Geometric Properties of Analytic Functions Defined by the (p, q) Derivative Operator Involving the Poisson Distribution

Abstract

The objective of the current paper is to find the necessary and sufficient condition for the function to belong to the subclass of of analytic functions involving the Poisson distribution defined by the derivative operator. Furthermore, distortion bounds, covering theorems, a radius of starlikeness, and convexity for functions belonging to this class are obtained.

1. Introduction

Let indicate the open unit disk’s normalised class function , which is provided by

Moreover, let be the subclass of contained in univalent functions. A function is starlike of order if and only if comfort of the analytic condition and its class of the function are denoted by , and is called the convex of the order . Let express the subclass of consisting of functions. Its coefficients are nonzero, appropriated as the second one, and negative, i.e., an univalent and analytic function , and it follows as

In [1], the subclass of M is indicated by and was obtained. Let indicate the class of the function which is given bywhich is analytic in the open unit disk . Altintas [2] studied the class , and the following condition is given byfor a few and for all . Srivastava et al. [3] (also by Chatterjea [4]) studied the classes, namely, and , as follows:

Also, Kamali and Akbulut [5] introduced the subclass convex function, and the following condition is given by

Applications of hypergeometric functions [6], generalized hypergeometric functions [7], confluent hypergeometric functions [8], generalized Bessel functions [9, 10], Wright function [11], and Fox–Wright function [12] are crucial to the development of geometric function theory. Derivations of specific geometric properties were recently conducted in [13, 14] to build links between geometric function theory and statistics by using a generalized discrete probability distribution. Porwal [15] introduced the Poisson distribution series in 2014 (see also [14, 16]); he also obtained necessary and sufficient conditions for a few classes of univalent functions and connected probability density functions and geometric function theory. Following the publication of this paper, many researchers developed hypergeometric distribution series [13], confluent hypergeometric distribution series [17], binomial distribution series [18], hypergeometric distribution series [13, 19], Borel distribution series [20], and Pascal distribution series [21] and discovered some intriguing properties of various classes of univalent functions.

Definition 1. A variable is called the Poisson distribution if the values are taken such as . along probabilities . respectively, where is a parameter. Thus,A power series with Poisson distribution probabilities as its coefficients was recently developed by Porwal [15].We observe that the above series radius of convergence is infinite according to the ratio test. The series was introduced by Porwal [15].Fractional q-calculus, which was developed in the early 20th century, is the q-extension of the common fractional calculus. Numerous scientific fields, including ordinary fractional calculus, optimal control, q-difference, and q-integral equations, as well as the geometric function notion of complex analysis, utilize the theory of q-calculus operators. The q-analogue of the derivative and the integral operator was first developed by Jackson [22] in 1908, along with some of its applications. Ismail et al. further developed the concept of the q-extension of the class of q-starlike functions [23], and Srivastava [24] examined q-calculus in the context of the theory of univalent functions. The q-analogue of the Ruscheweyh differential operator was first developed by Kanas and Raducanu [25], and Srivastava examined q-starlike functions associated with the generalized conic domain [26]. In addition to introducing the q-analogue of the Noor integral operator and studying some of its applications utilizing the convolution notion, Srivastava et al. [27] also published a series of publications that focused on the class of q-starlike functions from various angles (see [4, 28–31]), and Khan et al. [32] obtained uniformly convex functions and a corresponding subclass of starlike functions with a fixed coefficient associated with the -analogous of the Ruscheweyh operator. Zhang et al. [33] further introduced the new subclass of -starlike functions associated with the generalized conic domain, and Zhang et al. [34] examined the q-difference operator in harmonic univalent functions. In 2022, Khan et al. [35] studied a new subclass of analytic functions related to the generalized conic domain associated with the q-differential operator.
In this article, there are many applications of quantum calculus in the division of different functions and many other fields [36–42]. The approach of quantum calculus established as q-calculus is equal to classic infinitesimal calculus after the concept of limits.
Furthermore, the development of quantum calculus is called postquantum calculus, and it is indicated by calculus. Chakrabarti and Jagannathan [43] introduced a consideration of the integer in order to generalize or unify several forms of oscillator algebras well-known in the physics literature related to the representation theory of single-paramater quantum algebra, and Araci et al. [44] obtained a certain derivative involving several relations. Kavitha et al. [45] also identified the first coefficient bounds for the new classes on quantum calculus that has been defined. We use basic notations, and calculus is given by where .

