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

Authors: S. Santhiya; K. Thilagavathi
Journal: Journal of Mathematics
 (2023)

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## Abstract

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


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## Body

## 1. Introduction

Letℬ indicate the open unit disk’s normalised class function U, which is provided by(1)hζ=ζ+∑d=2∞bdζd.Moreover, letℱ be the subclass of ℬ contained in univalent functions. A function h∈ℱ is starlike of order β0≤β<1 if and only if comfort of the analytic condition ℛζh′ζ/hζ>βζ∈U and its class of the function are denoted by ℱ∗β, and ℛ1+ζh″ζ/h′ζ>βζ>1 is called the convex of the order β. Let M express the subclass of ℬ consisting of functions. Its coefficients are nonzero, appropriated as the second one, and negative, i.e., an univalent and analytic function h∈M, and it follows as(2)hζ=ζ−∑d=2∞bdζd.In [1], the subclass of M is indicated by M∗β and Cβ was obtained. Let ℬd indicate the class of the function which is given by(3)hζ=ζ−∑a=d+1∞baζaba≥0:d∈N=1,2,3…,which is analytic in the open unit disk U. Altintas [2] studied the class Pd,δ,β, and the following condition is given by(4)ℛζh′ζ+δζ2h′′ζδζh′ζ+1−δhζ>β,for a few δ0≤δ≤1,β0≤β<1 and for all ζ∈U. Srivastava et al. [3] (also by Chatterjea [4]) studied the classes, namely, Tβd and cβd, as follows:(5)pd,0,β=Mβd,pd,1,β=cβd,p1,0,β=m∗β,p1,1,β=cβ.Also, Kamali and Akbulut [5] introduced the subclass Cd,δ,β convex function, and the following condition is given by(6)ℛδζ2h′″ζ+2δ+1ζh″ζ+h′ζδζ2h″ζ+ζh′ζ>β.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 variabley is called the Poisson distribution if the values are taken such as 0,1,2,3,…. along probabilities e−μ,μe−mu/1!,μ2e−mu/2!,…. respectively, where μ is a parameter. Thus,(7)Py=k=μke−μk!,k=0,1,2,3,….
A power series with Poisson distribution probabilities as its coefficients was recently developed by Porwal [15].(8)Aμ,ζ=ζ+∑a=d+1∞μa−1a−1!e−μζa,ζ∈U,d∈N=1,2,….
We observe that the above series radius of convergence is infinite according to the ratio test. The series was introduced by Porwal [15].(9)Aμ,ζ=ζ−∑a=d+1∞μa−1a−1!e−μζa,ζ∈U,d∈N=1,2,…,kμ,ζ=2ζ−Aμ,ζ=ζ−∑a=d+1∞μa−1a−1!e−μζa,ζ∈U,d∈N=1,2,….
Fractionalq-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 q-analogous of the Ruscheweyh operator. Zhang et al. [33] further introduced the new subclass of q-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 byp,q calculus. Chakrabarti and Jagannathan [43] introduced a consideration of the p,q integer in order to generalize or unify several forms of q− 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 p,q derivative involving several relations. Kavitha et al. [45] also identified the first coefficient bounds for the new classes on p,q quantum calculus that has been defined. We use basic notations, and p,q calculus is given by ap,q=pa−qa/p−q,0p,q=0 where p>0,q>0.


