Basic Properties for Certain Subclasses of Meromorphic p-Valent Functions with Connected q-Analogue of Linear Differential Operator

Abstract

In this paper, we define three subclasses ${\mathcal{M}}_{p,\alpha }^{n,q}(\eta ,A,B),{\mathcal{I}}_{p,\alpha }^n(\lambda ,\mu ,\gamma ),$, R${}_p^{n,q}(\lambda ,\mu ,\gamma )$ connected with a q-analogue of linear differential operator ${\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}$ which consist of functions $\mathcal{F}$ of the form $\mathcal{F}\left(\zeta \right)={\zeta }^{-p}+{\displaystyle \sum _{j=1-p}^{\infty }}{a}_j{\zeta }^j$$(p\in \mathbb{N})$ satisfying the subordination condition $p-\frac{1}{\eta }\left\{\frac{\zeta {\left({\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}\mathcal{F}\left(\zeta \right)\right)}^{{}^{\prime }}}{{\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}\mathcal{F}\left(\zeta \right)}+p\right\}\prec p\frac{1+A\zeta }{1+B\zeta }.$Also, we study the various properties and characteristics of this subclass ${\mathcal{M}}_{p,\alpha }^{n,q,\ast }(\eta ,A,B)$ such as coefficients estimate, distortion bounds and convex family. Also the concept of $\delta$neighborhoods and partial sums of analytic functions to the class ${\mathcal{M}}_{p,\alpha }^{n,q}(\eta ,A,B)$.

1. Introduction

Let ${\mathcal{M}}_p$ is the class of p-valently meromorphic functions of the form:

$$\mathcal{F}\left(\zeta \right)={\zeta }^{-p}+\sum _{j=1-p}^{\infty }{a}_j{\zeta }^j(p\in \mathbb{N}=\{1,2,....\}),$$

(1)

which are analytic in the punctured open unit disk ${\Delta }^{*}:=\{\zeta \in \mathbb{C}:0<|\zeta |<1\}=\Delta \setminus \left\{0\right\}.$ Let $\mathcal{F}$ and $\mathcal{E}$ are analytic functions in $\Delta$, we say that $\mathcal{F}$ is subordinate to $\mathcal{E}$ if there exists an analytic function $\varpi \left(\zeta \right)$ with $\varpi \left(0\right)=0$ and $\left|\varpi \left(\zeta \right)\right|<1$$(\zeta \in \Delta )$ such that $\mathcal{F}=\mathcal{E}\left(\varpi \right(\zeta \left)\right).$ We denote by $\mathcal{F}\prec \mathcal{E}$ (see [1,2]):

Let the functions $\mathcal{F}\left(\zeta \right)\in {\mathcal{M}}_p$ defined by (1) and $\mathcal{G}\left(\zeta \right)\in {\mathcal{M}}_p$ defined by

$$\mathcal{G}\left(\zeta \right)={\zeta }^{-p}+\sum _{j=1-p}^{\infty }{b}_j{\zeta }^j(p\in \mathbb{N}).$$

(2)

The Hadamard product or convolution of $\mathcal{F}\left(\zeta \right)$ and $\mathcal{G}\left(\zeta \right)$ is defined by

$$(\mathcal{F}\ast \mathcal{G})\left(\zeta \right)={\zeta }^{-p}+\sum _{j=1-p}^{\infty }{a}_j{b}_j{\zeta }^j=(\mathcal{G}\ast \mathcal{F})\left(\zeta \right).$$

(3)

In this paper, we define some concepts of fractional derivative, for any non-negative integer j, the $q-$factorial ${\left[j\right]}_q!$ is defined by (see [3]):

Assume that $0<q<1,$ the q-number ${\left[j\right]}_q$ are defined by (see [3,4,5,6,7,8,9]). where

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

(4)

El-Deeb et al. [10] defined the q-derivative operator for $\mathcal{F}\ast \mathcal{G}$ as follows (see [11])

$${\mathcal{D}}_q\left(\mathcal{F}\ast \mathcal{G}\right)\left(\zeta \right):=\left\{\begin{array}{cc}\frac{\left(\mathcal{F}\ast \mathcal{G}\right)\left(q\zeta \right)-\left(\mathcal{F}\ast \mathcal{G}\right)\left(\zeta \right)}{\zeta (q-1)}\hfill & \zeta \ne 0\hfill \\ {\mathcal{F}}^{\prime }\left(0\right)\hfill & \zeta =0.\hfill \end{array}\right.$$

(5)

Also, we have

$$\underset{q\to 1^{-}}{lim}{\mathcal{D}}_q\left(\mathcal{F}\ast \mathcal{G}\right)\left(\zeta \right):=\underset{q\to 1^{-}}{lim}\frac{\left(\mathcal{F}\ast \mathcal{G}\right)\left(q\zeta \right)-\left(\mathcal{F}\ast \mathcal{G}\right)\left(\zeta \right)}{\zeta (q-1)}={\left(\left(\mathcal{F}\ast \mathcal{G}\right)\left(\zeta \right)\right)}^{{}^{\prime }}.$$

From (1) and (5), we get

$${\mathcal{D}}_q\left(\mathcal{F}\ast \mathcal{G}\right)\left(\zeta \right):=-\frac{{\left[p\right]}_q}{q^p}{\zeta }^{-p-1}+\sum _{j=1-p}^{\infty }{\left[j\right]}_q{a}_j{b}_j{\zeta }^{j-1},\zeta \ne 0.$$

(6)

Also, we define the linear differential operator ${\mathcal{D}}_{\alpha ,p,g}^{n,q}:{\mathcal{M}}_p\to {\mathcal{M}}_p$ as follows:

$${\mathcal{D}}_{\alpha ,p,g}^{0,q}\mathcal{F}\left(\zeta \right)=\left(\mathcal{F}\ast \mathcal{G}\right)\left(\zeta \right),$$

