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DOI: 10.32604/cmes.2022.016940
ARTICLE
Properties of Certain Subclasses of Analytic Functions Involving q-Poisson Distribution
1School of Mathematical Sciences and Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai, 200241, China
2Department of Mathematics, Abbottabad University of Science and Technology, Abbottabad, 22010, Pakistan
3Department of Mathematics, King Abdulaziz University, Jeddah, 21589, Saudi Arabia
4Department of Mathematics, FATA University, Akhorwal (Darra Adam Khel), FR Kohat, 26000, Pakistan
*Corresponding Author: Bilal Khan. Email: bilalmaths789@gmail.com
Received: 12 April 2021; Accepted: 24 November 2021
Abstract: By using the basic (or q)-Calculus many subclasses of analytic and univalent functions have been generalized and studied from different viewpoints and perspectives. In this paper, we aim to define certain new subclasses of an analytic function. We then give necessary and sufficient conditions for each of the defined function classes. We also study necessary and sufficient conditions for a function whose coefficients are probabilities of q-Poisson distribution. To validate our results, some known consequences are also given in the form of Remarks and Corollaries.
Keywords: Analytic function; q-difference operator; sufficient condition; q-Poisson distribution
1 Introduction, Basic Definitions and Motivation
A function f is said to be in the class
in the open unit disk:
is satisfied and having series representation:
Note that the notation
Furthermore, let us consider a subclass
Definition 1.1. [1] A function f is said to be in the class
Definition 1.2. [1] A function f is said to be in the class
It should be noted that for
where
For a parameter
The Poisson distribution is a statistical distribution that calculates the probability of a certain number of events occurring in a particular time period. The Poisson distribution is commonly used to model rate of random events that occur (arrive) in some fixed time interval. The Poisson distribution models packet arrival times as an independent identically distributed process with an exponential distribution. However, it has been demonstrated in reality that packet inter-arrival durations do not follow an exponential distribution, resulting in a considerable increase in the error caused by modeling them as a Poisson distribution. User-initiated TCP (Transmission Control Protocol) session arrivals, such as remote login and file transfer, are well-modeled as Poisson processes with fixed hourly rates, but other connection arrivals deviate significantly from Poisson; modeling TELNET (Teletype Network Protocol) packet inter arrivals as exponential greatly underestimates the burstiness of TELNET traffic, according to studies.
The power series
with its coefficients as probabilities of Poisson distribution which was introduced by Porwal [3] (see also [4–6]). The radius of convergence of
More about special functions and related topics, we may refer to [7–11].
The Basic (or q-) series and basic (or q-) polynomials, particularly basic (or q-) hypergeometric functions and basic (or q-) hypergeometric polynomials, are useful in a wide range of fields, including, for example, Non-Linear Electric Circuit Theory, Finite Vector Spaces, Combinatorial Analysis, Quantum Mechanics, Particle Physics, Mechanical Engineering, Lie Theory, Theory of Heat Conduction, Cosmology and Statistics (see for detail [12]).
In 1748 Euler studied a generating function for pn, the number of partitions of a positive integer n into positive integers by considering the infinite product.
which laid foundation of the study of basic hypergeometric series (also called q-hypergeometric series or q-series). Nevertheless it took a century to get the status of independent subject after the Heine’s conversion of a straightforward observation that
into a systematic theory of
Beside from the influential study of Rogers and Thomae the subject stayed moderately comatose throughout the latter part of the nineteenth century up till Jackson embarked on a long lasting program of developing the theory of basic hypergeometric series in an organized mode (see [13] and [14]).
We next recollect certain elementary and useful concept details of the q-difference calculus, we let throughout the paper that
Definition 1.3. A q-generalization, q-extension, q-analogue or a q-deformation of an arbitrary number
Definition 1.4. Let the q-factorial
Definition 1.5. In a given subset of the set
provided that
In a given subset of
whereas f is differentiable function. Furthermore, from (1) and (4) we obtain
In Geometric Function Theory of Complex Analysis, the role of the q-difference (or the q-derivative) operator Dq is remarkably significant. In his article published by Ismail et al. (see [15]), presented the q-deformation of the class of
More recently, Srivastava published a review article [12], in which the applications of the Dq operator in Geometric function Theory were survived. In the development of Geometric Function Theory, the works of Srivastava [16] and Ismail et al. [15] further motivate the researchers to give their finding to this field. For example, it were Wongsaijai et al. [17], who significantly studied certain subclasses of q-starlike functions. In particular, they studied the inclusion results, radius problems and certain sufficient conditions for their defined functions classes. More recently, the works of Wongsaijai et al. [17] have been generalized by Srivastava et al. (see [18,19]) in a systematic way. In fact Srivastava et al. (see [18,19]) make use of the q-Calculus and certain Janwoski functions in order to devolved their results, which essentially are the generalizations of [17]. Some more recent works related to q-calculus can be found in [20–28].
