{{\left[\mathbf{p}\right]}_q{z}^{\mathbf{p}-1}}-1\right]\right|<\text{β}\\ & (b\in {\mathbb{C}}^{\ast },n\in {\mathbb{N}}_0,\mathbf{p},i\in \mathbb{N},0<q<1,\varrho ,\varkappa \ge 0,\\ & 0\le \sigma \le 1,0<\text{β}\le 1).\end{array}$(10)

We note that:

1. $\underset{q\to 1^{-}}{lim}{K}_q^n(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β})=K^n(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β})$ $\begin{array}{rl}& \left\{f\in T_i\left(\mathbf{p}\right):\left|\frac{1}{b}\left[(1-\sigma )\frac{I_{\mathbf{p}}^n(\varkappa ,\varrho )f\left(z\right)}{z^{\mathbf{p}}}\right.\right.\right.\\ & \left.+\left.\sigma \frac{\left(I_{\mathbf{p}}^n\right(\varkappa ,\varrho \left)f\right(z){)}^{\mathrm{\prime }}}{\mathbf{p}{z}^{\mathbf{p}-1}}-1\right]\right|<\text{β}\\ & (b\in {\mathbb{C}}^{\ast },n\in {\mathbb{N}}_0,\mathbf{p},i\in \mathbb{N},\varrho ,\varkappa \ge 0,0\le \sigma \le 1,\\ & \left.0<\text{β}\le 1)\phantom{\frac{I_{\mathbf{p}}^n(\varkappa ,\varrho )f\left(z\right)}{z^{\mathbf{p}}}}\right\};\end{array}$

2. $K_q^n(i,\mathbf{p},1,\varrho ,\sigma ,b,\text{β})=K_q^n(i,\mathbf{p},\varrho ,\sigma ,b,\text{β})$ $\begin{array}{rl}& \left\{f\in T_i\left(\mathbf{p}\right):\left|\frac{1}{b}\left[(1-\sigma )\frac{I_{q,\mathbf{p}}^n\left(\varrho \right)f\left(z\right)}{z^{\mathbf{p}}}\right.\right.\right.\\ & \left.+\left.\sigma \frac{D_{q,\mathbf{p}}\left(I_{q,\mathbf{p}}^n\right(\varrho \left)f\right(z\left)\right)}{{\left[\mathbf{p}\right]}_q{z}^{\mathbf{p}-1}}-1\right]\right|<\text{β}\\ & (b\in {\mathbb{C}}^{\ast },n\in {\mathbb{N}}_0,\mathbf{p},i\in \mathbb{N},0<q<1,\varrho \ge 0,\\ & \left.0\le \sigma \le 1,0<\text{β}\le 1)\phantom{\frac{I_{\mathbf{p}}^n(\varkappa ,\varrho )f\left(z\right)}{z^{\mathbf{p}}}}\right\};\end{array}$

3. $K_q^n(i,\mathbf{p},1,0,\sigma ,b,\text{β})=K_q^n(i,\mathbf{p},\sigma ,b,\text{β})$ $\begin{array}{rl}& \left\{f\in T_i\left(\mathbf{p}\right):\left|\frac{1}{b}\left[(1-\sigma )\frac{D_{q,\mathbf{p}}^{n}f\left(z\right)}{z^{\mathbf{p}}}\right.\right.\right.\\ & \left.+\left.\sigma \frac{D_{q,\mathbf{p}}\left(D_{q,\mathbf{p}}^{n}f\right(z\left)\right)}{{\left[\mathbf{p}\right]}_q{z}^{\mathbf{p}-1}}-1\right]\right|<\text{β}\\ & (b\in {\mathbb{C}}^{\ast },n\in {\mathbb{N}}_0,\mathbf{p},i\in \mathbb{N},0<q<1,0\le \sigma \le 1,\\ & \left.0<\text{β}\le 1)\phantom{\frac{I_{\mathbf{p}}^n(\varkappa ,\varrho )f\left(z\right)}{z^{\mathbf{p}}}}\right\};\end{array}$

4. $K_q^n(i,\mathbf{p},\varkappa ,0,\sigma ,b,\text{β})=K_q^n(i,\mathbf{p},\varkappa ,\sigma ,b,\text{β})$ $\begin{array}{rl}& \left\{f\in T_i\left(\mathbf{p}\right):\left|\frac{1}{b}\left[(1-\sigma )\frac{D_{q,\mathbf{p}}^n\left(\varkappa \right)f\left(z\right)}{z^{\mathbf{p}}}\right.\right.\right.\\ & \left.+\left.\sigma \frac{D_{q,\mathbf{p}}\left(D_{q,\mathbf{p}}^n\right(\varkappa \left)f\right(z\left)\right)}{{\left[\mathbf{p}\right]}_q{z}^{\mathbf{p}-1}}-1\right]\right|<\text{β}\\ & (b\in {\mathbb{C}}^{\ast },n\in {\mathbb{N}}_0,\mathbf{p},i\in \mathbb{N},\varkappa \ge 0,\\ & \left.0<q<1,0\le \sigma \le 1,0<\text{β}\le 1)\phantom{\frac{D_{q,\mathbf{p}}\left(D_{q,\mathbf{p}}^n\right(\varkappa \left)f\right(z\left)\right)}{{\left[\mathbf{p}\right]}_q{z}^{\mathbf{p}-1}}}\right\}.\end{array}$