$$\begin{array}{ccc}\hfill {\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{1,q}\mathcal{F}\left(\zeta \right)& =& \frac{\alpha q^p}{{\left[p\right]}_q}\zeta {\mathcal{D}}_q\left({\mathcal{D}}_{\alpha ,p,g}^{0,q}\mathcal{F}\left(\zeta \right)\right)+(1-\alpha )\left(\mathcal{F}\ast \mathcal{G}\right)\left(\zeta \right)+2\alpha {\zeta }^{-p}\hfill \\ & =& {\zeta }^{-p}+\sum _{j=1-p}^{\infty }\left(\frac{\alpha q^p{\left[j\right]}_q+\left(1-\alpha \right){\left[p\right]}_q}{{\left[p\right]}_q}\right)a_j{b}_j{\zeta }^j\hfill \\ \hfill {\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{2,q}\mathcal{F}\left(\zeta \right)& =& \frac{\alpha q^p}{{\left[p\right]}_q}\zeta {\mathcal{D}}_q\left({\mathcal{D}}_{\alpha ,p,g}^{1,q}\mathcal{F}\left(\zeta \right)\right)+(1-\alpha ){\mathcal{D}}_{\alpha ,p,g}^{1,q}\mathcal{F}\left(\zeta \right)+2\alpha {\zeta }^{-p}\hfill \\ & =& {\zeta }^{-p}+\sum _{j=1-p}^{\infty }{\left(\frac{\alpha q^p{\left[j\right]}_q+\left(1-\alpha \right){\left[p\right]}_q}{{\left[p\right]}_q}\right)}^2{a}_j{b}_j{\zeta }^j\hfill \\ & & .\hfill \\ & & .\hfill \\ & & .\hfill \\ \hfill {\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}\mathcal{F}\left(\zeta \right)& =& \frac{\alpha q^p}{{\left[p\right]}_q}\zeta {\mathcal{D}}_q\left({\mathcal{D}}_{\alpha ,p,g}^{n-1,q}\mathcal{F}\left(\zeta \right)\right)+(1-\alpha ){\mathcal{D}}_{\alpha ,p,g}^{n-1,q}\mathcal{F}\left(\zeta \right)+2\alpha {\zeta }^{-p}\hfill \\ & =& {\zeta }^{-p}+\sum _{j=1-p}^{\infty }{\left(\frac{\alpha q^p{\left[j\right]}_q+\left(1-\alpha \right){\left[p\right]}_q}{{\left[p\right]}_q}\right)}^n{a}_j{b}_j{\zeta }^j\hfill \\ & & \left(p\in \mathbb{N},n\in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\},0<q<1,\alpha >0\right).\hfill \end{array}$$

(7)

From (7), we obtain the following relations:

$$\left(\mathrm{i}\right){\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n+1,q}\mathcal{F}\left(\zeta \right)=\frac{\alpha q^p}{{\left[p\right]}_q}\zeta D_q\left({\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}\mathcal{F}\left(\zeta \right)\right)+\left(1-\alpha \right){\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}\mathcal{F}\left(\zeta \right)+2\alpha {\zeta }^{-p},\zeta \in {\Delta }^{*};$$

(8)

$$\left(\mathrm{ii}\right){\mathcal{I}}_{\alpha ,p,\mathcal{G}}^n\mathcal{F}\left(\zeta \right):=\underset{q\to 1^{-}}{lim}{\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}\mathcal{F}\left(\zeta \right)={\zeta }^{-p}+\sum _{j=1-p}^{\infty }{\left(\frac{j\alpha +p\left(1-\alpha \right)}{p}\right)}^n{a}_j{b}_j{\zeta }^j,\zeta \in {\Delta }^{*}.$$

(9)

Remark 1.

(i) By taking $\mathcal{G}\left(\zeta \right)=\frac{{\zeta }^{-p}}{1-\zeta }($or$b_j=1)$in this operator ${\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}$, we have the linear differential operator ${\mathcal{D}}_{\alpha ,p,q}^n$defined by El-Deeb and El-Matary ([12], With $A=1$);

(ii) Put $\alpha =1$in the operator ${\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}$, we get the $(p,q)$-analogue of the operator${\mathcal{D}}_{p,\mathcal{G}}^{n,q}$defined as follows:

$${\mathcal{D}}_{p,\mathcal{G}}^{n,q}\mathcal{F}\left(\zeta \right)={\zeta }^{-p}+\sum _{j=1-p}^{\infty }{\left(\frac{q^p{\left[j\right]}_q}{{\left[p\right]}_q}\right)}^n{a}_j{b}_j{\zeta }^j\left(p\in \mathbb{N},n\in {\mathbb{N}}_0,0<q<1,\zeta \in {\Delta }^{*}\right);$$

(10)

(iii) Let$\alpha =1$and$q\to 1$in the operator ${\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}$, we have the operator${\mathcal{D}}_{p,\mathcal{G}}^n$defined as follows:

$${\mathcal{D}}_{p,\mathcal{G}}^n\mathcal{F}\left(\zeta \right):=\underset{q\to 1^{-}}{lim}{\mathcal{D}}_{1,p,q}^n\mathcal{F}\left(\zeta \right)={\zeta }^{-p}+\sum _{j=1-p}^{\infty }{\left(\frac{j}{p}\right)}^n{a}_j{b}_j{\zeta }^j,\left(p\in \mathbb{N},n\in {\mathbb{N}}_0,\zeta \in {\Delta }^{*}\right);$$

(11)

(iv) Taking $\alpha =1$and$\mathcal{G}\left(\zeta \right)=\frac{{\zeta }^{-p}}{1-\zeta }($or$b_j=1)$in the operator ${\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}$, we have the $(p,q)$-analogue of Salagean operator${\mathcal{D}}_{p,q}^n$defined as follows:

$${\mathcal{D}}_{p,q}^n\mathcal{F}\left(\zeta \right):={\zeta }^{-p}+\sum _{j=1-p}^{\infty }{\left(\frac{q^p{\left[j\right]}_q}{{\left[p\right]}_q}\right)}^n{a}_j{\zeta }^j\left(p\in \mathbb{N},n\in {\mathbb{N}}_0,0<q<1,\zeta \in {\Delta }^{*}\right);$$

(12)

(v) Putting $q\to 1^{-}$and $\alpha =1$in the operator ${\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}$, we get the operator in meromorphic${\mathcal{D}}_{p,\mathcal{G}}^n$defined as follows:

$${\mathcal{D}}_{p,\mathcal{G}}^n\mathcal{F}\left(\zeta \right):=\underset{q\to 1^{-}}{lim}{\mathcal{D}}_{1,p,\mathcal{G}}^{n,q}\mathcal{F}\left(\zeta \right)={\zeta }^{-p}+\sum _{j=1-p}^{\infty }{\left(\frac{j}{p}\right)}^n{a}_j{b}_j{\zeta }^j,\left(p\in \mathbb{N},n\in {\mathbb{N}}_0,\zeta \in {\Delta }^{*}\right).$$

(13)