## 2. Definition ofp,q

A natural generalization of theq− number is the twin-basic number, i.e., aq=1−qa/1−q,q≠0, and the p,q differential operator of a function h analytic in ζ<1 is defined as follows:(10)Dp,qhζ=hpζ−hqζζp−q,p≠q,ζ∈U,Dp,qhζ=1−∑a=d+1∞ap,qbaζa−1,and we note that limq⟶1−Dp,qhζ=h′ζ.We can refer to q calculus as a generalization of p,q calculus. The p,q factorial is defined by(11)0p,q!=1,ap,q!=∏k=1akp,q!,a≥1.Note that forp⟶1−p,q factorial decreases to the q factorial. Also, clearly limp⟶1−limq⟶1−ap,q=a and limp⟶1−limq⟶1−ap,q!=a!. To learn more about q− and related literature see [43, 44, 46, 47].In our paper, a functionDp,qJhζ∈ℬ is said to be in the class Cd,δ,μ,β when it belongs to the subclass Cd,δ,μ,β of convex functions in the unit disk and if it satisfies(12)ℛδ3ζDp,qJhζ′′′+2δ+1ζ2ζDp,qJhζ′′+ζζDp,qJhζ′δζ2ζDp,qJhζ′′+ζζDp,qJhζ′>β.Now, we define the linear operatorJhζ:ℬ⟶ℬ which is defined by the convolution product as(13)Jhζ=kμ,ζ∗hζ=ζ−∑a=d+1∞μa−1a−1!e−μbaζa,ζ∈U,d∈N=1,2,…,Dp,qJhζ=1−∑a=d+1∞μa−1a−1!ap,qe−μbaζa−1,ζ∈U,d∈N=1,2,…,ζDp,qJhζ=ζ−∑a=d+1∞μa−1a−1!ap,qe−μbaζa,ζ∈U,d∈N=1,2,….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 classCd,δ,μ,β which is given below.Theorem 1.
Letℬd indicate the class of functions of the form.(14)Jhζ=kμ,ζ∗hζ=ζ−∑a=d+1∞μa−1a−1!e−μbaζa,ζ∈U,d∈N=1,2,…,which are analytic in the unit disk. A function Jhζ∈ℬd is in the class Cd,δ,μ,β if and only if(15)∑a=d+1∞aa−βδa−δ+1ap,qμa−1a−1!e−μba≤1−β.Proof.
We prove that(16)∑a=d+1∞aa−βδa−δ+1ap,qμa−1a−1!e−μba≤1−β.
IfJhζ∈ℬd, it suffices to prove that the values for(17)ζδζ2ζDp,qJhζ′′′+2δ+1ζζDp,qJhζ′′+ζDp,qJhζ′δζ2ζDp,qJhζ″+ζζDp,qJhζ′,lie inside a circle whose radius is 1−β and whose centre is w=1. We have(18)δζ3ζDp,qJhζ′′′+2δ+1ζ2ζDp,qJhζ″+ζζDp,qJhζ′δζ2ζDp,qJhζ″+ζζDp,qJhζ′−1=δζ3ζDp,qJhζ′′′+δ+1ζ2ζDp,qJhζ″δζ2ζDp,qJhζ″+ζζDp,qJhζ′=−∑a=d+1∞δaa−1a−2+δ+1kk−1μa−1/a−1!e−μap,qbaζaζ−∑a=d+1∞δaa−1+aμa−1/a−1!e−μap,qbaζa≤∑a=d+1∞aa−1δa−δ+1μa−1/a−1!e−μap,qba1−∑a=d+1∞kδa−δ+1μa−1/a−1!e−μap,qba,and the calculation is bounded above by 1−β if(19)∑a=d+1∞aa−1δa−δ+1μa−1/a−1!e−μap,qba1−∑a=d+1∞kδa−δ+1μa−1/a−1!e−μap,qba≤1−β,and then, we get(20)∑a=d+1∞aa−βδa−δ+1μa−1a−1!e−μap,qba≤1−β.