Following the investigations by Ruscheweyh [25], Goodman [26], Altintaş et al. [27–29], Altintaş and Owa [30] and see also ([31–35]), we define the neighbourhood $(i,\delta )$ for$f\in T_i\left(\mathbf{p}\right)$ by (11) $\begin{array}{rl}{N}_{i,\delta }^{\mathbf{p}}\left(f\right)& =\left\{g:g\in T_i\left(\mathbf{p}\right),g\left(z\right)=z^{\mathbf{p}}-\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}{b}_k{z}^k\right.\\ & \left.\mathrm{a}\mathrm{n}\mathrm{d}\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}k\left|a_k-b_k\right|\le \delta \right\}.\end{array}$(11) In particular, if (12) $S\left(z\right)=z^{\mathbf{p}}(\mathbf{p}\in \mathbb{N}),$(12) we obtain (13) $\begin{array}{rl}{N}_{i,\delta }^{\mathbf{p}}\left(S\right)& =\left\{g:g\in T_i\left(\mathbf{p}\right),g\left(z\right)=z^{\mathbf{p}}-\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}{b}_k{z}^k\mathrm{a}\mathrm{n}\mathrm{d}\right.\\ & \left.\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}k\left|b_k\right|\le \delta \right\}.\end{array}$(13) Now we define the $(q,i,\delta )$ neighbourhood for $f\in T_i\left(\mathbf{p}\right)$ (see [36]) by (14) $\begin{array}{rl}{N}_{i,\delta }^{\mathbf{p},q}\left(f\right)& =\left\{g:g\in T_i\left(\mathbf{p}\right),g\left(z\right)=z^{\mathbf{p}}-\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}{b}_k{z}^k\right.\\ & \left.\mathrm{a}\mathrm{n}\mathrm{d}\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}{\left[k\right]}_q\left|a_k-b_k\right|\le \delta \right\}.\end{array}$(14) In particular, if $S\left(z\right)$ given by (12(12) $S\left(z\right)=z^{\mathbf{p}}(\mathbf{p}\in \mathbb{N}),$(12) ), we immediately have (15) $\begin{array}{rl}{N}_{i,\delta }^{\mathbf{p},q}\left(S\right)& =\left\{g:g\in T_i\left(\mathbf{p}\right),g\left(z\right)=z^{\mathbf{p}}-\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}{b}_k{z}^k\right.\\ & \left.\mathrm{a}\mathrm{n}\mathrm{d}\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}{\left[k\right]}_q\left|b_k\right|\le \delta \right\}.\end{array}$(15) We note that $\underset{q\to 1^{-}}{lim}{N}_{i,\delta }^{\mathbf{p},q}\left(f\right)=N_{i,\delta }^{\mathbf{p}}\left(f\right)$ and$\underset{q\to 1^{-}}{lim}{N}_{i,\delta }^{\mathbf{p},q}\left(S\right)=N_{i,\delta }^{\mathbf{p}}\left(S\right)$ (see [36]).

2. Coefficient estimates

In this research, we shall assume that $b\in {\mathbb{C}}^{\ast },n\in {\mathbb{N}}_0,\mathbf{p},i\in \mathbb{N},\varrho ,\varkappa \ge 0,0<q<1,0\le \sigma \le 1,0<\text{β}\le 1$ and ${\mathrm{\Psi }}_{q,\mathbf{p}}^n(k,\varkappa ,\varrho )$ is given by (8(8) $\begin{array}{rl}& {\mathrm{\Psi }}_{q,\mathbf{p}}^n(k,\varkappa ,\varrho )={\left[\frac{{\left[\mathbf{p}+\varrho \right]}_q+\varkappa \left({\left[k+\varrho \right]}_q-{\left[\mathbf{p}+\varrho \right]}_q\right)}{{\left[\mathbf{p}+\varrho \right]}_q}\right]}^n\\ & (n\in {\mathbb{N}}_0,i,\mathbf{p}\in \mathbb{N},\varrho ,\varkappa \ge 0).\end{array}$(8) ).

In our present investigation of the inclusion relations, we shall require Lemmas 1 and 2.

Lemma 1

If $f\in T_i\left(\mathbf{p}\right)$ is given by (2(2) $f\left(z\right)=z^{\mathbf{p}}-\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}{a}_k{z}^k(a_k\ge 0,i,\mathbf{p}\in \mathbb{N}).$(2) ), then $f\in S_q^n(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β}),$

if and only if (16) $\begin{array}{rl}& \sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}({\left[k\right]}_q+\text{β}\left|b\right|-{\left[\mathbf{p}\right]}_q\left)\right[1+\sigma ({\left[k\right]}_q{-1\left)\right]\mathrm{\Psi }}_{q,\mathbf{p}}^n\\ & \times {(k,\varkappa ,\varrho )a}_k\le \text{β}\left|b\right|[1+\sigma ({\left[\mathbf{p}\right]}_q-1\left)\right].\end{array}$(16)

Proof.