2. Definition of

A natural generalization of the number is the twin-basic number, i.e., , and the differential operator of a function h analytic in is defined as follows:and we note that .We can refer to calculus as a generalization of calculus. The factorial is defined by

Note that for factorial decreases to the factorial. Also, clearly and . To learn more about and related literature see [43, 44, 46, 47].

In our paper, a function is said to be in the class when it belongs to the subclass of convex functions in the unit disk and if it satisfies

Now, we define the linear operator which is defined by the convolution product as

Inspired by multiple findings that show relationships between different analytic univalent function subclasses utilizing hypergeometric functions [16, 48, 49], we obtained the coefficient estimate, distortion, covering theorems, extreme points, convex linear combination, a radius of starlike functions and convex functions, and neighborhood property.

3. Main Results

Now, the following theorem is in the class which is given below.

Theorem 1. Let indicate the class of functions of the form.which are analytic in the unit disk. A function is in the class if and only if

Proof. We prove thatIf , it suffices to prove that the values forlie inside a circle whose radius is and whose centre is . We haveand the calculation is bounded above by ifand then, we getConversely, supposeThen,The values of are chosen on the real axis so thatis real. After removing the denominator in (3) and allowing to flow through real values, we obtainIf (15) is written as , then we getThus, is in the class .
Taking and in Theorem 1, we have the following.

Corollary 1. A function is in the class if and only if

Theorem 2. If , thenwith equality for

Proof. We haveThis final inequality is following Theorem 1. ThusSimilarly,

Theorem 3. If , thenwith equality for

Proof. We haveIn view of Theorem 1, we haveand then,orA substitution of equation (34) into equation (26), yields the right-hand inequality.
On the other hand,This completes the proof.

Theorem 4. and ; then, if it can be expressed in the formwhere and .

Proof. Supposethenand thusWe have .
Conversely, suppose . Sincewe may setThen,This completes the proof.
Taking and in Theorem 4, we have the following.

Corollary 2. The extreme points of are given by and .

Theorem 5. The class is a closed linear combination.

Proof. Suppose that the functions and defined byare in the class setting.and we find (37) thatIn view of Theorem 1, we have

Theorem 6. Let , and then, we have the following:(i) is starlike of the order in the class , i.e., where(ii) is convex of the order in the class , i.e., where

Proof. (i)Given is a starlike order . We have in relation to the left side of (46), we have This implies using the fact that if and only if we say that equivalently,(ii)Given is a convex order . We have for the left hand side of (55), we have using the fact that if and only if from (58) and (59), which yields the convexity of the family.

3.1. Neighbourhood Property

Following [1, 28, 50, 51], we specified the -neighbourhood of the function by

Definition 2. The function is defined as the class if the function occurs in a manner that the function is .

Theorem 7. If and , then .

Proof. Let . We then find fromwhich easily implies the coefficient inequalitysince we have from (15) thatHence, the proof is completed.
Taking and in Theorem 7, we have the following.

Corollary 3. If and , then .

4. Conclusion

This paper deals with geometric properties of analytic functions involving the Poisson distribution defined by the (p, q) derivative operator. Also, we found that the necessary and sufficient condition for the function belongs to the subclass of of analytic functions involving the Poisson distribution defined by the derivative operator. We anticipate that this distribution series will be important in several fields related to mathematics, science, and technology.

Data Availability

No data were used to support the findings of this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All the authors equally contributed to deriving the results and approved the final manuscript.