# A New Subclass of Analytic Functions Related to Mittag-Leffler Type Poisson Distribution Series

Authors: Nazek Alessa; B. Venkateswarlu; P. Thirupathi Reddy; K. Loganathan; K. Tamilvanan
Journal: Journal of Function Spaces
 (2021)

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## Abstract

The object of this work is to an innovation of a classk−U~STsℏ,υ,τ,ι,ς in Y with negative coefficients, further determining coefficient estimates, neighborhoods, partial sums, convexity, and compactness of this specified class.


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## Body

## 1. Introduction

LetY=ω:∣ω∣<1 be an open unit disc in C. Consider the analytic class function A that indicates j specified on the unit disk along with normalization
(1)j0=0,j′0=1and has the form
(2)jω=ω+∑n=2∞onωn,indicated by S, the subclass of A lying of functions that are univalent in Y. A function j∈A is stated in k−USTι, and k−UCVι, “the class of k-uniformly starlike functions and convex functions of order ι,0≤ι<1,” if and only if
(3)Rvj′vjv>kvj′vjv−1+ι,k≥0,R1+vj′′vj′v>kvj′′vj′v+ι,k≥0.The classes UCV and UST were introduced by Goodman [1] and studied by Ronning [2]. Due to Sakaguchi [3], the class STs of starlike functions w.r.t. symmetric points are defined as follows.The functionj∈A is stated to be starlike w.r.t. symmetric points in Y(4)⇔R2ωj′ωjω−j−ω>0,ω∈Y.Owa et al. [4] defined the class STsα,ς as complies
(5)R1−ςωj′ωjω−jςω>α,ω∈Y,where 0≤α<1,∣ς∣≤1,ς≠1. Here, STs0,−1=STs and STsα,−1=STsα is named Sakaguchi function of order α.In recent years, binomial distribution series, Pascal distribution series, Poisson distribution series, etc., play important role in GFT. The sufficient ways were innovated for ST, UCV for some special functions in the GFT. By the motivation of the works [5–13], we develop this work.In [14], Porwal, Poisson distribution series, gives a gracious application on analytic functions; it exposed a new way of research in GFT. Subsequently, the authors turned on the distribution series of confluent hypergeometric, hypergeometric, binomial, and Pascal and prevail necessary and sufficient stipulation for certain classes of univalent functions.Lately, Porwal and Dixit [15] innovate Mittag-Leffler type Poisson distribution and prevailed moments, mgf, which is an abstraction of Poisson distribution using the definition of this distribution. Bajpai [16] innovated Mittag-Leffler type Poisson distribution series and discussed about necessary and sufficient conditions.The probability mass function for this is(6)Pℏ,τ,υ;nω=ℏnEτ,υℏΓτn+υ,n=0,1,2,⋯,where
(7)Eτ,υω=∑n=0∞ωnΓτn+υ,τ,υ∈C,Rτ>0,Rυ>0.The series (7) converges for all finite values of ω if Rτ>0,Rυ>0. This suggest that the series Eτ,υℏ is convergent for τ,υ,ℏ>0. For further details of the study, see [17]. It is easy to see that the series (7) are reduced to exponential series for τ=υ=1.A variablex is said to have Poisson distribution if it takes the values 0,1,2,3,⋯ with probabilities e−ℏ,ℏe−ℏ/1!,ℏ2e−ℏ/2!,ℏ3e−ℏ/3!,⋯, respectively, where ℏ is called the parameter.Thus,(8)Px=n=ℏne−ℏn!,n=0,1,2,⋯.This motivates researchers (see [15, 17, 18], etc.) to introduce a new probability distribution if it assumes nonnegative values and its probability mass function is given by (6). It is easy to see that Pℏ,τ,υ;nω given by (6) is the probability mass function because
(9)Pℏ,τ,υ;nω≥0,∑n=0∞Pℏ,τ,υ;nω=1.It is worthy to note that forα=β=1, it reduces to the Poisson distribution.Also note that(10)Eτ,υω=ωΓυEτ,τ+υω.In [18], Chakrabortya and Ong introduced and discussed about the Mittag-Leffler function distribution—a new generalization of hyper-Poisson distribution. The Mittag-Leffler type Poisson distribution series was innovated by Porwal and Dixit [15] and given as
(11)Kℏ,τ,υω=ω+∑n=2∞ℏn−1Γτn−1+υEτ,υℏωn.Equation (11) is a normalization function in S, since Kℏ,τ,υ0=0 and K′ℏ,τ,υ0=1. After that, in [19], Porwal et al. discussed about the geometric properties of (11).Forj∈A given by (2) and lω given by
(12)lω=ω+∑n=2∞bnωn,their convolution, indicated by j∗l, is given by
(13)j∗lω=ω+∑n=2∞onbnωn=l∗jω,ω∈Y.Note thatj∗l∈A.Next, we innovate the convolution operator(14)Iℏ,τ,υjω=Kℏ,τ,υ∗jω=ω+∑n=2∞φℏnτ,υonωn,where φℏnτ,υ=ℏn−1/Γτn−1+υEτ,υℏ.Then, using linear operatorIℏ,τ,υ, we exemplifier a contemporary subclass of functions in A.Definition 1.
Ifj∈A is named in the class k−USTsℏ,υ,τ,ι,ς if for all ω∈Y(15)R1−ςωIℏ,τ,υjω′Iℏ,τ,υjω−Iℏ,τ,υjςω≥k1−ςωIℏ,τ,υjω′Iℏ,τ,υjω−Iℏ,τ,υjςω−1+ι,for k≥0,∣ς∣≤1,ς≠1,0≤ι<1.Moreover, we named thatj∈k−USTsℏ,υ,τ,ι,ς is in the subclass k−U~STsℏ,υ,τ,ι,ς if jω is of the compiling form
(16)jω=ω−∑n=2∞onωn,on≥0,n∈ℕ,ω∈Y.In this work, we analyze the bounds for coefficient, partial sums, and some neighborhood outcomes of the classk−U~STsℏ,υ,τ,ι,ς.To claim our outcomes, we adopt lemmas [20].Lemma 2.
Letw be a complex number. Then, α≤Rw⇔∣w−1+α∣≤∣w+1−α∣.Lemma 3.
Suppose a complex numberw with real numbers α,ι. Then,
(17)Rw>α∣w−1∣+ι⇔Rw1+αeiρ−αeiρ>ι,−π<ρ≤π.