Let $f\in S_q^n(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β}).$ Then we have (17) $\begin{array}{rl}& \mathrm{\Re }\left\{\frac{\begin{array}{c}(1-\sigma )z{D}_{q,\mathbf{p}}\left(I_{q,\mathbf{p}}^n\right(\varkappa ,\varrho \left)f\right(z\left)\right)\\ +\sigma z{D}_{q,\mathbf{p}}\left(z{D}_{q,\mathbf{p}}\right(I_{q,\mathbf{p}}^n(\varkappa ,\varrho )f\left(z\right)\left)\right)\end{array}}{(1-\sigma )I_{q,\mathbf{p}}^n(\varkappa ,\varrho )f\left(z\right)+\sigma z{D}_{q,\mathbf{p}}\left(I_{q,\mathbf{p}}^n\right(\varkappa ,\varrho \left)f\right(z\left)\right)}\right.\\ & -\left.{\left[\mathbf{p}\right]}_q\phantom{\frac{\begin{array}{c}(1-\sigma )z{D}_{q,\mathbf{p}}\left(I_{q,\mathbf{p}}^n\right(\varkappa ,\varrho \left)f\right(z\left)\right)\\ +\sigma z{D}_{q,\mathbf{p}}\left(z{D}_{q,\mathbf{p}}\right(I_{q,\mathbf{p}}^n(\varkappa ,\varrho )f\left(z\right)\left)\right)\end{array}}{(1-\sigma )I_{q,\mathbf{p}}^n(\varkappa ,\varrho )f\left(z\right)+\sigma z{D}_{q,\mathbf{p}}\left(I_{q,\mathbf{p}}^n\right(\varkappa ,\varrho \left)f\right(z\left)\right)}}\right\}>-\text{β}\left|b\right|(z\in \mathbb{U}),\end{array}$(17) or, equivalently, (18) $\mathrm{\Re }\left\{\frac{\begin{array}{c}-\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}({\left[k\right]}_q-{\left[\mathbf{p}\right]}_q)\\ [1+\sigma ({\left[k\right]}_q-1\left)\right]\\ {\mathrm{\Psi }}_{q,\mathbf{p}}^n(k,\varkappa ,\varrho )a_k{z}^{k-\mathbf{p}}\end{array}}{\begin{array}{c}[1+\sigma ({\left[\mathbf{p}\right]}_q-1\left)\right]-\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}\\ [1+\sigma ({\left[k\right]}_q-1\left)\right]\\ {\mathrm{\Psi }}_{q,\mathbf{p}}^n(k,\varkappa ,\varrho )a_k{z}^{k-\mathbf{p}}\end{array}}\right\}>-\text{β}\left|b\right|.$(18)

Setting $z=r(0\le r<1)$ in (18(18) $\mathrm{\Re }\left\{\frac{\begin{array}{c}-\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}({\left[k\right]}_q-{\left[\mathbf{p}\right]}_q)\\ [1+\sigma ({\left[k\right]}_q-1\left)\right]\\ {\mathrm{\Psi }}_{q,\mathbf{p}}^n(k,\varkappa ,\varrho )a_k{z}^{k-\mathbf{p}}\end{array}}{\begin{array}{c}[1+\sigma ({\left[\mathbf{p}\right]}_q-1\left)\right]-\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}\\ [1+\sigma ({\left[k\right]}_q-1\left)\right]\\ {\mathrm{\Psi }}_{q,\mathbf{p}}^n(k,\varkappa ,\varrho )a_k{z}^{k-\mathbf{p}}\end{array}}\right\}>-\text{β}\left|b\right|.$(18) ), so in the left side the denominator will be non negative for r = 0 and also for $(0\le r<1)$. Now, by Assuming $r⟶1^{-}$ through real values, (18(18) $\mathrm{\Re }\left\{\frac{\begin{array}{c}-\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}({\left[k\right]}_q-{\left[\mathbf{p}\right]}_q)\\ [1+\sigma ({\left[k\right]}_q-1\left)\right]\\ {\mathrm{\Psi }}_{q,\mathbf{p}}^n(k,\varkappa ,\varrho )a_k{z}^{k-\mathbf{p}}\end{array}}{\begin{array}{c}[1+\sigma ({\left[\mathbf{p}\right]}_q-1\left)\right]-\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}\\ [1+\sigma ({\left[k\right]}_q-1\left)\right]\\ {\mathrm{\Psi }}_{q,\mathbf{p}}^n(k,\varkappa ,\varrho )a_k{z}^{k-\mathbf{p}}\end{array}}\right\}>-\text{β}\left|b\right|.$(18) ) leads up to the proof of Lemma 1.

