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A certain subclass of meromorphically qstarlike functions associated with the Janowski functions
Journal of Inequalities and Applications volume 2019, Article number: 88 (2019)
Abstract
In this paper, the authors introduce a new subclass of meromorphic qstarlike functions which are associated with the Janowski functions. A characterization of meromorphically qstarlike functions associated with the Janowski functions has been obtained when the coefficients in their Laurent series expansion about the origin are all positive. This leads to a study of coefficient estimates, distortion theorems, partial sums, and the radius of starlikeness estimates for this class. It is seen that the class considered demonstrates, in some respects, properties analogous to those possessed by the corresponding class of univalent analytic functions with negative coefficients.
Introduction and definitions
Let the class of analytic functions in the open unit disk
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
Thus, each \(f\in \mathcal{A}\) has a Taylor–Maclaurin series representation as follows:
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
such that
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
Definition 2
The qderivative (or qdifference) \(D_{q}\) of a function f is defined in a given subset of \(\mathbb{C}\) by
provided that \(f^{\prime } ( 0 ) \) exists.
From Definition 2, we can observe that
for a differentiable function f in a given subset of \(\mathbb{C}\). It is readily known from (1.1) and (1.3) that
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
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
that are analytic in the punctured open unit disk
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
The sufficient condition for a function f to be in the class \(\mathcal{MS}^{\ast } ( \alpha ) \) is given by
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
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
Remark 1
First of all, it is easily seen that
where \(\mathcal{MS}^{\ast } [ A,B ] \) is a function class, introduced and studied by Ali et al. [2]. Secondly we have
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
when the coefficients are sufficiently small. Throughout this paper, unless otherwise mentioned, we will assume that
Coefficient estimates
In this section, we give a sufficient condition for a function f to be in the class \(\mathcal{MS}_{q}^{\ast } [ A,B ] \), which will work as one of the key results to find other results of this paper.
Theorem 1
A function \(f\in \mathcal{M}\) of the form given by (1.5) is in the class \(\mathcal{MS}_{q}^{\ast } [ A,B ] \) if it satisfies the following condition:
where
and
Proof
Assuming that (2.1) holds, it suffices to show that
Now we have
The last expression in (2.4) is bounded above by \(\frac{1}{1q}\) if
where \(\varLambda ( n,A,B,q ) \) and \(\varUpsilon ( A,B,q ) \) are given by (2.2) and (2.3) respectively. The proof of Theorem 1 is thus completed. □
It is easy to deduce the following consequence of Theorem 1.
Corollary 1
If a function \(f\in \mathcal{M}\) of the form given by (1.5) is in the class \(\mathcal{MS}_{q}^{\ast } [ A,B ] \), then
with equality for each n, with the function of the form
where \(\varUpsilon ( A,B,q ) \) and \(\varLambda ( n,A,B,q ) \) are given by (2.2) and (2.3) respectively.
Distortion inequalities
Theorem 2
If \(f\in \mathcal{MS}_{q}^{\ast } [ A,B ] \), then
where equality holds for the function
with \(\varLambda ( n,A,B,q ) \) and \(\varUpsilon ( A,B,q ) \) given by (2.2) and (2.3) respectively.
Proof
Let \(f\in \mathcal{MS}_{q}^{\ast } [ A,B ] \). Then, in view of Theorem 1, we have
which yields
Similarly, we have
We have thus completed the proof of Theorem 2. □
The following result (Theorem 3) can be proved by using arguments similar to those that have already been presented in the proof of Theorem 2. So we choose to omit the details of our proof of Theorem 3.
Theorem 3
If \(f\in \mathcal{MS}_{q}^{\ast } [ A,B ] \), then
and \(\varLambda ( k,A,B,q ) \) and \(\varUpsilon ( A,B,q ) \) are given by (2.2) and (2.3) respectively.