Motivated by the above-mentioned works of Srivastava [12,16] and Ismail et al. [15], in this article we shall mainly generalize the works presented in [1] and [3]. We shall study a number of sufficient conditions here. A necessary and sufficient condition shall be encounter for a certain function
2 The Subclasses
In this section, making use of the concept of q-calculus and the aforementioned works, we first define certain new subclasses
Definition 2.1. A function f with series representation (1) is to be placed in the class
Remark 1. First of all, it is easy to see that, if we put
where
where
where
Definition 2.2. A function f with series representation (1) is to be placed in the class
2.1 Sufficient Conditions for the Class
Theorem 2.1. A function f of the form (1) is in the class
For the following function
Proof. If we assume that
If the complex number z lie on the real axis side, then
is real. And so in limiting case if we take
We see that the last inequality is equivalent to (6).
Conversely, suppose that (6) hold true. Then adding
to both sides of (6), we get
On the other hand, we see that
It is now easy to see from (7) that the last expression in (8) is bounded above by
Corollary 1. If
Remark 2. First of all, we see that if we put
Theorem 2.2. A function f of the form (1) is in the class
The equality holds for the function
Proof. This theorem can be proved by using the arguments similar which we used in Theorem 2.1.
2.2 Closure Theorems for the Class
We define the function
Furthermore, we now give certain closure results for the function involved in (10).
Theorem 2.3 Let the functions fϰ(z) (ϰ = 1,2,3,...,l) defined by (10) be in the class
Proof. From (10), we have
Now making use of Theorem 2.1, we have
Now by Theorem 2.1, the proof of Theorem 2.3 is completed.
Theorem 2.4. The class
Proof. To prove our result, we let that the given functions
is in the class
By Theorem 2.1, we have
which evidently complete the proof of Theorem 2.4.
3 The Classes
The Euler distribution or Heine distribution, as shown in [29] is a q-deformed Poisson distribution that is known in the literature. All of these, of course, are natural deformations in some sense, but they are all discrete. Kemp [30] demonstrated that the Euler and Heine distributions are, respectively, the limiting forms of a q-analogue of the negative binomial distribution and one of the binomial distribution. Jing [29] proposed a new type of q-analogue of the binomial distribution that uses the Euler distribution as its limiting form.
Furthermore, the theory of q-discrete distributions is quite significant because of theoretical probabilistic and statistical interest. In quantum probability it appears at Brownian motion, further from fascinating Markov chains with discrete finite or infinite state spaces it arises as a steady state distributions (see for example [29–31]). Our aim in this section is to consider a necessary and sufficient condition for a function
where
The graph of
In other words, if
Figure 1: Graph of
In the following equations we settled a power series so that its coefficients are probabilities the of q-Poisson distribution as
It could be seen that by ratio test the radius of convergence of
Theorem 3.1. A function
Proof. Since
According to Theorem 2.1, we must show that
Therefore, we now consider
The last expression in (12) is evidently bounded by functional
Thus we complete the required proof of our Theorem.
Corollary 2. A function
Remark 3. If we set
Theorem 3.2. A function
Proof. Since
According to Theorem 2.2, we must show that
Therefore, we now consider
We in last expression see that the
Thus we have completed the proof of our Theorem.
Remark 4. If in Theorem 3.2, we let
4 Concluding Remarks and Observations
In our present work, by using the q-calculus, we have first defined certain new subclasses of an analytic function. We then gave necessary and sufficient conditions for each of the defined function classes. We have also studied necessary and sufficient conditions for a function whose coefficients are probabilities of q-Poisson distribution. We have also given particular known and new consequences in the form of Remarks and Corollaries.
The usage of basic (or q-) series in many diverse areas of Mathematics and Physics makes it very important. By using the basic (or q-) series, some wonderful works have been done. As we described in Section 1, the Srivastava’s observation [12] about the so-called (p, q)-calculus, we arrived at the point that indeed the result presented in this paper can be produced for the rather straightforward (p, q)-variations.
Authors’ Contributions: All authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.
Availability of Data and Materials: Not applicable.
Funding Statement: The authors received no specific funding for this study.
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.
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