Let the class of analytic functions in the open unit disk
$$ \mathbb{U}= \bigl\{ z:z\in \mathbb{C}\text{ and } \vert z \vert < 1 \bigr\} $$
be denoted by \(\mathcal{H} ( \mathbb{U} ) \), and let \(\mathcal{A }\) denote the class of all functions f, which are analytic in the unit disk \(\mathbb{U}\) and normalized by
$$ f ( 0 ) =0 \quad \text{and}\quad f^{\prime } ( 0 ) =1. $$
Thus, each \(f\in \mathcal{A}\) has a Taylor–Maclaurin series representation as follows:
$$ f ( z ) =z+\sum_{n=2}^{\infty }a_{n}z^{n} \quad ( \forall z\in \mathbb{U} ) . $$
(1.1)
Also, let \(\mathcal{S}\) be the subclass of analytic function class \(\mathcal{A}\), consisting of all univalent functions in \(\mathbb{U}\).
Furthermore, let \(\mathcal{P}\) denote the wellknown Carathéodory class of functions p, analytic in the open unit disk \(\mathbb{U}\), which are normalized by
$$ p ( z ) =1+\sum_{n=1}^{\infty }c_{n}z^{n}, $$
(1.2)
such that
$$ \Re \bigl\{ p ( z ) \bigr\} >0 \quad ( \forall z\in \mathbb{U} ) . $$
The intrinsic properties of qanalogs, including the applications in the study of quantum groups and qdeformed superalgebras, study of fractals and multifractal measures, and in chaotic dynamical systems, are known in the literature. Some integral transforms in the classical analysis have their qanalogues in the theory of qcalculus. This has led various researchers in the field of qtheory to extending all the important results involving the classical analysis to their qanalogs.
For the convenience, we provide some basic definitions and concept details of qcalculus which are used in this paper. Throughout this paper, we will assume that q satisfies the condition \(0< q<1\). We shall follow the notation and terminology of [7]. We first recall the definitions of fractional qcalculus operators of complex valued function f.
Definition 1
(see [7])
Let \(q\in ( 0,1 ) \) and define the qnumber \([ \lambda ] _{q}\) by
$$ [ \lambda ] _{q}= \textstyle\begin{cases} \frac{1q^{\lambda }}{1q} & ( \lambda \in \mathbb{C} ), \\ \sum_{k=0}^{n1}q^{k}=1+q+q^{2}+\cdots+q^{n1} & ( \lambda =n\in \mathbb{N} ) .\end{cases} $$
Definition 2
(see [9] and [10])
The qderivative (or qdifference) \(D_{q}\) of a function f is defined in a given subset of \(\mathbb{C}\) by
$$ ( D_{q}f ) ( z ) = \textstyle\begin{cases} \frac{f ( qz ) f ( z ) }{ ( q1 ) z}& ( z\neq0 ) ,\\ f^{\prime } ( 0 ) & ( z=0 ) \end{cases} $$
(1.3)
provided that \(f^{\prime } ( 0 ) \) exists.
From Definition 2, we can observe that
$$ \lim_{q\rightarrow 1} ( D_{q}f ) ( z ) = \lim _{q\rightarrow 1}\frac{f ( qz ) f ( z ) }{ ( q1 ) z}=f^{\prime } ( z ) $$
for a differentiable function f in a given subset of \(\mathbb{C}\). It is readily known from (1.1) and (1.3) that
$$ ( D_{q}f ) ( z ) =1+\sum_{n=2}^{\infty } [ n ] _{q}a_{n}z^{n1}. $$
(1.4)
A number of subclasses of normalized analytic function class \(\mathcal{A}\) in geometric function theory have been studied already from different viewpoints. The above defined qcalculus provides an important tool in order to investigate several subclasses of class \(\mathcal{A}\). A firm footing usage of the qcalculus in the context of geometric function theory was presented mainly and basic (or q) hypergeometric functions were first used in geometric function theory in a book chapter by Srivastava (see, for details, [17, pp. 347 et seq.]; see also [18]).
Recently Srivastava et al. [20] successfully combined the concept of Janowski [11] and the above mentioned qcalculus and defined the following.
Definition 3
A function \(f\in \mathcal{A}\) is said to belong to the class \(\mathcal{S}_{q}^{\ast } [ A,B ] \) if and only if
$$ \biggl\vert \frac{ ( B1 ) ( \frac{z ( D_{q}f ) ( z ) }{f ( z ) } )  ( A1 ) }{ ( B+1 ) ( \frac{z ( D_{q}f ) ( z ) }{f ( z ) } )  ( A+1 ) }\frac{1}{1q} \biggr\vert < \frac{1}{1q}. $$
From Definition 3, one can easily observe that

1.
\(\lim_{q\rightarrow 1}\mathcal{S}_{q}^{\ast } [ A,B ] = \mathcal{S}^{\ast } [ A,B ] \), where \(\mathcal{S}^{\ast } [ A,B ] \) is the function class introduced and studied by Janowski [11].

2.
\(\mathcal{S}_{q}^{\ast } [ 1,1 ] =\mathcal{S}_{q}^{\ast }\), where \(\mathcal{S}_{q}^{\ast }\) is the function class introduced and studied by Ismail et al. [8].