## 2. Coefficient Bounds

Theorem 4.
A functionj given by (16) is in k−U~STsℏ,υ,τ,ι,ς(18)⇔∑n=2∞φℏnτ,υ∣nk+1−jnk+ι∣on≤1−ι,here k≥0,∣ς∣≤1,ς≠1,0≤ι<1 and jn=1+ς+⋯+ςn−1. The result is sharp for jω is
(19)jω=ω−1−ιφℏnτ,υ∣nk+1−jnk+ι∣ωn.Proof.
By Definition1, we have
(20)R1−ςωIℏ,τ,υjωIℏ,τ,υjω−Iℏ,τ,υjςω1+keiρ−keiρ≥ι,−π<ρ≤π.
LetHω=1−ςωIℏ,τ,υjω1+keiρ−keiρIℏ,τ,υjω−Iℏ,τ,υjςω and Kω=Iℏ,τ,υjω−Iℏ,τ,υjςω.
By Lemma2, (20) becomes
(21)Hω+1−ιKω≥Hω−1+ιKω,for0≤ι<1.
But(22)∣Hω+1−ιKω∣=∣1−ς2−ιω−∑n=2∞φℏnτ,υn+jn1−ιonωn−keiρ∑n=2∞φℏnτ,υn−jnonωn∣≥∣1−ς∣2−ι∣ω∣−∑n=2∞φℏnτ,υ∣n+jn1−ι∣on∣ωn∣−k∑n=2∞φℏnτ,υ∣n−jn∣on∣ωn∣.
Also(23)∣Hω−1+ιKω∣=∣1−ς−ιω−∑n=2∞φℏnτ,υn−jn1+ιonωn−keiρ∑n=2∞φℏnτ,υn−jnonωn∣≤∣1−ς∣ι∣ω∣+∑n=2∞φℏnτ,υ∣n−jn1+ι∣on∣ωn∣+k∑n=2∞φℏnτ,υ∣n−jn∣on∣ωn∣.
So(24)∣Hω+1−ιKω∣−∣Hω−1+ιKω∣≥∣1−ς∣21−ι∣ω∣−∑n=2∞φℏnτ,υ∣n+jn1−ι∣+∣n−jn1+ι∣+2k∣n−jn∣on∣ωn∣≥21−ι∣ω∣−∑n=2∞2φℏnτ,υ∣nk+1−jnk+ι∣on∣ωn∣≥0.
Conversely, suppose (18) holds. Then, we have
(25)R1−ςωIℏ,τ,υjω1+keiρ−keiρIℏ,τ,υjω−Iℏ,τ,υjςωIℏ,τ,υjω−Iℏ,τ,υjςω≥ι.
Optingω values on the +ve real axis, where 0≤∣ω∣=r<1, then
(26)R1−ι−∑n=2∞φℏnτ,υn1+keiρ−jnι+keiρonωn−11−∑n=2∞φℏnτ,υjnonωn−1≥0.
SinceR−eiρ≥−∣eiρ∣=−1, then
(27)R1−ι−∑n=2∞φℏnτ,υn1+k−jnι+konrn−11−∑n=2∞φℏnτ,υjnonrn−1≥0.
Taking limitr tends to 1−, we obtain our needed result.Corollary 5.
Ifjω∈k−U~STsℏ,υ,τ,ι,ς, then
(28)on≤11−ι−1φℏnτ,υ∣nk+1−jnk+ι∣,where k≥0,∣ς∣≤1,ς≠1,0≤ι<1 and jn=1+ς+⋯+ςn−1.


