# The Sufficient and Necessary Conditions for the Poisson Distribution Series to Be in Some Subclasses of Analytic Functions

Authors: Abdel Moneim Y. Lashin; Abeer O. Badghaish; Amani Z. Bajamal
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, we introduce new subclasses of analytic functions in the open unit disc. Furthermore, the necessary and sufficient conditions for the Poisson distribution series to be in these new subclasses are found.


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## Body

## 1. Introduction

LetA be the class of all analytic functions f in the open unit disc U=z∈ℂ:z<1 and normalized by f0=0 and f′0=1. A function f∈A has the Taylor series expansion of the form
(1)fz=z+∑k=2∞akzk.We denote byS the subclass of A consisting of normalized functions of the form (1) which are univalent in U. Further, we denote by T the subclass of S consisting of functions with negative coefficients of the form
(2)fz=z−∑k=2∞akzk,ak≥0.Iff,g∈A such that f is given by (1) and g is given by gz=z+∑k=2∞bkzk, then, the Hadamard product f∗gz is defined by
(3)f∗gz=z+∑k=2∞akbkzk.In 1837, the French mathematician Siméon Denis Poisson created the Poisson distribution which is a popular distribution expresses the probability of a given number of events occurring in a fixed interval of time or space. In [1], Porwal introduced a power series such that its coefficients are probabilities of the Poisson distribution
(4)Nm,z=z+∑k=2∞mk−1k−1!e−mzk,m>0,z∈U.In addition, he introduced the series(5)Rm,z=2z−Nm,z=z−∑k=2∞mk−1k−1!e−mzk,m>0,z∈U.In [2], Porwal and Kumar introduced a new linear operator defined by
(6)Nm,z∗fz=z+∑k=2∞mk−1k−1!e−makzk,m>0,z∈U.In [3, 4], El-Ashwah and Kota presented the functions Hυm,z and Hυ∗m,z as below:
(7)Hυm,z=1−υNm,z+υzNm,z′=z+∑k=2∞1+υk−1mk−1k−1!e−mzk,and
(8)Hυ∗m,z=2z−Hυm,z=z−∑k=2∞1+υk−1mk−1k−1!e−mzk,where m>0,0≤υ≤1, and z∈U. Suppose the functions ϕ,ψ, and ζ are given by
(9)ϕz=z+∑k=2∞γkzk,γk≥0,ψz=z+∑k=2∞μkzk,μk≥0,(10)ζz=z+∑k=2∞ρkzk,ρk≥0,for every z∈U.Definition 1.
Let0≤λ,β<1. A function f∈A is said to be in the class Aϕ,ψ,ζ,λ,β if the following condition is satisfied
(11)R1−λf∗ϕz+λf∗ψzf∗ζz>β.Further, we define the classTϕ,ψ,ζ,λ,β by
(12)Tϕ,ψ,ζ,λ,β=Aϕ,ψ,ζ,λ,β∩T.Indeed, we have(1)(13)Az1−z2,z+z21−z3,z,λ,β≡Fλ,β,=f∈A:Rf′z+λzf′′z>β,0≤λ,β<1,where the class Fλ,β was studied by Chichra in [5]. We define the class Fλ,β∩T=R1λ,β which was introduced and studied by Orhan [6].(2)(14)Az1−z2,z+z21−z3,z1−z,λ,β≡Hλ,β,=f∈A:Rzf′z+λz2f′′zfz>β,0≤λ,β<1,where the class Hλ,β was studied by Obradovic and Joshi in [7], and the class Hλ,0 was intoduced and studied by Ramesha et al. in [8]. We define the class Hλ,β∩T=H¯λ,β which was studied by Lashin [9].(3)(15)Az1−z,z1−z2,z,λ,β≡Jλ,β,=f∈A:R1−λfzz+λf′z>β,0≤λ,β<1,where the class Jλ,β was studied by Ding et al. [10]. We define the class Jλ,β∩T=J∗λ,β which was introduced by Hassan [11].(4)(16)Az1−z,z+z21−z3,z1−z2,λ,β≡Xλ,β,=f∈A:R1−λfzzf′z+λ1+zf′′zf′z>β,0≤λ,β<1,where the classes Xλ,β and Xλ,β∩T≪X∗λ,β were introduced by Lashin et al. in [12].In this paper, we find the necessary and sufficient conditions for the Poisson distribution series to be in the classesJ∗λ,β,H¯λ,β, and X∗λ,β.