Conversely, by applying (16(16) $\begin{array}{rl}& \sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}({\left[k\right]}_q+\text{β}\left|b\right|-{\left[\mathbf{p}\right]}_q\left)\right[1+\sigma ({\left[k\right]}_q{-1\left)\right]\mathrm{\Psi }}_{q,\mathbf{p}}^n\\ & \times {(k,\varkappa ,\varrho )a}_k\le \text{β}\left|b\right|[1+\sigma ({\left[\mathbf{p}\right]}_q-1\left)\right].\end{array}$(16) ) and letting $\left|z\right|=1$, we observe from (18(18) $\mathrm{\Re }\left\{\frac{\begin{array}{c}-\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}({\left[k\right]}_q-{\left[\mathbf{p}\right]}_q)\\ [1+\sigma ({\left[k\right]}_q-1\left)\right]\\ {\mathrm{\Psi }}_{q,\mathbf{p}}^n(k,\varkappa ,\varrho )a_k{z}^{k-\mathbf{p}}\end{array}}{\begin{array}{c}[1+\sigma ({\left[\mathbf{p}\right]}_q-1\left)\right]-\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}\\ [1+\sigma ({\left[k\right]}_q-1\left)\right]\\ {\mathrm{\Psi }}_{q,\mathbf{p}}^n(k,\varkappa ,\varrho )a_k{z}^{k-\mathbf{p}}\end{array}}\right\}>-\text{β}\left|b\right|.$(18) ) that $\begin{array}{rl}& \left|\frac{\begin{array}{c}(1-\sigma )z{D}_{q,\mathbf{p}}\left(I_{q,\mathbf{p}}^n\right(\varkappa ,\varrho \left)f\right(z\left)\right)\\ +\sigma z{D}_{q,\mathbf{p}}\left(z{D}_{q,\mathbf{p}}\right(I_{q,\mathbf{p}}^n(\varkappa ,\varrho )f\left(z\right)\left)\right)\end{array}}{(1-\sigma )I_{q,\mathbf{p}}^n(\varkappa ,\varrho )f\left(z\right)+\sigma z{D}_{q,\mathbf{p}}\left(I_{q,\mathbf{p}}^n\right(\varkappa ,\varrho \left)f\right(z\left)\right)}-{\left[\mathbf{p}\right]}_q\right|\\ & =\left|\frac{\begin{array}{c}\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}({\left[k\right]}_q-{\left[\mathbf{p}\right]}_q)[1+\sigma ({\left[k\right]}_q-1\left)\right]\\ {\mathrm{\Psi }}_{q,\mathbf{p}}^n(k,\varkappa ,\varrho )a_k{z}^{k-\mathbf{p}}\end{array}}{\begin{array}{c}[1+\sigma ({\left[\mathbf{p}\right]}_q-1\left)\right]-\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}[1+\sigma ({\left[k\right]}_q-1\left)\right]\\ {\mathrm{\Psi }}_{q,\mathbf{p}}^n(k,\varkappa ,\varrho )a_k{z}^{k-\mathbf{p}}\end{array}}\right|\\ & \le \frac{\begin{array}{c}\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}({\left[k\right]}_q-{\left[\mathbf{p}\right]}_q)[1+\sigma ({\left[k\right]}_q-1\left)\right]\\ {\mathrm{\Psi }}_{q,\mathbf{p}}^n(k,\varkappa ,\varrho )a_k{\left|z\right|}^{k-\mathbf{p}}\end{array}}{\begin{array}{c}[1+\sigma ({\left[\mathbf{p}\right]}_q-1\left)\right]-\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}[1+\sigma ({\left[k\right]}_q-1\left)\right]\\ {\mathrm{\Psi }}_{q,\mathbf{p}}^n(k,\varkappa ,\varrho )a_k{\left|z\right|}^{k-\mathbf{p}}\end{array}}\\ & \le \frac{\begin{array}{c}\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}({\left[k\right]}_q-{\left[\mathbf{p}\right]}_q)[1+\sigma ({\left[k\right]}_q-1\left)\right]\\ {\mathrm{\Psi }}_{q,\mathbf{p}}^n(k,\varkappa ,\varrho )a_k\end{array}}{\begin{array}{c}[1+\sigma ({\left[\mathbf{p}\right]}_q-1\left)\right]\\ -\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}[1+\sigma ({\left[k\right]}_q-1\left)\right]\\ {\mathrm{\Psi }}_{q,\mathbf{p}}^n(k,\varkappa ,\varrho )a_k\end{array}}=\text{β}\left|b\right|.\end{array}$ Thus we have $f\in S_q^n(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β}),$ by applying the maximum modulus theorem, which evidently completes the proof.

We can prove the lemma below like the proof of Lemma 1.

Lemma 2

Let $f\in T_i\left(\mathbf{p}\right)$ is given by (2(2) $f\left(z\right)=z^{\mathbf{p}}-\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}{a}_k{z}^k(a_k\ge 0,i,\mathbf{p}\in \mathbb{N}).$(2) ). Then $f\in K_q^n(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β}),$

if and only if (19) $\begin{array}{rl}& \sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}[{\left[\mathbf{p}\right]}_q+\sigma ({\left[k\right]}_q-{\left[\mathbf{p}\right]}_q{\left)\right]\mathrm{\Psi }}_{q,\mathbf{p}}^n{(k,\varkappa ,\varrho )a}_k\\ & \le \text{β}\left|b\right|{\left[\mathbf{p}\right]}_q.\end{array}$(19)

3. Neighbourhood properties for two new classes

$S_q^n(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β})$ and $K_q^n(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β})$

In this part, we determine inclusion relations for each of the classes $S_q^n(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β})$ and $K_q^n(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β})$ involving $(q,i,\delta )$ neighbourhood defined by (14(14) $\begin{array}{rl}{N}_{i,\delta }^{\mathbf{p},q}\left(f\right)& =\left\{g:g\in T_i\left(\mathbf{p}\right),g\left(z\right)=z^{\mathbf{p}}-\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}{b}_k{z}^k\right.\\ & \left.\mathrm{a}\mathrm{n}\mathrm{d}\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}{\left[k\right]}_q\left|a_k-b_k\right|\le \delta \right\}.\end{array}$(14) ) and (15(15) $\begin{array}{rl}{N}_{i,\delta }^{\mathbf{p},q}\left(S\right)& =\left\{g:g\in T_i\left(\mathbf{p}\right),g\left(z\right)=z^{\mathbf{p}}-\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}{b}_k{z}^k\right.\\ & \left.\mathrm{a}\mathrm{n}\mathrm{d}\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}{\left[k\right]}_q\left|b_k\right|\le \delta \right\}.\end{array}$(15) ).