Conversely, suppose(21)∑a=d+1∞aa−βδa−δ+1μa−1a−1!e−μap,qba≤1−β.
Then,(22)ζδζ2ζDp,qJhζ′′′+2δ+1ζζDp,qJhζ″+ζDp,qJhζ′δζ2ζDp,qJhζ″+ζζDp,qJhζ′=ζ−∑a=d+1∞δaa−1a−2+2δ+1aa−1+aμa−1/a−1!e−μap,qbaζaζ−∑a=d+1∞δaa−1+kμa−1/a−1!e−μap,qbaζa=1−∑a=d+1∞a2δa−a+1μa−1/a−1!e−μap,qbaζa−11−∑a=d+1∞aδa−a+1μa−1/a−1!e−μap,qbaζa−1.
The values ofζ are chosen on the real axis so that(23)ζδζ2ζDp,qJhζ′″+2δ+1ζζDp,qJhζ″+ζDp,qJhζ′δζ2ζDp,qJhζ″+ζζDp,qJhζ′,is real. After removing the denominator in (3) and allowing ζ⟶1− to flow through real values, we obtain(24)ℛζδζ2ζDp,qJhζ′″+2δ+1ζζDp,qJhζ″+ζDp,qJhζ′δζ2ζDp,qJhζ″+ζζDp,qJhζ′=1−∑a=d+1∞a2δa−a+1μa−1/a−1!e−μap,qba1−∑a=d+1∞aδa−a+1μa−1/a−1!e−μap,qba.
If (15) is written as 1−∑a=d+1∞a2δa−a+1μa−1/a−1!e−μap,qba≤1−β1−∑a=d+1∞aδa−a+1μa−1/a−1!e−μap,qba, then we get(25)ℛζδζ2ζDp,qJhζ′″+2δ+1ζζDp,qJhζ″+ζDp,qJhζ′δζ2ζDp,qJhζ″+ζζDp,qJhζ′≥β−β∑a=d+1∞aδa−a+1μa−1/a−1!e−μap,qba1−∑a=d+1∞aδa−a+1μa−1/a−1!e−μap,qba=β.
Thus,Jhζ∈ℬd is in the class Cd,δ,μ,β.
Takingδ=0 and d=1 in Theorem 1, we have the following.Corollary 1.
A functionJhζ∈ℬd is in the class C1,0,μ,β if and only if(26)∑a=2∞aa−βap,qμa−1a−1!e−μba≤1−β.Theorem 2.
IfJhζ∈Cd,δ,μ,β, then(27)n−1−βd+1d+1−βδd+1d+1p,qnd+1≤Jhζ≤n+1−βd+1d+1−βδd+1d+1p,qnd+1,with equality for(28)Jhζ=ζ−1−βd+1d+1−βδd+1d+1p,qζd+1,ζ∈U,d∈N=1,2,….Proof.
We have(29)d+1d+1−βδd+1∑a=d+1∞μa−1a−1!e−μap,qba≤∑a=d+1∞aa−βδa−δ+1μa−1a−1!e−μap,qba≤1−β.This final inequality is following Theorem 1. Thus(30)Jhζ≤n+∑a=d+1∞μa−1a−1!e−μbana≤n+nd+1∑a=d+1∞μa−1a−1!e−μba≤n+nd+1.1−βd+1d+1−βδd+1d+1p,q.
Similarly,Jhζ≥n−∑a=d+1∞μa−1/a−1!e−μbana(31)≥n−nd+1∑a=d+1∞μa−1a−1!e−μba≥n−nd+1.1−βd+1d+1−βδd+1d+1p,q.Theorem 3.
IfJhζ∈Cd,δ,μ,β, then(32)1−1−βd+1d+1−βδd+1d+1p,qnd≤Jhζ′≤1−1−βd+1d+1−βδd+1d+1p,qnd,with equality for(33)Jhζ=ζ−1−βd+1d+1−βδd+1d+1p,qζd+1,ζ∈U,d∈N=1,2,….Proof.