3.
When
$$ A=12\alpha \quad ( 0\leq \alpha < 1 ) \quad \text{and}\quad B=1 $$
and let \(q\rightarrow 1\), the class \(\mathcal{S}_{q}^{\ast } [ A,B ] \) reduces to the function class \(\mathcal{S}^{\ast }( \alpha )\), which was introduced and studied by Silverman (see [15]). Moreover, its worthy of note that \(\mathcal{S}^{\ast }(0)=\mathcal{S}^{\ast }\), where \(\mathcal{S}^{ \ast }\) is a wellknown function class of starlike functions.

4.
When
$$ A=12\alpha \quad ( 0\leq \alpha < 1 ) \quad \text{and}\quad B=1, $$
the class \(\mathcal{S}_{q}^{\ast } [ A,B ] \) reduces to the function class \(\mathcal{S}_{q}^{\ast }(\alpha )\), which was introduced and studied recently by Agrawal and Sahoo (see [1]).
Let \(\mathcal{M}\) denote the class of functions f of the form
$$ f ( z ) =\frac{1}{z}+\sum_{n=1}^{\infty }a_{n}z^{n}, $$
(1.5)
that are analytic in the punctured open unit disk
$$ \mathbb{U}^{\ast }= \bigl\{ z:z\in \mathbb{C}\text{ and }0< \vert z \vert < 1 \bigr\} =\mathbb{U}\backslash \{ 0 \} $$
with a simple pole at the origin with residue 1 there. Let \(\mathcal{MS}^{\ast } ( \alpha ) \) (\(0\leq \alpha <1\)) denote the subclasses of \(\mathcal{M}\) meromorphically starlike of order α. Analytically f of the form (1.5) is in \(\mathcal{MS}^{\ast } ( \alpha ) \) if and only if
$$ \Re \biggl( \frac{zf^{\prime } ( z ) }{f ( z ) } \biggr) >\alpha \quad ( z\in \mathbb{U} ) . $$
The sufficient condition for a function f to be in the class \(\mathcal{MS}^{\ast } ( \alpha ) \) is given by
$$ \sum_{n=1}^{\infty } ( n+\alpha ) \vert a_{n} \vert \leq 1\alpha \quad ( 0\leq \alpha < 1 ) . $$
(1.6)
The class \(\mathcal{MS}^{\ast } ( \alpha ) \) and similar other classes have been extensively studied by Pommerenke [13], Clunie [5], Miller [12], Royster [14], and others.
Since to a certain extent the work in the meromorphic univalent case has paralleled that of the analytic univalent case, one is tempted to search results analogous to those of Silverman [16] for meromorphic univalent functions in \(\mathbb{U}^{\ast }\). Several different subclasses of meromorphic univalent function class \(\mathcal{M}\) were introduced and studied analogously by the many authors; see, for example, [3, 4, 6, 19, 21]. However, analogous to Definition 2, we extend the idea of qdifference operator to a function f given by (1.5) from the class \(\mathcal{M}\) and also define analogous of meromorphic analogy of the function class \(\mathcal{S}_{q}^{\ast } [ A,B ] \).
Definition 4
For \(f\in \mathcal{M}\), let the qderivative operator (or qdifference operator) be defined by
$$ ( D_{q}f ) ( z ) =\frac{f ( qz ) f ( z ) }{ ( q1 ) z}=\frac{1}{qz^{2}}+\sum _{n=0}^{\infty } [ n ] _{q}a_{n}z^{n1} \quad \bigl( \forall z\in \mathbb{U}^{\ast } \bigr) . $$
Definition 5
A function \(f\in \mathcal{A}\) is said to belong to the class \(\mathcal{MS}_{q}^{\ast } [ A,B ] \) if and only if
$$ \biggl\vert \frac{ ( B1 ) ( \frac{z ( D_{q}f ) ( z ) }{f ( z ) } )  ( A1 ) }{ ( B+1 ) ( \frac{z ( D_{q}f ) ( z ) }{f ( z ) } )  ( A+1 ) }\frac{1}{1q} \biggr\vert < \frac{1}{1q}. $$
Remark 1
First of all, it is easily seen that
$$ \lim_{q\rightarrow 1}\mathcal{MS}_{q}^{\ast } [ A,B ] = \mathcal{MS}^{\ast } [ A,B ], $$
where \(\mathcal{MS}^{\ast } [ A,B ] \) is a function class, introduced and studied by Ali et al. [2]. Secondly we have
$$ \lim_{q\rightarrow 1}\mathcal{MS}_{q}^{\ast } [ 1,1 ] = \mathcal{MS}^{\ast }, $$
where \(\mathcal{MS}^{\ast }\) is the wellknown function class of meromorphic starlike functions. This function class and similar other classes have been extensively studied by Pommerenke [13], Clunie [5], Miller [12], Royster [14], and others.
In the present paper, we give a sufficient condition for a function f to be in the class \(\mathcal{MS}_{q}^{\ast } [ A,B ] \), which will be used as a supporting result for further investigation the remainder of this article. Distortion inequalities, as well as results concerning the radius of starlikeness, are obtained. We will investigate the ratio of a function of the form (1.5) to its sequence of partial sums
$$ f_{k} ( z ) =\frac{1}{z}+\sum_{n=1}^{k}a_{n}z^{n} \quad ( k\in \mathbb{N} ) , $$
(1.7)
when the coefficients are sufficiently small. Throughout this paper, unless otherwise mentioned, we will assume that
$$ 1\leq B< A\leq 1 \quad \text{and}\quad q\in ( 0,1 ) . $$