A function $\mathcal{F}\in {\mathcal{M}}_p$ is said to be in the subclass ${\mathcal{MS}}^{\ast }\left(\gamma \right)$ of meromorphic starlike functions of order $\gamma$ in ${\Delta }^{*},$ if it satisfies the following condition (see [13,14,15,16]):

$$\Re \left(\frac{\zeta {\mathcal{F}}^{{}^{\prime }}\left(\zeta \right)}{\mathcal{F}\left(\zeta \right)}\right)<-\gamma \left(\zeta \in {\Delta }^{*};0\le \gamma <1\right).$$

(14)

A function $\mathcal{F}\in {\mathcal{M}}_p$ is said to be in the subclass $\mathcal{MC}\left(\gamma \right)$ of meromorphic convex functions of order $\gamma$ in ${\Delta }^{*},$ if it satisfies the following condition (see [17]):

$$\Re \left(1+\frac{\zeta {\mathcal{F}}^{{}^{\prime \prime }}\left(\zeta \right)}{{\mathcal{F}}^{{}^{\prime }}\left(\zeta \right)}\right)<-\gamma \left(\zeta \in {\Delta }^{*};0\le \gamma <1\right).$$

(15)

It is easy to observe from (14) and (15) that

$$\mathcal{F}\in \mathcal{MC}\left(\gamma \right)\iff -\zeta {\mathcal{F}}^{{}^{\prime }}\in {\mathcal{MS}}^{\ast }\left(\gamma \right).$$

(16)

We will generalize these classes by using the new operator ${\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q},$ we define the new class ${\mathcal{M}}_{p,\alpha }^{n,q}(\lambda ,\mu ,\gamma )$ and study some theorems for this class.

Definition 1.

Assume that $\mathcal{F}\in {\mathcal{M}}_p$ be in the class ${\mathcal{M}}_{p,\alpha }^{n,q}(\eta ,A,B)$ if

$$p-\frac{1}{\eta }\left\{\frac{\zeta {\left({\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}\mathcal{F}\left(\zeta \right)\right)}^{{}^{\prime }}}{{\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}\mathcal{F}\left(\zeta \right)}+p\right\}\prec p\frac{1+A\zeta }{1+B\zeta }$$

(17)

or, equivalently, to

$$\left|\frac{{{\displaystyle \frac{\zeta \left({\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}\mathcal{F}\left(\zeta \right)\right)}{{\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}\mathcal{F}\left(\zeta \right)}}}^{{}^{\prime }}+p}{B{\displaystyle \frac{\zeta {\left({\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}\mathcal{F}\left(\zeta \right)\right)}^{{}^{\prime }}}{{\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}\mathcal{F}\left(\zeta \right)}}+p\left[\left(A-B\right)\eta +B\right]}\right|<1$$

(18)

$$\left(p\in \mathbb{N},n\in {\mathbb{N}}_0,0<q<1,\alpha >0,\eta \in {\mathbb{C}}^{\ast },-1\le B<A\le 1,\zeta \in {\Delta }^{*}\right).$$

Let ${\mathcal{M}}_p^{\ast }$ is subclass of ${\mathcal{M}}_p$ which contains functions on the form:

$$\mathcal{F}\left(\zeta \right):={\zeta }^{-p}+\sum _{j=p}^{\infty }{a}_j{\zeta }^j(p\in \mathbb{N}).$$

(19)

Also, we can write

$${\mathcal{M}}_{p,\alpha }^{n,q,\ast }(\eta ,A,B)={\mathcal{M}}_{p,\alpha }^{n,q}(\eta ,A,B)\cap {\mathcal{M}}_p^{\ast }.$$

Remark 2.

(i) Taking $q\to 1^{-},$ we get $\underset{q\to 1^{-}}{lim}{\mathcal{M}}_{p,\alpha }^{n,q}(\lambda ,\mu ,\gamma )=:{\mathcal{I}}_{p,\alpha }^n(\lambda ,\mu ,\gamma )$, where ${\mathcal{I}}_{p,\alpha }^n(\lambda ,\mu ,\gamma )$ represents the function $\mathcal{F}\in {\mathcal{M}}_p$ that satisfies (18) for ${\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}$ replaced with ${\mathcal{I}}_{\alpha ,p,\mathcal{G}}^n$ given by (9);

(ii) Putting $\alpha =1$, we get the subclass R${}_p^{n,q}(\lambda ,\mu ,\gamma )$ represents the function $\mathcal{F}\in {\mathcal{M}}_p$ that satisfies (18) for ${\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}$ replaced with ${\mathcal{D}}_{p,\mathcal{G}}^{n,q}$ given by (10).

2. Basic Properties of the Subclass ${\mathcal{M}}_{\mathbf{p},\mathbf{\alpha }}^{\mathbf{n},\mathbf{q},\ast }$(η, A, B)

Theorem 1.

The function $\mathcal{F}$ defined by (19) belongs to the subclass ${\mathcal{M}}_{p,\alpha }^{n,q,\ast }(\eta ,A,B)$ if and only if

$$\sum _{j=p}^{\infty }\left[\left(j+p\right)(1-B)-p\left|\eta \right|(A-B)\right]{\left(\frac{\alpha q^p{\left[j\right]}_q+\left(1-\alpha \right){\left[p\right]}_q}{{\left[p\right]}_q}\right)}^n{b}_j\left|a_j\right|\le p\left|\eta \right|(A-B).$$

(20)

Proof. 

Let (20) holds true, we get

$$\begin{array}{ccc}& & \left|\zeta {\left({\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}\mathcal{F}\left(\zeta \right)\right)}^{{}^{\prime }}+p{\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}\mathcal{F}\left(\zeta \right)\right|-\left|B\zeta {\left({\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}\mathcal{F}\left(\zeta \right)\right)}^{{}^{\prime }}+\left[Bp(1-\eta )+Ap\eta \right]{\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}\mathcal{F}\left(\zeta \right)\right|\hfill \\ & =& \left|\sum _{j=p}^{\infty }\left(j+p\right){\left(\frac{\alpha q^p{\left[j\right]}_q+\left(1-\alpha \right){\left[p\right]}_q}{{\left[p\right]}_q}\right)}^n{a}_j{b}_j{\zeta }^{j+p}\right|-\hfill \\ & & \left|p\eta (A-B)+\sum _{j=p}^{\infty }\left[B\left(j+p\right)+p\eta (A-B)\right]{\left(\frac{\alpha q^p{\left[j\right]}_q+\left(1-\alpha \right){\left[p\right]}_q}{{\left[p\right]}_q}\right)}^n{a}_j{b}_j{\zeta }^{j+p}\right|\hfill \\ & \le & \sum _{j=p}^{\infty }\left(j+p\right){\left(\frac{\alpha q^p{\left[j\right]}_q+\left(1-\alpha \right){\left[p\right]}_q}{{\left[p\right]}_q}\right)}^n{b}_j\left|a_j\right|r^{j+p}-p\eta (A-B)\hfill \\ & & -\sum _{j=p}^{\infty }\left[B\left(j+p\right)+p\eta (A-B)\right]{\left(\frac{\alpha q^p{\left[j\right]}_q+\left(1-\alpha \right){\left[p\right]}_q}{{\left[p\right]}_q}\right)}^n{b}_j\left|a_j\right|r^{j+p}\hfill \\ & =& \sum _{j=p}^{\infty }\left[\left(1-B\right)\left(j+p\right)-p\eta (A-B)\right]{\left(\frac{\alpha q^p{\left[j\right]}_q+\left(1-\alpha \right){\left[p\right]}_q}{{\left[p\right]}_q}\right)}^n{b}_j\left|a_j\right|r^{j+p}-p\eta (A-B).\hfill \end{array}$$