We have(34)Jhζ′≤1−∑a=d+1∞μa−1a−1!e−μbaaζa−1≤1−nd+1−1∑a=d+1∞μa−1a−1!e−μa.baJhζ′≤1−nd∑a=d+1∞μa−1a−1!e−μa.ba.
In view of Theorem1, we have(35)∑a=d+1∞aa−βδa−δ+1μa−1a−1!e−μap,qba≤1−β,and then,(36)d+1d+1−βδd+1d+1p,q∑a=d+1∞μa−1a−1!e−μba≤∑a=d+1∞aa−βδa−δ+1μa−1a−1!e−μap,qba≤1−β,or(37)∑a=d+1∞μa−1a−1!e−μaba≤1−βd+1d+1−βδd+1d+1p,q.
A substitution of equation (34) into equation (26), yields the right-hand inequality.
On the other hand,(38)Jhζ′≥1−nd∑a=d+1∞μa−1a−1!e−μa.ba≥1−1−βd+1d+1−βδd+1d+1p,qnd.
This completes the proof.Theorem 4.
Jhdζ=ζ and Jhaζ=ζ−1−β/aa−βδa−δ+1ap,qζaa=d+1,d+2,…; then, Jhζ∈Cd,δ,μ,β if it can be expressed in the form(39)Jhζ=∑a=d∞ηaJhaζ,where ηa≥0 and ∑a=d∞ηa=1.Proof.
Suppose(40)Jhζ=∑a=d∞ηaJhaζ,then(41)Jhζ=ηdJhdζ+∑a=d+1∞ηaJhaζJhζ=ηdJhdζ+∑a=d+1∞ηaζ−1−βaa−βδa−δ+1ap,qζa=ηdζ+∑a=d+1∞ηaζ−∑a=d+1∞ηa1−βaa−βδa−δ+1ap,qζa=∑a=d∞ηaζ−∑a=d+1∞ηa1−βaa−βδa−δ+1ap,qζa=ζ−∑a=d+1∞ηa1−βaa−βδa−δ+1ap,qζa,and thus(42)=∑a=d+1∞ηa.1−βaa−βδa−δ+1ap,q.aa−βδa−δ+1ap,q1−β=∑a=d+1∞ηa=∑a=d∞ηa−ηd=1−ηd≤1.We have Jhζ∈Cd,δ,μ,β.
Conversely, supposeJhζ∈Cd,δ,μ,β. Since(43)ba≤1−βaa−βδa−δ+1ap,q,a=d+1,d+2,…,we may set(44)ηa=aa−βδa−δ+1ap,q1−βba,a=d+1,d+2,…,ηd=1−∑a=d+1∞ηa.
Then,(45)Ea,ζh=ζ−∑a=d+1∞μa−1a−1!e−μbaζkEa,ζh=ζ−∑a=d+1∞1−βa+1a+1−βδa+1ap,qηaζa=ζ−∑a=d+1∞ηaζ−Jhaζ=ζ−∑a=d+1∞ηaζ+∑a=d+1∞ηaJhaζ=1−∑a=d+1∞ηaζ+∑a=d+1∞ηaJhaζ=ηdζ+∑a=d+1∞ηaJhaζ=ηdJhdζ+∑a=d+1∞ηaJhaζ=∑a=d∞ηaJhaζ.
This completes the proof.
Takingδ=0 and d=1 in Theorem 4, we have the following.Corollary 2.
The extreme points ofC1,0,μ,β are given by Jhdζ=ζ and Jhaζ=ζ−1−β/aa−βap,qζaa=2,3,….Theorem 5.
The classCd,δ,μ,β is a closed linear combination.Proof.
Suppose that the functionsJh1ζ and Jh2ζ defined by(46)Jhζ=ζ−∑a=d+1∞μa−1a−1!e−μbaζa,are in the class Cd,δ,μ,β setting.(47)Jhζ=ηJh1ζ+1−ηJh2ζ,and we find (37) that(48)Jhζ=ζ−∑a=d+1∞η∑a=d+1∞μa−1a−1!e−μba,1+1−η∑a=d+1∞μa−1a−1!e−μba,2ζa.