## 2. Coefficient Inequalities and Related Properties

We first derive the sufficient and necessary conditions for the functionf to be in the aforementioned classes.Theorem 2.
The sufficient condition forf to be in the class Aϕ,ψ,ζ,λ,β is
(17)∑k=2∞γk−ρk+λμk−γk+1−βρkak≤1−β.Proof.
We need to show that(18)R1−λf∗ϕz+λf∗ψzf∗ζz>β.
Then, we have(19)1−λf∗ϕz+λf∗ψzf∗ζz−1=∑k=2∞γk−ρk+λμk−γkakzk−11+∑k=2∞ρkakzk−1,≤∑k=2∞γk−ρk+λμk−γkak1−∑k=2∞ρkak≤1−β,if condition (17) holds. This implies that f∈Aϕ,ψ,ζ,λ,β which completes the proof.Theorem 3.
LetFk=γk−ρk+λμk−γk be an increasing function of k and λ≥max0,ρ2−γ2/μ2−γ2. Then, the necessary and sufficient condition for the function f to be in the class Tϕ,ψ,ζ,λ,β is
(20)∑k=2∞γk−ρk+λμk−γk+1−βρkak≤1−β.Proof.
In view of Theorem2, it suffices to show the necessary condition only. Assume that f∈Tϕ,ψ,ξ,λ,β. Then
(21)R1−λf∗ϕz+λf∗ψzf∗ζz>β,which is equivalent to
(22)R1−∑k=2∞γk+λμk−γkakzk−11−∑k=2∞ρkakzk−1>β.
Choosingz on the real axis, then
(23)1−∑k=2∞γk+λμk−γkakzk−11−∑k=2∞ρkakzk−1,is also real. Let z⟶1− through real values, we get
(24)1−∑k=2∞γk+λμk−γkak≥β1−∑k=2∞ρkak,which is equivalent to (20), and this completes the proof.Puttingϕ=z/1−z2,ψ=z+z2/1−z3, and ζ=z/1−z in Theorems 3, we get the following corollary due to Lashin [9].Corollary 4 (see [9]).
A functionf∈T is in the class H¯λ,β if and only if
(25)∑k=2∞k−1λk+1+1−βak≤1−β,0≤λ,β<1.Puttingϕ=z/1−z,ψ=z/1−z2, and ζ=z in Theorem 3, we get the following corollary due to Hassan [11].Corollary 5 (see [11]).
A functionf∈T is in the class J∗λ,β if and only if
(26)∑k=2∞λk−1+1ak≤1−β,0≤λ,β<1.Puttingϕ=z/1−z,ψ=z+z2/1−z3, and ζ=z/1−z2 in Theorem 3, we get the following corollary.Corollary 6 (see [12]).
Let1/3≤λ<1. A function f∈T is in the class X∗λ,β if and only if
(27)∑k=2∞k−1λk+1−1+1−βkak≤1−β,0≤β<1.Making use of the techniques and methodology given by Porwal [1] (see also [13–18]), we get the following theorems.Theorem 7.
The sufficient condition forHυm,z to be in the class Hλ,β is
(28)λυm3+4υλ+υ+λm2+1+2λ1+υ+υ1−βm≤1−βe−m.Also, condition (28) is necessary and sufficient for Hυ∗m,z to be in the class H¯λ,β.Proof.
According to Theorem2, we need to show that
(29)∑k=2∞k−1λk+1+1−β1+vk−1mk−1k−1!e−m≤1−β.
Thus,(30)∑k=2∞k−1λk+1+1−β1+υk−1mk−1k−1!e−m=e−m∑k=2∞k−1λk−2+1+2λ+1−βmk−1k−1!+e−m∑k=2∞υλk−1k−2k−3+2υλk−1k−2mk−1k−1!+e−m∑k=2∞υ1+2λk−1k−2+υ1+2λk−1+υ1−βk−1mk−1k−1!,=υλe−m∑k=4∞mk−1k−4!+4υλ+υ+λe−m∑k=3∞mk−1k−3!+1+2λ1+υ+υ1−βe−m∑k=2∞mk−1k−2!+1−βe−m∑k=2∞mk−1k−1!,=λυm3+4υλ+υ+λm2+1+2λ1+υ+υ1−βm+1−β1−e−m≤1−β,if condition (28) holds. Then, from Theorem 3, it follows that the condition (28) is necessary and sufficient for Hυ∗m,z∈H¯λ,β. Hence, the proof is completed.In Theorem7, if we put υ=0, then, we get the following corollary which was obtained by Murugusundaramoorthy et al. [19].Corollary 8.