(21)

Since (21) holds for all $r\in \left(0,1\right).$Letting $r\to 1^{-},$we obtain

$$\begin{array}{ccc}& & \left|\zeta {\left({\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}\mathcal{F}\left(\zeta \right)\right)}^{{}^{\prime }}+p{\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}\mathcal{F}\left(\zeta \right)\right|-\left|B\zeta {\left({\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}\mathcal{F}\left(\zeta \right)\right)}^{{}^{\prime }}+\left[Bp(1-\eta )+Ap\eta \right]{\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}\mathcal{F}\left(\zeta \right)\right|\hfill \\ & \le & \sum _{j=p}^{\infty }\left[\left(1-B\right)\left(j+p\right)-p\eta (A-B)\right]{\left(\frac{\alpha q^p{\left[j\right]}_q+\left(1-\alpha \right){\left[p\right]}_q}{{\left[p\right]}_q}\right)}^n{b}_j\left|a_j\right|-p\eta (A-B)\hfill \\ & \le & 0\hfill \end{array}$$

(by (20))

Hence, we get $\mathcal{F}\left(\zeta \right)\in {\mathcal{M}}_{p,\alpha }^{n,q}(\eta ,A,B).$

Conversely, Let $\mathcal{F}\left(\zeta \right)$ belongs to ${\mathcal{M}}_{p,\alpha }^{n,q}(\eta ,A,B)$ with $\mathcal{F}\left(\zeta \right)$ of the form (19), we find from (18), that

$$\left|\frac{\zeta {\left({\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}\mathcal{F}\left(\zeta \right)\right)}^{{}^{\prime }}+p{\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}\mathcal{F}\left(\zeta \right)}{B\zeta {\left({\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}\mathcal{F}\left(\zeta \right)\right)}^{{}^{\prime }}+\left[Bp\left(1-b\right)+Apb\right]{\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}\mathcal{F}\left(\zeta \right)}\right|$$

$$=\left|\frac{{\displaystyle \sum _{j=p}^{\infty }}\left(j+p\right){\left(\frac{\alpha q^p{\left[j\right]}_q+\left(1-\alpha \right){\left[p\right]}_q}{{\left[p\right]}_q}\right)}^n{a}_j{b}_j{\zeta }^{j+p}}{p\eta (A-B)+{\displaystyle \sum _{j=p}^{\infty }}\left[B\left(j+p\right)+p\eta (A-B)\right]{\left(\frac{\alpha q^p{\left[j\right]}_q+\left(1-\alpha \right){\left[p\right]}_q}{{\left[p\right]}_q}\right)}^n{a}_j{b}_j{\zeta }^{j+p}}\right|<1.$$

(22)

Using the fact that$\Re \left\{\zeta \right\}\le \left|\zeta \right|$ for all $\zeta ,$we get

$$\Re \left\{\frac{\frac{\zeta {\left({\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}\mathcal{F}\left(\zeta \right)\right)}^{{}^{\prime }}}{{\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}\mathcal{F}\left(\zeta \right)}+p}{\frac{B\zeta {\left({\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}\mathcal{F}\left(\zeta \right)\right)}^{{}^{\prime }}}{{\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}\mathcal{F}\left(\zeta \right)}+\left[Bp\left(1-\eta \right)+Ap\eta \right]}\right\}<1,\zeta \in {\Delta }^{*}.$$

(23)

If we take $\zeta$ on real axis, so that $\frac{\zeta {\left({\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}\mathcal{F}\left(\zeta \right)\right)}^{{}^{\prime }}}{{\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}\mathcal{F}\left(\zeta \right)}$ is real. Upon clearing the denominator in (23) and letting $\zeta \to 1^{-}$, we get

$$\sum _{j=p}^{\infty }\left[\left(j+p\right)(1-B)-p\left|\eta \right|(A-B)\right]{\left(\frac{\alpha q^p{\left[j\right]}_q+\left(1-\alpha \right){\left[p\right]}_q}{{\left[p\right]}_q}\right)}^n{b}_j\left|a_j\right|\le p\left|\eta \right|(A-B),$$

(24)

which we’ve got the assertion (20) of Theorem 1. □

Corollary 1.

The function $\mathcal{F}\left(\zeta \right)$ be defined by (19) belongs to ${\mathcal{M}}_{p,\alpha }^{n,q,\ast }(\eta ,A,B)$, then

$$\left|a_j\right|\le {\displaystyle \frac{p\left|\eta \right|(A-B)}{\left[\left(j+p\right)(1-B)-p\left|\eta \right|(A-B)\right]{\left(\frac{\alpha q^p{\left[j\right]}_q+\left(1-\alpha \right){\left[p\right]}_q}{{\left[p\right]}_q}\right)}^n{b}_j}}(j\ge p).$$

(25)

This result is sharp for $\mathcal{F}$ given by

$$\mathcal{F}\left(\zeta \right)={\zeta }^{-p}+{\displaystyle \frac{p\left|\eta \right|(A-B)}{\left[\left(j+p\right)(1-B)-p\left|\eta \right|(A-B)\right]{\left(\frac{\alpha q^p{\left[j\right]}_q+\left(1-\alpha \right){\left[p\right]}_q}{{\left[p\right]}_q}\right)}^n{b}_j}}{\zeta }^j(j\ge p).$$

(26)

Theorem 2.