## 3. Neighborhood Properties

The notion ofβ-neighbourhood was innovated and studied by Goodman [21] and Ruscheweyh [22].Definition 6.
We define theβ-neighborhood of a mapping j∈A and indicate by Nβj lying of all mappings gω=ω−∑n=2∞bnωn∈Sbn≥0,n∈ℕ satisfies the condition
(29)∑n=2∞φℏnτ,υ∣nk+1−jnk+ι∣1−ιon−bn≤1−β,where k≥0,∣ς∣≤1,ς≠1,0≤ι<1,β≥0 and jn=1+ς+⋯+ςn−1.Theorem 7.
Letjω∈k−U~STsℏ,υ,τ,ι,ς and every real ρ we get ιeiρ−1−2eiρ≠0. For any ε∈ℂ with ∣ε∣<ββ≥0, if j fulfills
(30)jω+εω1+ε∈k−U~STsℏ,υ,τ,ι,ς,then, Nβj⊂k−U~STsℏ,υ,τ,ι,ς.Proof.
It is evident thatj∈k−U~STsℏ,υ,τ,ι,ς(31)⇔1−ςωIℏ,τ,υjω1+keiρ−keiρ+1+ιIℏ,τ,υjω−Iℏ,τ,υjςω1−ςωIℏ,τ,υjω1+keiρ+1−keiρ−ιIℏ,τ,υjω−Iℏ,τ,υjςω<1,where −π≤ρ≤π for some s∈ℂ and ∣s∣=1, we obtain
(32)1−ςωIℏ,τ,υjω1+keiρ−keiρ+1+ιIℏ,τ,υjω−Iℏ,τ,υjςω1−ςωIℏ,τ,υjω1+keiρ+1−keiρ−ιIℏ,τ,υjω−Iℏ,τ,υjςω≠s.
In other words,(33)1−s1−ςωIℏ,τ,υjω1+keiρ−keiρ+1+ι+s−1+keiρ+ι×Iℏ,τ,υjω−Iℏ,τ,υjςω≠0⇒ω−∑n=2∞φℏnτ,υn−jn1+keiρ−skeiρ−sn+jn−jnι1−sιs−1−2sωn≠0.
However,j∈k−U~STsℏ,υ,τ,ι,ς⇔j∗h/ω≠0,ω∈Y−0, where hω=ω−∑n=2∞cnωn and
(34)cn=φℏnτ,υn−jn1+keiρ−skeiρ−sn+jn−jnι1−sιs−1−2s⇒cn≤φℏnτ,υn1+k−jnk+ι1−ι,since jω+εω/1+ε∈k−U~STsℏ,υ,τ,ι,ς; therefore, ω−1jω+εω/1+ε∗hω≠0, which implies
(35)j∗hω1+εω+ε1+ε≠0.
Now, supposej∗hω/ω<β. Then, by (35),
(36)j∗hω1+εω+ε1+ε>∣ε∣−β∣1+ε∣≥0,which contradicts by ∣ε∣<β, and thus, we arrive j∗hω/ω≥β.
Ifgω=ω−∑2≤n≤∞bnωn∈Nβj, then
(37)β−g∗hωω≤j−g∗hωω<∑n=2∞φℏnτ,υ∣n1+k−jnk+ι∣1−ι∣on−bn∣≤β.