Theorem 1

Let $f\in T_i\left(\mathbf{p}\right)$ be in the class $S_q^n(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β}),$ then (20) $S_q^n{(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β})\subset N}_{i,\theta }^{\mathbf{p},q}\left(S\right),$(20) since $S\left(z\right)$ is defined by (12(12) $S\left(z\right)=z^{\mathbf{p}}(\mathbf{p}\in \mathbb{N}),$(12) ) and θ is given by (21) $\theta =\frac{{\left[i+\mathbf{p}\right]}_q\text{β}\left|b\right|[1+\sigma ({\left[\mathbf{p}\right]}_q-1\left)\right]}{\begin{array}{c}({\left[i+\mathbf{p}\right]}_q+\text{β}\left|b\right|-{\left[\mathbf{p}\right]}_q)\\ [1+\sigma ({\left[i+\mathbf{p}\right]}_q-1\left)\right]\\ {\mathrm{\Psi }}_{q,\mathbf{p}}^n(i+\mathbf{p},\varkappa ,\varrho )\end{array}}({\left[\mathbf{p}\right]}_q>\left|b\right|).$(21)

Proof.

If $f\in S_q^n(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β})$, then by using (16(16) $\begin{array}{rl}& \sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}({\left[k\right]}_q+\text{β}\left|b\right|-{\left[\mathbf{p}\right]}_q\left)\right[1+\sigma ({\left[k\right]}_q{-1\left)\right]\mathrm{\Psi }}_{q,\mathbf{p}}^n\\ & \times {(k,\varkappa ,\varrho )a}_k\le \text{β}\left|b\right|[1+\sigma ({\left[\mathbf{p}\right]}_q-1\left)\right].\end{array}$(16) ) we obtain (22) $\begin{array}{rl}& ({\left[i+\mathbf{p}\right]}_q+\text{β}\left|b\right|-{\left[\mathbf{p}\right]}_q)[1+\sigma ({\left[i+\mathbf{p}\right]}_q-1\left)\right]\\ & \times {\mathrm{\Psi }}_{q,\mathbf{p}}^n(i+\mathbf{p},\varkappa ,\varrho )\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}{a}_k\\ & \le \sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}({\left[k\right]}_q+\text{β}\left|b\right|-{\left[\mathbf{p}\right]}_q)\\ & \times [1+\sigma ({\left[k\right]}_q-1\left)\right]{\mathrm{\Psi }}_{q,\mathbf{p}}^n(k,\varkappa ,\varrho )a_k\\ & \le \text{β}\left|b\right|[1+\sigma ({\left[\mathbf{p}\right]}_q-1\left)\right],\end{array}$(22) which readily yields (23) $\begin{array}{r}\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}{a}_k\le \frac{\text{β}\left|b\right|[1+\sigma ({\left[\mathbf{p}\right]}_q-1\left)\right]}{\begin{array}{c}({\left[i+\mathbf{p}\right]}_q+\text{β}\left|b\right|-{\left[\mathbf{p}\right]}_q)\\ [1+\sigma ({\left[i+\mathbf{p}\right]}_q-1\left)\right]{\mathrm{\Psi }}_{q,\mathbf{p}}^n(i+\mathbf{p},\varkappa ,\varrho )\end{array}}.\end{array}$(23) Making use of (16(16) $\begin{array}{rl}& \sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}({\left[k\right]}_q+\text{β}\left|b\right|-{\left[\mathbf{p}\right]}_q\left)\right[1+\sigma ({\left[k\right]}_q{-1\left)\right]\mathrm{\Psi }}_{q,\mathbf{p}}^n\\ & \times {(k,\varkappa ,\varrho )a}_k\le \text{β}\left|b\right|[1+\sigma ({\left[\mathbf{p}\right]}_q-1\left)\right].\end{array}$(16) ) with (23(23) $\begin{array}{r}\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}{a}_k\le \frac{\text{β}\left|b\right|[1+\sigma ({\left[\mathbf{p}\right]}_q-1\left)\right]}{\begin{array}{c}({\left[i+\mathbf{p}\right]}_q+\text{β}\left|b\right|-{\left[\mathbf{p}\right]}_q)\\ [1+\sigma ({\left[i+\mathbf{p}\right]}_q-1\left)\right]{\mathrm{\Psi }}_{q,\mathbf{p}}^n(i+\mathbf{p},\varkappa ,\varrho )\end{array}}.\end{array}$(23) ), we obtain $\begin{array}{rl}& [1+\sigma ({\left[i+\mathbf{p}\right]}_q-1\left)\right]{\mathrm{\Psi }}_{q,\mathbf{p}}^n(i+\mathbf{p},\varkappa ,\varrho )\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}{\left[k\right]}_q{a}_k\\ & \le \text{β}\left|b\right|[1+\sigma ({\left[\mathbf{p}\right]}_q-1\left)\right]+({\left[\mathbf{p}\right]}_q-\text{β}\left|b\right|)\\ & \times [1+\sigma ({\left[i+\mathbf{p}\right]}_q-1\left)\right]{\mathrm{\Psi }}_{q,\mathbf{p}}^n(i+\mathbf{p},\varkappa ,\varrho )\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}{a}_k\\ & \le \text{β}\left|b\right|[1+\sigma ({\left[\mathbf{p}\right]}_q-1\left)\right]\\ & +\frac{({\left[\mathbf{p}\right]}_q-\text{β}\left|b\right|)\text{β}\left|b\right|[1+\sigma ({\left[\mathbf{p}\right]}_q-1\left)\right]}{({\left[i+\mathbf{p}\right]}_q+\text{β}\left|b\right|-{\left[\mathbf{p}\right]}_q)}\\ & =\frac{{\left[i+\mathbf{p}\right]}_q\text{β}\left|b\right|[1+\sigma ({\left[\mathbf{p}\right]}_q-1\left)\right]}{{\left[i+\mathbf{p}\right]}_q+\text{β}\left|b\right|-{\left[\mathbf{p}\right]}_q}.\end{array}$ Hence (24) $\begin{array}{rl}\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}{\left[k\right]}_q{a}_k& \le \frac{{\left[i+\mathbf{p}\right]}_q\text{β}\left|b\right|[1+\sigma ({\left[\mathbf{p}\right]}_q-1\left)\right]}{\begin{array}{c}({\left[i+\mathbf{p}\right]}_q+\text{β}\left|b\right|-{\left[\mathbf{p}\right]}_q)\\ [1+\sigma ({\left[i+\mathbf{p}\right]}_q-1\left)\right]\\ {\mathrm{\Psi }}_{q,\mathbf{p}}^n(i+\mathbf{p},\varkappa ,\varrho )\end{array}}\\ & =\theta ({\left[\mathbf{p}\right]}_q>\left|b\right|),\end{array}$(24) which, by means of the definition (14(14) $\begin{array}{rl}{N}_{i,\delta }^{\mathbf{p},q}\left(f\right)& =\left\{g:g\in T_i\left(\mathbf{p}\right),g\left(z\right)=z^{\mathbf{p}}-\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}{b}_k{z}^k\right.\\ & \left.\mathrm{a}\mathrm{n}\mathrm{d}\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}{\left[k\right]}_q\left|a_k-b_k\right|\le \delta \right\}.\end{array}$(14) ), established the inclusion (20(20) $S_q^n{(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β})\subset N}_{i,\theta }^{\mathbf{p},q}\left(S\right),$(20) ) asserted by Theorem 1.