The function $\mathcal{F}\left(\zeta \right)$ defined by (19) belongs ${\mathcal{M}}_{p,\alpha }^{n,q,\ast }(\eta ,A,B),$ then for $\left|\zeta \right|=r<1$, we have

$$\begin{array}{cc}\hfill & \left\{\frac{(p+m-1)!}{(p-1)!}-\frac{p!\left|\eta \right|(A-B)}{\left[2(1-B)-\left|\eta \right|(A-B)\right]{\left(1+\alpha \left(q^p-1\right)\right)}^n(p-m)!b_p}{r}^{2p}\right\}{r}^{-(p+m)}\hfill \\ \hfill & \le \left|{\mathcal{F}}^{\left(m\right)}\left(\zeta \right)\right|\le \left\{\frac{(p+m-1)!}{(p-1)!}+\frac{p!\left|\eta \right|(A-B)}{\left[2(1-B)-\left|\eta \right|(A-B)\right]{\left(1+\alpha \left(q^p-1\right)\right)}^n(p-m)!b_p}{r}^{2p}\right\}{r}^{-(p+m)}.\hfill \end{array}$$

(27)

This result is sharp for $\mathcal{F}$ given by

$$\mathcal{F}\left(\zeta \right)={\zeta }^{-p}+\frac{\left|\eta \right|(A-B)}{\left[2(1-B)-\left|\eta \right|(A-B)\right]{\left(1+\alpha \left(q^p-1\right)\right)}^n{b}_p}{\zeta }^p.$$

(28)

Proof. 

Let $\mathcal{F}\left(\zeta \right)\in {\mathcal{M}}_{p,\alpha }^{n,q,\ast }(\eta ,A,B),$ then

$$\begin{array}{ccc}& & \frac{p\left[2(1-B)-\left|\eta \right|(A-B)\right]{\left(1+\alpha \left(q^p-1\right)\right)}^n(p-m)!b_p}{p!}\sum _{j=p}^{\infty }\frac{j!}{(j-m)!}\left|a_j\right|\hfill \\ & \le & \sum _{j=p}^{\infty }\left[\left(j+p\right)(1-B)-p\left|\eta \right|(A-B)\right]{\left(\frac{\alpha q^p{\left[j\right]}_q+\left(1-\alpha \right){\left[p\right]}_q}{{\left[p\right]}_q}\right)}^n{b}_j.\left|a_j\right|\le p\left|\eta \right|(A-B),\hfill \end{array}$$

which yields

$$\sum _{j=p}^{\infty }\frac{j!}{(j-m)!}\left|aj\right|\le \frac{\left|\eta \right|(A-B)}{\left[2(1-B)-\left|\eta \right|(A-B)\right]{\left(1+\alpha \left(q^p-1\right)\right)}^n{b}_p}\frac{p!}{(p-m)!}.$$

(29)

Differentiating both sides of (19) m times with respect to $\zeta$, we get

$${\mathcal{F}}^{\left(m\right)}\left(\zeta \right)={(-1)}^m\frac{(p+m-1)!}{(p-1)!}{\zeta }^{-(p+m)}+\sum _{j=p}^{\infty }\frac{j!}{(j-m)!}\left|a_j\right|{\zeta }^{j-m}(p\in \mathbb{N},0\le m<p)$$

(30)

and Theorem 2 follows easily from (29) and (30), and it is easy to have the bounds in (27) are attained for $\mathcal{F}$ given by (28). □

Theorem 3.

The function$\mathcal{F}$defined by (19) belings to ${\mathcal{M}}_{p,\alpha }^{n,q,\ast }(\eta ,A,B),$ then

(i) $\mathcal{F}$ is meromorphically p-valent q-starlike of order ρ $(0\le \rho <{\left[p\right]}_q)$ in the disc $\left|\zeta \right|<r_1,$ that is,

$$\Re \left\{-\frac{\zeta {\mathcal{D}}_q\mathcal{F}\left(\zeta \right)}{\mathcal{F}\left(\zeta \right)}\right\}>\rho (\left|\zeta \right|<r_1,0\le \rho <{\left[p\right]}_q,p\in \mathbb{N}),$$

(31)

where

$$r_1=\underset{j\ge p}{inf}{\left\{{\displaystyle \frac{\left[\left(j+p\right)(1-B)-p\left|\eta \right|(A-B)\right]{\left(\frac{\alpha q^p{\left[j\right]}_q+\left(1-\alpha \right){\left[p\right]}_q}{{\left[p\right]}_q}\right)}^n}{p\left|\eta \right|(A-B)}}\frac{\left(\frac{{\left[p\right]}_q}{q^p}-\rho \right)b_j}{\left({\left[j\right]}_q+\rho \right)}\right\}}^{{\displaystyle \frac{1}{j+p}}},$$

(32)

(ii) $\mathcal{F}$is meromorphically p-valent q-convex of order ρ $(0\le \rho <{\left[p\right]}_q)$ in the disc $\left|\zeta \right|<r_2,$ that is,

$$\Re \left\{-\left(\frac{{\mathcal{D}}_q\left(\zeta {\mathcal{D}}_q\mathcal{F}\left(\zeta \right)\right)}{{\mathcal{D}}_q\mathcal{F}\left(\zeta \right)}\right)\right\}>\rho (\left|\zeta \right|<r_2,0\le \rho <{\left[p\right]}_q,p\in \mathbb{N}),$$

(33)

where

$$r_2=\underset{j\ge p}{inf}{\left\{\frac{\left[\left(j+p\right)(1-B)-p\left|\eta \right|(A-B)\right]{\left(\frac{\alpha q^p{\left[j\right]}_q+\left(1-\alpha \right){\left[p\right]}_q}{{\left[p\right]}_q}\right)}^n\left(\frac{{\left[p\right]}_q}{q^p}-\rho \right){\left[p\right]}_q{b}_j}{p{q}^p{\left[j\right]}_q\left({\left[j\right]}_q+\rho \right)\left|\eta \right|(A-B)}\right\}}^{{\displaystyle \frac{1}{j+p}}}.$$

(34)

Each of these results is sharp for the function $\mathcal{F}\left(\zeta \right)$ given by (26).

Proof .