In view of Theorem1, we have(49)∑a=d+1∞μa−1a−1!e−μap,qaa−βδa−δ+1ηba,1+1−ηba,2=η∑a=d+1∞μa−1a−1!e−μap,qaa−βδa−δ+1ba,1+1−η∑a=d+1∞μa−1a−1!e−μap,q×aa−βδa−δ+1ba,2≤η1−β+1−η1−β≤η1−β+1−β−η1−β=1−β.Theorem 6.
LetJhζ∈Cd,δ,μ,β, and then, we have the following:(i)
Jhζ is starlike of the order ρ0≤ρ<0 in the class ζ<r2, i.e., ℛζζ∂p,qJhζ′/ζ∂p,qJhζ>ρ,ζ<r2;0≤ρ<0 where(50)r2=1−ρρ−a+2aa−βδa−δ+11−β.(ii)
Jhζ is convex of the order ρ0≤ρ<0 in the class ζ<r3, i.e., ℛ1+ζζ∂p,qJhζ″/ζ∂p,qJhζ′>ρ,ζ<r3;0≤ρ<0 where(51)r3=1−ρa2−aρaa−βδa−δ+11−β.Proof.
(i)
GivenJhζ∈τ is a starlike order ρ. We have(52)ζζDp,qJhζ′ζDp,qJhζ−1<1−ρ,in relation to the left side of (46), we have(53)ζζDp,qJhζ′ζDp,qJhζ−1≤ζ1−∑a=d+1∞aμa−1/a−1!e−μap,qbaζa−1ζ−∑a=d+1∞μa−1/a−1!e−μap,qbaζa−1≤∑a=d+1∞1−aμa−1/a−1!e−μap,qbaζa−11−∑a=d+1∞μa−1/a−1!e−μap,qbaζa−1∑a=d+1∞1−aμa−1a−1!e−μap,qbaζa−1≤1−ρ1−∑a=d+1∞μa−1a−1!e−μap,qbaζa−1≤1−ρ−∑a=d+1∞μa−1a−1!e−μap,qbaζa−1+ρ∑a=d+1∞μa−1a−1!e−μap,qbaζa−1.This implies(54)ρ−a+21−ρμa−1a−1!e−μap,qbaζa−1<1,using the fact thatJhζ∈Cd,δ,μ,β if and only if(55)∑a=d+1∞aa−βδa−δ+1μa−1a−1!e−μap,qba≤1−β∑a=d+1∞aa−βδa−δ+1μa−1/a−1!e−μap,qba1−β≤1,we say that(56)ρ−a+21−ρμa−1a−1!e−μap,qbaζa−1≤aa−βδa−δ+1μa−1/a−1!e−μap,qba1−β,equivalently,(57)ζa−1=1−ρρ−a+2aa−βδa−δ+11−β.(ii)
GivenJhζ∈τ is a convex order ρ. We have(58)ζζ∂p,qJhζ″ζ∂p,qJhζ′<1−ρ,for the left hand side of (55), we have(59)ζζDp,qJhζ″ζDp,qJhζ′≤ζ∑a=d+1∞aμa−1/a−1!e−μap,qbaa−1ζa−21−∑a=d+1∞μa−1/a−1!e−μap,qbaaζa−1∑a=d+1∞aμa−1/a−1!e−μap,qbaa−1ζa−11−∑a=d+1∞μa−1/a−1!e−μap,qbaaζa−1≤1−ρ∑a=d+1∞aμa−1a−1!e−μap,qbaa−1ζa−1≤1−ρ1−∑a=d+1∞μa−1a−1!e−μap,qbaaζa−1≤1−ρ−∑a=d+1∞μa−1a−1!e−μap,qbaaζa−1+ρ∑a=d+1∞μa−1a−1!e−μap,qbaaζa−1∑a=d+1∞μa−1a−1!e−μap,qbaa2−aζa−1≤1−ρ−∑a=d+1∞μa−1a−1!e−μap,qbaaζa−1+ρ∑a=d+1∞μa−1a−1!e−μap,qbaaζa−1∑a=d+1∞μa−1a−1!a2−ae−μap,qbaζa−1≤∑a=d+1∞μa−1a−1!a2−aρ1−ρe−μap,qbaζa−1≤1,using the fact thatJhζ∈Cd,δ,μ,β if and only if(60)∑a=d+1∞aa−βδa−δ+11−βμa−1a−1!e−μap,qba≤1,from (58) and (59),(61)a2−aρ1−ρζa−1<aa−βδa−δ+11−βζa−1=1−ρa2−aρaa−βδa−δ+11−β,which yields the convexity of the family.