In analogous manner, by applying (19(19) $\begin{array}{rl}& \sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}[{\left[\mathbf{p}\right]}_q+\sigma ({\left[k\right]}_q-{\left[\mathbf{p}\right]}_q{\left)\right]\mathrm{\Psi }}_{q,\mathbf{p}}^n{(k,\varkappa ,\varrho )a}_k\\ & \le \text{β}\left|b\right|{\left[\mathbf{p}\right]}_q.\end{array}$(19) ) of Lemma 2 instead of (16(16) $\begin{array}{rl}& \sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}({\left[k\right]}_q+\text{β}\left|b\right|-{\left[\mathbf{p}\right]}_q\left)\right[1+\sigma ({\left[k\right]}_q{-1\left)\right]\mathrm{\Psi }}_{q,\mathbf{p}}^n\\ & \times {(k,\varkappa ,\varrho )a}_k\le \text{β}\left|b\right|[1+\sigma ({\left[\mathbf{p}\right]}_q-1\left)\right].\end{array}$(16) ) of Lemma 1 to functions in the class $K_q^n(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β}),$ we can prove the next inclusion relationship.

Theorem 2

Let $f\in T_i\left(\mathbf{p}\right)$ belongs to $K_q^n(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β}),$ then (25) $K_q^n{(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β})\subset N}_{i,\delta }^{\mathbf{p},q}\left(S\right),$(25) where $S\left(z\right)$ is defined by (12(12) $S\left(z\right)=z^{\mathbf{p}}(\mathbf{p}\in \mathbb{N}),$(12) ) and δ is introduced by (26) $\delta =\frac{{\left[i+\mathbf{p}\right]}_q\text{β}\left|b\right|{\left[\mathbf{p}\right]}_q}{[{\left[\mathbf{p}\right]}_q+\sigma ({\left[i+\mathbf{p}\right]}_q-{\left[\mathbf{p}\right]}_q\left)\right]{\mathrm{\Psi }}_{q,\mathbf{p}}^n(i+\mathbf{p},\varkappa ,\varrho )}.$(26)

4. Neighbourhood properties for two new classes $S_q^{n\left(\gamma \right)}(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β})$ and $K_q^{n\left(\gamma \right)}(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β})$

In this part, we determine the neighbourhood for each of the classes $S_q^{n\left(\gamma \right)}(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β})$ and $K_q^{n\left(\gamma \right)}(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β})$, which we define as follows. A function $f\in T_i\left(\mathbf{p}\right)$ belongs to the class $S_q^{n\left(\gamma \right)}(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β})$ if there exists a function $\rho \left(z\right)\in S_q^{n\left(\gamma \right)}(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β})$ such that (27) $\left|\frac{f\left(z\right)}{\rho \left(z\right)}-1\right|<{\left[\mathbf{p}\right]}_q-\gamma (z\in \mathbb{U};0\le \gamma <{\left[\mathbf{p}\right]}_q).$(27) Parallelly, $f\left(z\right)\in T_i\left(\mathbf{p}\right)$ belongs to the class $K_q^{n\left(\gamma \right)}(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β}),$ if there exists a function $\rho \left(z\right)\in K_q^{n\left(\gamma \right)}(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β})$ such that (27(27) $\left|\frac{f\left(z\right)}{\rho \left(z\right)}-1\right|<{\left[\mathbf{p}\right]}_q-\gamma (z\in \mathbb{U};0\le \gamma <{\left[\mathbf{p}\right]}_q).$(27) ) holds true.