(i) From the definition (19), we easily get

$$\left|{\displaystyle \frac{{\displaystyle \frac{\zeta {\mathcal{D}}_q\mathcal{F}\left(\zeta \right)}{\mathcal{F}\left(\zeta \right)}}+\frac{{\left[p\right]}_q}{q^p}}{{\displaystyle \frac{\zeta {\mathcal{D}}_q\mathcal{F}\left(\zeta \right)}{\mathcal{F}\left(\zeta \right)}}-\frac{{\left[p\right]}_q}{q^p}+2\rho }}\right|\le \frac{{\displaystyle \sum _{j=p}^{\infty }}({\left[j\right]}_q+\frac{{\left[p\right]}_q}{q^p})\left|a_j\right|{\left|\zeta \right|}^{j+p}}{2(\frac{{\left[p\right]}_q}{q^p}-\rho )-{\displaystyle \sum _{j=p}^{\infty }}({\left[j\right]}_q-\frac{{\left[p\right]}_q}{q^p}+2\rho )\left|a_j\right|{\left|\zeta \right|}^{j+p}}.$$

(35)

We have the inequality

$$\left|{\displaystyle \frac{{\displaystyle \frac{\zeta {\mathcal{D}}_q\mathcal{F}\left(\zeta \right)}{\mathcal{F}\left(\zeta \right)}}+\frac{{\left[p\right]}_q}{q^p}}{{\displaystyle \frac{\zeta {\mathcal{D}}_q\mathcal{F}\left(\zeta \right)}{\mathcal{F}\left(\zeta \right)}}-\frac{{\left[p\right]}_q}{q^p}+2\rho }}\right|\le 1(0\le \rho <{\left[p\right]}_q;p\in \mathbb{N}),$$

(36)

if

$$\sum _{j=p}^{\infty }\left(\frac{{\left[j\right]}_q+\rho }{\frac{{\left[p\right]}_q}{q^p}-\rho }\right)\left|a_j\right|{\left|\zeta \right|}^{j+p}\le 1.$$

(37)

Hence, by Theorem 1, (37) will be true

$$\frac{\left({\left[j\right]}_q+\rho \right)}{\left(\frac{{\left[p\right]}_q}{q^p}-\rho \right)}{\left|\zeta \right|}^{j+p}\le {\displaystyle \frac{\left[\left(j+p\right)(1-B)-p\left|\eta \right|(A-B)\right]{\left(\frac{\alpha q^p{\left[j\right]}_q+\left(1-\alpha \right){\left[p\right]}_q}{{\left[p\right]}_q}\right)}^n{b}_j}{p\left|\eta \right|(A-B)}}$$

$$\left|\zeta \right|\le {\left\{{\displaystyle \frac{\left[\left(j+p\right)(1-B)-p\left|\eta \right|(A-B)\right]{\left(\frac{\alpha q^p{\left[j\right]}_q+\left(1-\alpha \right){\left[p\right]}_q}{{\left[p\right]}_q}\right)}^n{b}_j}{p\left|\eta \right|(A-B)}}\frac{\left(\frac{{\left[p\right]}_q}{q^p}-\rho \right)}{\left({\left[j\right]}_q+\rho \right)}\right\}}^{{\displaystyle \frac{1}{j+p}}},$$

(38)

the inequality leads us immediately to the disc $\left|\zeta \right|<r_1,$ where $r_1$ is given by (32).

(ii) To prove the second assertion of Theorem 3, we get from the definition (19) that

$$\left|{\displaystyle \frac{\frac{{\mathcal{D}}_q\left(\zeta {\mathcal{D}}_q\mathcal{F}\left(\zeta \right)\right)}{{\mathcal{D}}_q\mathcal{F}\left(\zeta \right)}+\frac{{\left[p\right]}_q}{q^p}}{\frac{D_q\left(\zeta {\mathcal{D}}_q\mathcal{F}\left(\zeta \right)\right)}{{\mathcal{D}}_q\mathcal{F}\left(\zeta \right)}-\frac{{\left[p\right]}_q}{q^p}+2\rho }}\right|\le \frac{{\displaystyle \sum _{j=p}^{\infty }}{\left[j\right]}_q({\left[j\right]}_q+\frac{{\left[p\right]}_q}{q^p})\left|a_j\right|{\left|\zeta \right|}^{j+p}}{2\frac{{\left[p\right]}_q}{q^p}(\frac{{\left[p\right]}_q}{q^p}-\rho )-{\displaystyle \sum _{j=p}^{\infty }}{\left[j\right]}_q({\left[j\right]}_q-\frac{{\left[p\right]}_q}{q^p}+2\rho )\left|a_j\right|{\left|\zeta \right|}^{j+p}}.$$

(39)

Thus, we have the desired inequality

$$\left|{\displaystyle \frac{\frac{{\mathcal{D}}_q\left(\zeta {\mathcal{D}}_q\mathcal{F}\left(\zeta \right)\right)}{{\mathcal{D}}_q\mathcal{F}\left(\zeta \right)}+\frac{{\left[p\right]}_q}{q^p}}{\frac{{\mathcal{D}}_q\left(\zeta {\mathcal{D}}_q\mathcal{F}\left(\zeta \right)\right)}{{\mathcal{D}}_q\mathcal{F}\left(\zeta \right)}-\frac{{\left[p\right]}_q}{q^p}+2\rho }}\right|\le 1(0\le \rho <{\left[p\right]}_q,p\in \mathbb{N}),$$

(40)

if

$$\sum _{j=p}^{\infty }\frac{q^p{\left[j\right]}_q}{{\left[p\right]}_q}\left(\frac{{\left[j\right]}_q+\rho }{\frac{{\left[p\right]}_q}{q^p}-\rho }\right)\left|a_j\right|{\left|\zeta \right|}^{j+p}\le 1.$$

(41)

From Theorem 1, (41) will be true if

$$\frac{q^p{\left[j\right]}_q}{{\left[p\right]}_q}\left(\frac{{\left[j\right]}_q+\rho }{\frac{{\left[p\right]}_q}{q^p}-\rho }\right){\left|\zeta \right|}^{j+p}\le {\displaystyle \frac{\left[\left(j+p\right)(1-B)-p\left|\eta \right|(A-B)\right]{\left(\frac{\alpha q^p{\left[j\right]}_q+\left(1-\alpha \right){\left[p\right]}_q}{{\left[p\right]}_q}\right)}^n{b}_j}{p\left|\eta \right|(A-B)}}.$$

(42)

The inequality (42) readily yields the disc $\left|\zeta \right|<r_2,$ where $r_2$ defined by (34), and the proof of Theorem 3 is completed. □