Partial sums for the function class \(\mathcal{MS}_{q}^{\ast}[A,B]\)
In this section, we examine the ratio of a function of the form (1.5) to its sequence of partial sums
when the coefficients of f are sufficiently small to satisfy condition (2.1). We will determine sharp lower bounds for
Unless otherwise stated, we will assume that f is of the form (1.5) and that its sequence of partial sums is denoted by
Theorem 4
If f of the form (1.5) satisfies condition (2.1), then
and
where
and \(\varLambda ( k,A,B,q ) \) and \(\varUpsilon ( A,B,q ) \) are given by (2.2) and (2.3) respectively.
Proof
In order to prove inequality (4.1), we set
If we set
then we find, after some suitable simplification, that
Thus, clearly, we find that
and
Now one can see that
if and only if
which implies that
Finally, to prove the inequality in (4.1), it suffices to show that the lefthand side of (4.4) is bounded above by \(\sum_{n=1}^{\infty }\xi _{n} \vert a_{n} \vert \), which is equivalent to
By virtue of (4.5), the proof of inequality in (4.1) is now completed.
Next, in order to prove inequality (4.2), we set
where
This last inequality in (4.6) is equivalent to
Finally, we can see that the lefthand side of the inequality in (4.7) is bounded above by \(\sum_{n=1}^{\infty }\xi _{n} \vert a_{n} \vert \), and so we have completed the proof of (4.2), which completes the proof of Theorem 4. □
We next turn to ratios involving derivatives.
Theorem 5
If f of the form (1.1) satisfies condition (2.1), then
and
where \(\xi _{k}\) is given by (4.3).
Proof
The proof of Theorem 5 is similar to that of Theorem 4, we here choose to omit the analogous details. □
Radius of starlikeness
In the following theorem we obtain the radius of qstarlikeness for the class \(\mathcal{MS}_{q}^{\ast } [ A,B ] \), we say that f given by (1.5) is meromorphically starlike of order α (\(0\leq \alpha <1\)) in \(\vert z \vert < r\) when it satisfies condition (1.6) in \(\vert z \vert < r\).
Theorem 6
Let the function f given by (1.5) be in the class \(\mathcal{MS}_{q}^{\ast } [ A,B ] \). Then, if
is positive, then f is meromorphically starlike of order α in \(\vert z \vert \leq r\), where \(\varLambda ( n,A,B,q ) \) and \(\varUpsilon ( A,B,q ) \) are given by (2.2) and (2.3) respectively.
Proof
In order to prove the result, we must show that
We have
Hence (5.1) holds true if
The inequality in (5.2) can be written as
With the aid of (2.1), inequality (5.3) is true if
Solving (5.4) for \(\vert z \vert \), we have
In view of (5.5), the proof of our theorem is now completed. □
Concluding remarks and observations
In our present investigation, we have introduced and studied systematically a new subclass of the class of the meromorphically qstarlike functions, which is associated with the Janowski functions. We have given a characterization of these meromorphically qstarlike functions associated with the Janowski functions when the coefficients in the Laurent series expansion about the origin are all positive. This has led us to a study of coefficient estimates, distortion theorems, partial sums and estimates of the radius of qstarlikeness for this meromorphic function class. We have observed that the class considered in this article demonstrates, in some respects, properties analogous to those possessed by the corresponding class of univalent analytic functions with negative coefficients.
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Acknowledgements
The first author would like to acknowledge Prof. Dr. Salim ur Rehman, V.C. Sarhad University of Science & I. T, for providing excellent research and academic environment.
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Sarhad University of Science & I. T Peshawar, Pakistan.
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Mahmood, S., Ahmad, Q.Z., Srivastava, H.M. et al. A certain subclass of meromorphically qstarlike functions associated with the Janowski functions. J Inequal Appl 2019, 88 (2019). https://doi.org/10.1186/s136600192020z
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DOI: https://doi.org/10.1186/s136600192020z
MSC
 05A30
 30C45
 11B65
 47B38
Keywords
 Meromorphically starlike functions
 Janowski functions
 qderivative (or qdifference)
 Distortion theorems
 Partial sums
 Radius of qstarlikeness