Theorem 3

Let $f\in T_i\left(\mathbf{p}\right)$ belongs to $S_q^{n\left(\gamma \right)}(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β})$ and (28) $\gamma ={\left[\mathbf{p}\right]}_q-\frac{\begin{array}{c}\theta ({\left[i+\mathbf{p}\right]}_q+\text{β}\left|b\right|-{\left[\mathbf{p}\right]}_q)\\ [1+\sigma ({\left[i+\mathbf{p}\right]}_q-1\left)\right]\\ {\mathrm{\Psi }}_{q,\mathbf{p}}^n(i+\mathbf{p},\varkappa ,\varrho )\end{array}}{\begin{array}{c}{\left[i+\mathbf{p}\right]}_q\left\{({\left[i+\mathbf{p}\right]}_q+\text{β}\left|b\right|-{\left[\mathbf{p}\right]}_q)\right.\\ [1+\sigma ({\left[i+\mathbf{p}\right]}_q-1\left)\right]\\ {\mathrm{\Psi }}_{q,\mathbf{p}}^n(i+\mathbf{p},\varkappa ,\varrho )\\ \left.-\text{β}\left|b\right|[1+\sigma ({\left[\mathbf{p}\right]}_q-1\left)\right]\right\}\end{array}},$(28) then (29) $N_{i,\theta }^{\mathbf{p},q}{\left(S\right)\subset S}_q^{n\left(\gamma \right)}(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β}),$(29) where $\theta \le [\mathbf{p}{]}_q{\left[i+\mathbf{p}\right]}_q\left\{1-\frac{\text{β}\left|b\right|[1+\sigma ({\left[\mathbf{p}\right]}_q-1\left)\right]}{\begin{array}{c}({\left[i+\mathbf{p}\right]}_q+\text{β}\left|b\right|-{\left[\mathbf{p}\right]}_q)\\ [1+\sigma ({\left[i+\mathbf{p}\right]}_q-1\left)\right]\\ {\mathrm{\Psi }}_{q,\mathbf{p}}^n(i+\mathbf{p},\varkappa ,\varrho )\end{array}}\right\}.$

Proof.

Assume $f\in N_{i,\theta }^{\mathbf{p},q}\left(S\right).$ From (14(14) $\begin{array}{rl}{N}_{i,\delta }^{\mathbf{p},q}\left(f\right)& =\left\{g:g\in T_i\left(\mathbf{p}\right),g\left(z\right)=z^{\mathbf{p}}-\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}{b}_k{z}^k\right.\\ & \left.\mathrm{a}\mathrm{n}\mathrm{d}\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}{\left[k\right]}_q\left|a_k-b_k\right|\le \delta \right\}.\end{array}$(14) ) we find that (30) $\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}{\left[k\right]}_q\left|a_k-b_k\right|\le \theta ,$(30) which readily implies that (31) $\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}\left|a_k-b_k\right|\le \frac{\theta }{{\left[i+\mathbf{p}\right]}_q}(\mathbf{p},i\in \mathbb{N}).$(31) Next, since $\rho \left(z\right)\in S_q^{n\left(\gamma \right)}(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β}),$ by using (23(23) $\begin{array}{r}\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}{a}_k\le \frac{\text{β}\left|b\right|[1+\sigma ({\left[\mathbf{p}\right]}_q-1\left)\right]}{\begin{array}{c}({\left[i+\mathbf{p}\right]}_q+\text{β}\left|b\right|-{\left[\mathbf{p}\right]}_q)\\ [1+\sigma ({\left[i+\mathbf{p}\right]}_q-1\left)\right]{\mathrm{\Psi }}_{q,\mathbf{p}}^n(i+\mathbf{p},\varkappa ,\varrho )\end{array}}.\end{array}$(23) ), we have (32) $\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}{b}_k\le \frac{\text{β}\left|b\right|[1+\sigma ({\left[\mathbf{p}\right]}_q-1\left)\right]}{\begin{array}{c}({\left[i+\mathbf{p}\right]}_q+\text{β}\left|b\right|-{\left[\mathbf{p}\right]}_q)\\ [1+\sigma ({\left[i+\mathbf{p}\right]}_q-1\left)\right]{\mathrm{\Psi }}_{q,\mathbf{p}}^n(i+\mathbf{p},\varkappa ,\varrho )\end{array}},$(32) so $\begin{array}{rl}& \left|\frac{f\left(z\right)}{\rho \left(z\right)}-1\right|\le \frac{\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}\left|a_k-b_k\right|}{1-\sum _{k=i+\mathbf{p}}^{\mathrm{\infty }}{b}_k}\\ & \le \frac{\begin{array}{c}\theta ({\left[i+\mathbf{p}\right]}_q+\text{β}\left|b\right|-{\left[\mathbf{p}\right]}_q)\\ [1+\sigma ({\left[i+\mathbf{p}\right]}_q-1\left)\right]\\ {\mathrm{\Psi }}_{q,\mathbf{p}}^n(i+\mathbf{p},\varkappa ,\varrho )\end{array}}{\begin{array}{c}{\left[i+\mathbf{p}\right]}_q\left\{({\left[i+\mathbf{p}\right]}_q+\text{β}\left|b\right|-{\left[\mathbf{p}\right]}_q)\right.\\ [1+\sigma ({\left[i+\mathbf{p}\right]}_q-1\left)\right]{\mathrm{\Psi }}_{q,\mathbf{p}}^n(i+\mathbf{p},\varkappa ,\varrho )\\ \left.-\text{β}\left|b\right|[1+\sigma ({\left[\mathbf{p}\right]}_q-1\left)\right]\right\}\end{array}}\\ & ={\left[\mathbf{p}\right]}_q-\gamma ,\end{array}$ provided that γ is given by (28(28) $\gamma ={\left[\mathbf{p}\right]}_q-\frac{\begin{array}{c}\theta ({\left[i+\mathbf{p}\right]}_q+\text{β}\left|b\right|-{\left[\mathbf{p}\right]}_q)\\ [1+\sigma ({\left[i+\mathbf{p}\right]}_q-1\left)\right]\\ {\mathrm{\Psi }}_{q,\mathbf{p}}^n(i+\mathbf{p},\varkappa ,\varrho )\end{array}}{\begin{array}{c}{\left[i+\mathbf{p}\right]}_q\left\{({\left[i+\mathbf{p}\right]}_q+\text{β}\left|b\right|-{\left[\mathbf{p}\right]}_q)\right.\\ [1+\sigma ({\left[i+\mathbf{p}\right]}_q-1\left)\right]\\ {\mathrm{\Psi }}_{q,\mathbf{p}}^n(i+\mathbf{p},\varkappa ,\varrho )\\ \left.-\text{β}\left|b\right|[1+\sigma ({\left[\mathbf{p}\right]}_q-1\left)\right]\right\}\end{array}},$(28) ). Thus, with the above definition, $f\in S_q^{n\left(\gamma \right)}(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β})$. This completes the proof.