3. Neighborhoods and Partial Sums

By following the earlier works based upon the familiar concept of neighborhoods of analytic functions by Goodman [15] and Ruscheweyh [18] and (more recently) by Altintas et al. [19,20,21], Liu [22], Liu and Srivastava [23] and El-Ashwah et al. [24], we introduce here the $\delta$-neighborhoods of a function $\mathcal{F}\in {\mathcal{M}}_p$ has the form (1) by means of the definition given by:

$$\begin{array}{cc}\hfill {\mathcal{N}}_{\delta }\left(\mathcal{F}\right)& =\left\{h:h\in {\mathcal{M}}_p,h\left(\zeta \right)={\zeta }^{-p}+\sum _{j=1-p}^{\infty }{c}_j{z}^j\mathrm{and}\right.\hfill \\ \hfill & \sum _{j=1-p}^{\infty }\frac{\left[\left(j+p\right)(1-B)-p\left|\eta \right|(A-B)\right]{\left(\frac{\alpha q^p{\left[j\right]}_q+\left(1-\alpha \right){\left[p\right]}_q}{{\left[p\right]}_q}\right)}^n{b}_j}{p\left|\eta \right|(A-B)}\left|c_j-a_j\right|\le \delta \hfill \\ \hfill & \left.{\begin{array}{c}(n\in {\mathbb{N}}_0,0<q<1,\alpha >0,\eta \in {\mathbb{C}}^{\ast },-1\le B<A\le 1)\end{array}}_{\begin{array}{c}\end{array}}^{\begin{array}{c}\\ \\ \end{array}}\right\}.\hfill \end{array}$$

(43)

Using the definition (43), we will obtain the following theorem:

Theorem 4.

The function $\mathcal{F}$defined by (1) belongs to $M_{p,\alpha }^{n,q}(\eta ,A,B)$. If $\mathcal{F}$ satisfies the condition

$$\frac{\mathcal{F}\left(\zeta \right)+ϵ{\zeta }^{-p}}{1+ϵ}\in M_{p,\alpha }^{n,q}(\eta ,A,B)(ϵ\in \mathbb{C},\left|ϵ\right|<\delta ,\delta >0)$$

(44)

then

$$N_{\delta }\left(\mathcal{F}\right)\subset M_{p,\alpha }^{n,q}(\eta ,A,B).$$

(45)

Proof. 

From (18), we obtain $h\in M_{p,\alpha }^{n,q}(\eta ,A,B)$ if, for $\sigma \in \mathbb{C}$ with $\left|\sigma \right|=1,$ we have

$$\frac{\zeta {\left({\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}h\left(\zeta \right)\right)}^{{}^{\prime }}+p{\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}h\left(\zeta \right)}{B\zeta {\left({\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}h\left(\zeta \right)\right)}^{{}^{\prime }}+\left[Bp\left(1-b\right)+Apb\right]{\mathcal{D}}_{\alpha ,p,\mathcal{G}}^{n,q}h\left(\zeta \right)}\ne \sigma (\zeta \in \Delta ),$$

(46)

which is equivalent to

$$\frac{(h\ast \psi )\left(\zeta \right)}{{\zeta }^{-p}}\ne 0(\zeta \in {\Delta }^{\ast }),$$

(47)

where, for convenience,

$$\begin{array}{cc}\hfill \psi \left(\zeta \right)& ={\zeta }^{-p}+\sum _{j=1-p}^{\infty }{y}_j{\zeta }^j\hfill \\ \hfill & ={\zeta }^{-p}+\sum _{j=1-p}^{\infty }\frac{\left[\left(j+p\right)(1-B\sigma )-p\left|\eta \right|\sigma (A-B)\right]{\left(\frac{\alpha q^p{\left[j\right]}_q+\left(1-\alpha \right){\left[p\right]}_q}{{\left[p\right]}_q}\right)}^n{b}_j}{p\eta \sigma (A-B)}{\zeta }^j.\hfill \end{array}$$

(48)

From (48), we get

$$\begin{array}{cc}\hfill \left|y_j\right|& =\left|\frac{\left[\left(j+p\right)(1-B\sigma )-p\left|\eta \right|\sigma (A-B)\right]{\left(\frac{\alpha q^p{\left[j\right]}_q+\left(1-\alpha \right){\left[p\right]}_q}{{\left[p\right]}_q}\right)}^n{b}_j}{p\eta \sigma (A-B)}\right|\hfill \\ \hfill & \le \frac{\left[\left(j+p\right)(1+\left|B\right|)-p\left|\eta \right|(A-B)\right]}{p\left|\eta \right|(A-B)}{\left(\frac{\alpha q^p{\left[j\right]}_q+\left(1-\alpha \right){\left[p\right]}_q}{{\left[p\right]}_q}\right)}^n{b}_j(j\ge p,p\in \mathbb{N}).\hfill \end{array}$$

(49)

If $\mathcal{F}\left(\zeta \right)={\zeta }^{-p}+\sum _{j=1-p}^{\infty }{a}_j{\zeta }^j\in {\mathcal{M}}_p$ holds the condition (44), then (47) yields

$$\left|\frac{(\mathcal{F}\ast \psi )\left(\zeta \right)}{{\zeta }^{-p}}\right|>\delta (\zeta \in {\Delta }^{*},\delta >0).$$

(50)

Let

$$\mathrm{\Phi }\left(\zeta \right)={\zeta }^{-p}+\sum _{j=1-p}^{\infty }{d}_j{\zeta }^j\in N_{\delta }\left(\mathcal{F}\right)$$

(51)

we have

$$\left|\frac{\left[\mathrm{\Phi }\left(\zeta \right)-\mathcal{F}\left(\zeta \right)\right]\ast \psi \left(\zeta \right)}{{\zeta }^{-p}}\right|=\left|\sum _{j=1-p}^{\infty }(d_j-a_j)y_j{\zeta }^{j+p}\right|$$

$$\begin{array}{cc}\hfill & \le \left|\zeta \right|\sum _{j=1-p}^{\infty }\frac{\left[\left(j+p\right)(1+\left|B\right|)-p\left|\eta \right|(A-B)\right]}{p\left|\eta \right|(A-B)}{\left(\frac{\alpha q^p{\left[j\right]}_q+\left(1-\alpha \right){\left[p\right]}_q}{{\left[p\right]}_q}\right)}^n{b}_j\left|d_j-a_j\right|\hfill \\ \hfill & <\delta (\zeta \in \Delta ,\delta >0).\hfill \end{array}$$

(52)

We have (47), and hence also (46) for any $\sigma ,$ which implies that $\mathrm{\Phi }\in M_{p,\alpha }^{n,q}(\eta ,A,B).$ This evidently proves the assertion (45) of Theorem 4. □

Theorem 5.