The sufficient condition forNm,z to be in the class Hλ,β is
(31)λm2+1+2λm≤1−βe−m.Also, condition (31) is necessary and sufficient for Rm,z to be in the class H¯λ,β.Theorem 9.
The sufficient condition forHυm,z to be in the class Jλ,β is
(32)λυm2+λ+υ1+λm+1−e−m≤1−β.Also, condition (32) is necessary and sufficient for Hυ∗m,z to be in the class J∗λ,β.Proof.
According to Theorem2, we need to show that
(33)∑k=2∞1+λk−11+υk−1mk−1k−1!e−m≤1−β.
Thus,(34)∑k=2∞1+λk−11+υk−1mk−1k−1!e−m=e−m∑k=2∞λυk−1k−2+λυ+λ+υk−1+1mk−1k−1!,=e−mλυ∑k=3∞mk−1k−3!+λ+υ1+λ∑k=2∞mk−1k−2!+∑k=2∞mk−1k−1!,=λυm2+λ+υ1+λm+1−e−m≤1−β,if condition (32) holds. Then, from Theorem 3, it follows that condition (32) is necessary and sufficient for Hυ∗m,z∈J∗λ,β. Hence, the proof is completed.In Theorem9, if we put υ=0, then, we get the following corollary which was obtained by Frasin [20].Corollary 10.
The sufficient condition forNm,z to be in the class Jλ,β is
(35)λm+1−e−m≤1−β.Also, condition (35) is necessary and sufficient for Rm,z to be in the class J∗λ,β.Theorem 11.
The sufficient condition forNm,z to be in the class Xλ,β is
(36)λm2+3λ−βm≤1−βe−m,where 1/3≤λ<1. Also, condition (36) is necessary and sufficient for Rm,z to be in the class X∗λ,β.Proof.
According to Theorem2, we need to show that
(37)∑k=2∞k−1λk+1−1+1−βkmk−1k−1!e−m≤1−β.
Thus,(38)∑k=2∞k−1λk+1−1+1−βkmk−1k−1!e−m=e−m∑k=2∞λk−1k−2+3λ−βk−1+1−βmk−1k−1!,=e−mλ∑k=3∞mk−1k−3!+3λ−β∑k=2∞mk−1k−2!+1−β∑k=2∞mk−1k−1!,=λm2+3λ−βm+1−β1−e−m≤1−β,if condition (36) holds. Then, from Theorem 3, it follows that the condition (36) is necessary and sufficient for Rm,z∈X∗λ,β. Hence, the proof is completed.Theorem 12.
The sufficient condition forHυm,z to be in the class Xλ,β is
(39)λυm3+λ+υ5λ−βm2+3λ−βυ+1+υ1−βm≤1−βe−m,where 1/3≤λ<1. Also, condition (39) is necessary and sufficient for Hυ∗m,z to be in the class X∗λ,β.Proof.
According to Theorem2, we need to show that
(40)∑k=2∞k−1λk+1−1+1−βk1+υk−1mk−1k−1!e−m≤1−β.
Thus,(41)∑k=2∞k−1λk+1−1+1−βk1+υk−1mk−1k−1!e−m=e−m∑k=2∞λυk−1k−2k−3+λ+υ5λ−βk−1k−2+3λ−βυ+1+υ1−βk−1+1−βmk−1k−1!,=e−mλυ∑k=4∞mk−1k−4!+λ+υ5λ−β∑k=3∞mk−1k−3!+3λ−βυ+1+υ1−β∑k=2∞mk−1k−2!+1−β∑k=2∞mk−1k−1!,=λυm3+λ+υ5λ−βm2+3λ−βυ+1+υ1−βm+1−β1−e−m≤1−β, if condition (39) holds. Then, from Theorem 3, it follows that the condition (39) is necessary and sufficient for Hυ∗m,z∈X∗λ,β. Hence, the proof is completed.



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