## 4. Partial Sums

Theorem 8.
If the functionj is of the form (2) fulfill (18) then
(38)Rjωjmω≥1−1χm+1,(39)χn=1,if2≤n≤m;χm+1,ifm+1≤n≤∞,where
(40)χn=φℏnτ,υ∣n1+k−jnk+ι∣1−ι.
The estimate (38) is sharp, for every m, with
(41)jω=ω+ωm+1χm+1.Proof.
Now, we define℘; we can define
(42)1+℘ω1−℘ω=χm+1jωjmω−1−1χm+1=1+∑2≤n≤monωn−1+χm+1∑n=m+1∞onωn−11+∑n=2monωn−1.
Then, from (42), we attain
(43)℘ω=χm+1∑n=m+1∞onωn−12+2∑n=2monωn−1+χm+1∑n=m+1∞onωn−1,∣℘ω∣≤χm+1∑n=m+1∞on2−2∑n=2mon−χm+1∑n=m+1∞on.
Now,∣℘ω∣≤1 if
(44)2χm+1∑n=m+1∞on≤2−2∑n=2mon⇒∑n=2mon+χm+1∑m+1∞on≤1.
It is enough to prove that the LHS of (44) is bounded above by ∑n=2∞χnon, which implies
(45)∑n=2mχn−1on+∑n=m+1∞χn−χm+1on≥0.
To show that the mapping disposed by (41) gives the exact result, we notice that for ω=reiπ/n,(46)jωjmω=1+ωmχm+1.
Taking limitω tends to 1−, we have
(47)jωjmω=1−1χm+1.
Hence, the proof is completed.Theorem 9.
Ifj of the form (2) which fulfill (18) then
(48)Rjmωjω≥χm+11+χm+1.
The result is sharp with (41).Proof.
Define(49)1+℘ω1−℘ω=1+χm+1jmωjω−χm+11+χm+1=1+∑n=2monωn−1−χm+1∑n=m+1∞onωn−11+∑n=2∞onωn−1,where
(50)℘ω=1+χm+1∑n=m+1∞onωn−1−2+2∑n=2monωn−1−1−χm+1∑n=m+1∞onωn−1,∣℘ω∣≤1+χm+1∑n=m+1∞on2−2∑n=2mon+1−χm+1∑n=m+1∞on≤1.
This last inequality is(51)∑n=2mon+χm+1∑n=m+1∞on≤1.
It is enough to prove that the LHS of (51) is bounded above by ∑n=2∞χnon, which implies
(52)∑2≤n≤mχn−1on+∑n=m+1∞χn−χm+1on≥0.
This completes the proof.Theorem 10.
Ifj of the form (2) fulfill (18) then
(53)Rj′ωjm′ω≥1−m+1χm+1,(54)Rjm′ωj′ω≥χm+11+m+χm+1,where
(55)χn≥1,if1≤n≤mnχm+1m+1,ifm+1≤n≤∞.and χn is given by (40). The computation in (53) and (54) are sharp with (41).Theorem 11.
k−U~STsℏ,υ,ι,ς is a convex and compact subset of T.Proof.
Supposejd∈k−U~STsℏ,υ,τ,ι,ς,
(56)jdω=ω−∑n=2∞∣ad,n∣ωn.
Then, for0≤ψ<1, let j1,j2∈k−U~STsℏ,υ,τ,ι,ς be given by (56). Then,
(57)ξω=ψj1ω+1−ψj2ω=ψω−∑n=2∞a1,nωn+1−ψω−∑n=2∞a2,nωn=ω−∑n=2∞ψa1,n+1−ψa2,nωn,∑n=2∞φℏnτ,υnk+1−jnk+ιψ∣a1,n∣+1−ψ∣a2,n∣≤ψ1−ι+1−ψ1−ι=1−ι.
Then,ξω=ψj1ω+1−ψj2ω∈k−U~STsℏ,υ,τ,ι,ς. Therefore, k−U~STsℏ,υ,τ,ι,ς is convex. Now, we have to show k−U~STsℏ,υ,τ,ι,ς is compact.
Forjd∈k−U~STsℏ,υ,τ,ι,ς,ς∈ℕ and ∣ω∣<r0<r<1, then we arrive
(58)∣jdω∣≤r+∑n=2∞∣ad,n∣rn≤r+∑n=2∞φℏnτ,υnk+1−jnk+ι∣ad,n∣rn≤r+1+rrn.
Therefore,k−U~STsℏ,υ,τ,ι,ς is uniformly bounded.
Letjdω=ω−∑n=2∞∣ad,n∣ωn,ω∈Y,d∈ℕ.
Also, letjω=ω−∑n=2∞onωn. Then, by Theorem 4, we get
(59)∑n=2∞φℏnτ,υnk+1−jnk+ι∣on∣≤1−ι.
Assumingjd⟶j, then we have ad,n⟶on as n⟶∞,d∈ℕ.
Letρn be the array of partial sums of the series
(60)∑n=2∞φℏnτ,υnk+1−jnk+ιon.
Then,ρn is a nondecreasing array and by (59), it is bounded above by 1−ι.
Thus, it is convergent and(61)∑n=2∞φℏnτ,υnk+1−jnk+ι∣ad,n∣=limn→∞ρn≤1−ι.
Therefore,j∈k−U~STsℏ,υ,τ,ι,ς and the class k−U~STsℏ,υ,τ,ι,ς is closed.



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