Let $\mathcal{F}\in {\mathcal{M}}_p$ defined by (1) and $-1\le B\le 0,$ the partial sums ${\mathcal{S}}_1\left(\zeta \right)$ and ${\mathcal{S}}_m\left(\zeta \right)$ are given by

$${\mathcal{S}}_1\left(\zeta \right)={\zeta }^{-p}and{\mathcal{S}}_m\left(\zeta \right)={\zeta }^{-p}+\sum _{j=1-p}^{m-1}{a}_j{\zeta }^j(m\in \mathbb{N}\setminus \left\{1\right\}).$$

(53)

Also, suppose that

$$\sum _{j=1-p}^{\infty }{y}_{j+p}\left|a_j\right|\le 1\left(y_{j+p}=\frac{\left[\left(j+p\right)(1+\left|B\right|)-p\left|\eta \right|(A-B)\right]}{p\left|\eta \right|(A-B)}{\left(\frac{\alpha q^p{\left[j\right]}_q+\left(1-\alpha \right){\left[p\right]}_q}{{\left[p\right]}_q}\right)}^n{b}_j\right),$$

(54)

then

$$\mathsf{\left(}i\mathsf{\right)}\mathcal{F}\left(\zeta \right)\in M_{p,\alpha }^{n,q}(\eta ,A,B)$$

$$\mathsf{\left(}\mathit{ii}\mathsf{\right)}Re\left\{{\displaystyle \frac{\mathcal{F}\left(\zeta \right)}{{\mathcal{S}}_m\left(\zeta \right)}}\right\}>1-{\displaystyle \frac{1}{y_q}}(\zeta \in \Delta ,m\in \mathbb{N})$$

(55)

and

$$\mathsf{\left(}\mathit{iii}\mathsf{\right)}Re\left\{{\displaystyle \frac{{\mathcal{S}}_m\left(\zeta \right)}{\mathcal{F}\left(\zeta \right)}}\right\}>{\displaystyle \frac{y_q}{1+y_q}}(\zeta \in \Delta ,m\in \mathbb{N}).$$

(56)

The estimates in (55) and (56) are sharp.

Proof. 

Since ${\displaystyle \frac{{\zeta }^{-p}+\epsilon {\zeta }^{-p}}{1+\epsilon }}={\zeta }^{-p}\in M_{p,\alpha }^{n,q}(\eta ,A,B),$$\left|\epsilon \right|<1,$ then by Theorem 4, we have $N_{\delta }\left(\mathcal{F}\right)\subset M_{p,\alpha }^{n,q}(\eta ,A,B),$$p\in \mathbb{N}.$$N_1\left({\zeta }^{-p}\right)$ denoting the 1-neighbourhood). Now since

$$\sum _{j=1-p}^{\infty }{y}_j\left|a_j\right|\le 1,$$

(57)

then $\mathcal{F}\in N_1\left({\zeta }^{-p}\right)$ and $\mathcal{F}\in M_{p,\alpha }^{n,q}(\eta ,A,B).$ Since $\left\{y_j\right\}$ is an increasing sequence, we get

$$\sum _{j=1-p}^{m-p-1}\left|a_j\right|+y_m\sum _{j=m-p}^{\infty }\left|a_j\right|\le \sum _{j=1-p}^{\infty }{y}_{j+p}\left|a_j\right|\le 1,$$

(58)

we have used the hypothesis (54). Putting

$$h_1\left(\zeta \right)=y_m\left\{{\displaystyle \frac{\mathcal{F}\left(\zeta \right)}{{\mathcal{S}}_m\left(\zeta \right)}}-(1-{\displaystyle \frac{1}{y_m}})\right\}=1+\frac{y_m{\displaystyle \sum _{j=m-p}^{\infty }}\left|a_j\right|{\zeta }^{j+p}}{1+{\displaystyle \sum _{j=1-p}^{m-p-1}}\left|a_j\right|{\zeta }^{j+p}}$$

and applying (58), we find that

$$\left|\frac{h_1\left(\zeta \right)-1}{h_1\left(\zeta \right)+1}\right|\le \frac{y_m{\displaystyle \sum _{j=m-p}^{\infty }}\left|a_j\right|}{2-2{\displaystyle \sum _{j=1-p}^{m-p-1}}\left|a_j\right|-y_m{\displaystyle \sum _{j=m-p}^{\infty }}\left|a_j\right|}\le 1(\zeta \in \Delta ),$$

(59)

which readily yields the assertion (55) of Theorem 5. If we take

$$\mathcal{F}\left(\zeta \right)={\zeta }^{-p}-\frac{{\zeta }^m}{y_m},$$

(60)

then

$${\displaystyle \frac{\mathcal{F}\left(\zeta \right)}{{\mathcal{S}}_m\left(\zeta \right)}}=1-\frac{{\zeta }^{p+m}}{y_m}\to 1-\frac{1}{y_m},\mathrm{as}\zeta \to 1^{-},$$

which shows that the bound in (55) is the best possible for each $m\in \mathbb{N}.$

If we put

$$h_2\left(\zeta \right)=(1+y_m)\left\{{\displaystyle \frac{{\mathcal{S}}_m\left(\zeta \right)}{\mathcal{F}\left(\zeta \right)}}-{\displaystyle \frac{y_m}{1+y_m}}\right\}=1-\frac{(1+y_m){\displaystyle \sum _{j=m-p}^{\infty }}\left|a_j\right|{\zeta }^{j+p}}{1+{\displaystyle \sum _{j=1-p}^{\infty }}\left|a_j\right|{\zeta }^{j+p}},$$

(61)

and make use of (58), we can deduce that

$$\left|\frac{h_2\left(\zeta \right)-1}{h_2\left(\zeta \right)+1}\right|\le \frac{(1+y_m){\displaystyle \sum _{j=m-p}^{\infty }}\left|a_j\right|}{2-2{\displaystyle \sum _{j=1-p}^{m-p-1}}\left|a_j\right|-(1-y_m){\displaystyle \sum _{j=m-p}^{\infty }}\left|a_j\right|}\le 1,$$

leads us to the assertion (56) of Theorem 5. The bound in (56) is sharp. The proof of Theorem 5 is completed. □

4. Concluding Remarks and Observations

In our present investigation, we have introduced and studied the properties of some new subclasses of the class of meromorphic p-valent functions in the open unit disk ${\Delta }^{*}$ by using the combination of q-derivative and convolution and obtain the new operator ${\mathcal{D}}_{\alpha ,p,g}^{n,q}$. Among other properties and results such as coefficients estimate, distortion bounds and convex family. Also the concept of $\delta$neighborhoods and partial sums of analytic functions to the class ${\mathcal{M}}_{p,\alpha }^{n,q}(\eta ,A,B)$.

Interesting results about meromorphic functions can be found in the works [25,26,27,28,29,30,31].