We delete the details for proving Theorem 4 as they like the details for proving Theorem 3.

Theorem 4

Let $f\in T_i\left(\mathbf{p}\right)$ belongs to $K_q^{n\left(\gamma \right)}(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β})$ and (33) $\gamma ={\left[\mathbf{p}\right]}_q-\frac{\begin{array}{c}\delta [{\left[\mathbf{p}\right]}_q+\sigma ({\left[i+\mathbf{p}\right]}_q-{\left[\mathbf{p}\right]}_q\left)\right]\\ {\mathrm{\Psi }}_{q,\mathbf{p}}^n(i+\mathbf{p},\varkappa ,\varrho )\end{array}}{\begin{array}{c}{\left[i+\mathbf{p}\right]}_q\left\{[{\left[\mathbf{p}\right]}_q+\sigma ({\left[i+\mathbf{p}\right]}_q-{\left[\mathbf{p}\right]}_q\left)\right]\right.\\ \left.{\mathrm{\Psi }}_{q,\mathbf{p}}^n(i+\mathbf{p},\varkappa ,\varrho )-\text{β}{\left[\mathbf{p}\right]}_q\left|b\right|\right\}\end{array}},$(33) then (34) $N_{i,\delta }^{\mathbf{p},q}{\left(S\right)\subset K}_q^{n\left(\gamma \right)}(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β}),$(34) where (35) $\delta \le {\left[\mathbf{p}\right]}_q{\left[i+\mathbf{p}\right]}_q\left\{1-\frac{\text{β}{\left[\mathbf{p}\right]}_q\left|b\right|}{\begin{array}{c}({\left[\mathbf{p}\right]}_q+\sigma ({\left[i+\mathbf{p}\right]}_q-{\left[\mathbf{p}\right]}_q\left)\right]\\ {\mathrm{\Psi }}_{q,\mathbf{p}}^n(i+\mathbf{p},\varkappa ,\varrho )\end{array}}\right\}.$(35)

Remark

1. Letting $q\to 1^{-}$ in Theorems 1–4, respectively, we obtain new results for the classes $S^n(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β}),$ $K^n(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β}),$ $S^{n\left(\gamma \right)}(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β})$ and $K^{n\left(\gamma \right)}(i,\mathbf{p},\varkappa ,\varrho ,\sigma ,b,\text{β}),$ respectively;

2. Putting (i) $\varkappa =1,$ (ii) $\varrho =0,$ (iii) $\varkappa =1$ and $\varrho =0$ in Lemma 1, Theorems 1 and 3, respectively, we obtain new results for the classes $S_q^n(i,\mathbf{p},\varrho ,\sigma ,b,\text{β}),S_q^n(i,\mathbf{p},\varkappa ,\sigma ,b,\text{β})$

   and$S_q^n(i,\mathbf{p},\sigma ,b,\text{β}),$ respectively;

3. Putting (i) $\varkappa =1,$ (ii)$\varrho =0,$ (iii) $\varkappa =1$ and $\varrho =0$ in Lemma 2, Theorems 2 and 4, respectively, we obtain new results for the classes $K_q^n(i,\mathbf{p},\varrho ,\sigma ,b,\text{β}),K_q^n(i,\mathbf{p},\varkappa ,\sigma ,b,\text{β})$

   and $K_q^n(i,\mathbf{p},\sigma ,b,\text{β}),$ respectively.

Acknowledgments

The authors are grateful to the referees for their valuable suggestions.

Availability of data and materials

During the current study, the data sets are derived arithmetically.

Disclosure statement

No potential conflict of interest was reported by the authors.