### 3.1. Neighbourhood Property

Following [1, 28, 50, 51], we specified the α -neighbourhood of the function Jhζ∈C by(62)ℱαJhζ=Jh∈C:Jhζ=ζ−∑a=d+1∞μa−1a−1!e−μbaζa&∑a=d+1∞λ∑a=d+1∞μa−1a−1!e−μba−ea≤α.Definition 2.
The functionJhζ∈ℬ is defined as the class C if the function Ghζ∈Cd,δ,μ,β occurs in a manner that the function is Ghζ∈Cd,δ,μ,β.(63)JhζGhζ−1<1−β,ζ∈D,0≤β<1.Theorem 7.
IfGh∈Cd,δ,μ,β and γ=1−αd+1−βδd+1d+1p,q/d+1d+1−βδd+1d+1p,q−1+β, then ℱαGhζ⊆Cd,δ,μ,β.Proof.
LetJh∈ℱαGhζ. We then find from(64)∑a=d+1∞aμa−1a−1!e−μba−ea≤α,which easily implies the coefficient inequality(65)∑a=d+1∞μa−1a−1!e−μba−ea≤αd+1,since Gh∈Cd,δ,μ,β we have from (15) that(66)∑a=d+1∞μa−1a−1!e−μba≤1−βd+1d+1−βδd+1d+1p,q,JhζGhζ−1<∑a=d+1∞aμa−1/a−1!e−μba−ea1−∑a=d+1∞μa−1/a−1!e−μba≤αd+1−βδd+1d+1p,qd+1d+1−βδd+1d+1p,q−1+β=1−γ.
Hence, the proof is completed.
Takingδ=0 and d=1 in Theorem 7, we have the following.Corollary 3.
IfGh∈C1,0,μ,β and γ=1−α2−β2p,q/22−β2p,q−1+β, then ℱαGhζ⊆C1,0,μ,β.


## 3.1. Neighbourhood Property

Following [1, 28, 50, 51], we specified the α -neighbourhood of the function Jhζ∈C by(62)ℱαJhζ=Jh∈C:Jhζ=ζ−∑a=d+1∞μa−1a−1!e−μbaζa&∑a=d+1∞λ∑a=d+1∞μa−1a−1!e−μba−ea≤α.Definition 2.
The functionJhζ∈ℬ is defined as the class C if the function Ghζ∈Cd,δ,μ,β occurs in a manner that the function is Ghζ∈Cd,δ,μ,β.(63)JhζGhζ−1<1−β,ζ∈D,0≤β<1.Theorem 7.
IfGh∈Cd,δ,μ,β and γ=1−αd+1−βδd+1d+1p,q/d+1d+1−βδd+1d+1p,q−1+β, then ℱαGhζ⊆Cd,δ,μ,β.Proof.
LetJh∈ℱαGhζ. We then find from(64)∑a=d+1∞aμa−1a−1!e−μba−ea≤α,which easily implies the coefficient inequality(65)∑a=d+1∞μa−1a−1!e−μba−ea≤αd+1,since Gh∈Cd,δ,μ,β we have from (15) that(66)∑a=d+1∞μa−1a−1!e−μba≤1−βd+1d+1−βδd+1d+1p,q,JhζGhζ−1<∑a=d+1∞aμa−1/a−1!e−μba−ea1−∑a=d+1∞μa−1/a−1!e−μba≤αd+1−βδd+1d+1p,qd+1d+1−βδd+1d+1p,q−1+β=1−γ.
Hence, the proof is completed.
Takingδ=0 and d=1 in Theorem 7, we have the following.Corollary 3.
IfGh∈C1,0,μ,β and γ=1−α2−β2p,q/22−β2p,q−1+β, then ℱαGhζ⊆C1,0,μ,β.


## 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 Cd,δ,μ,β of analytic functions involving the Poisson distribution defined by the p,q derivative operator. We anticipate that this distribution series will be important in several fields related to mathematics, science, and technology.



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