# Generalized Distribution and Its Geometric Properties Associated with Univalent Functions

Authors: Saurabh Porwal
Journal: Journal of Complex Analysis
 (2018)

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## Abstract

The purpose of the present paper is to introduce a generalized discrete probability distribution and obtain some results regarding moments, mean, variance, and moment generating function for this distribution. Further, we show that for specific values it reduces to various well-known distributions. Finally, we give a beautiful application of this distribution on certain analytic univalent functions.


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## Body

## 1. Introduction

Let the series∑n=0∞an, where an≥0,∀n∈N is convergent and its sum is denoted by S, that is, (1)S=∑n=0∞an.Now, we introduce the generalized discrete probability distribution whose probability mass function is (2)pn=anS,n=0,1,2,….Obviously p(n) is a probability mass function because p(n)≥0 and ∑npn=1.Now, we introduce the series(3)ϕx=∑n=0∞anxn.From (1) it is easy to see that the series given by (3) is convergent for x<1 and for x=1 it is also convergent.Definition 1.
IfX is a discrete random variable which can take the values x1,x2,x3,… with respective probabilities p1,p2,p3,… then expectation of X, denoted by E(X), is defined as (4)EX=∑n=1∞pnxn.Definition 2.
Therth moment of a discrete probability distribution about X=0 is defined by (5)μr′=EXr.Here μ1′ is known as mean of the distribution and variance of the distribution is given by μ2′-μ1′2.Moments about the Origin(1)(6)μ1′=∑n=0∞npn=∑n=0∞nanS=1S∑n=1∞nan=ϕ′1S.(2)(7)μ2′=∑n=0∞n2pn=∑n=0∞n2anS=1S∑n=0∞nn-1+nan=1S∑n=2∞nn-1an+∑n=1∞nan=1Sϕ′′1+ϕ′1.(3)(8)μ3′=∑n=0∞n3pn=∑n=0∞nn-1n-2+3nn-1+nanS=1S∑n=3∞nn-1n-2an+3∑n=2∞nn-1an+∑n=1∞nan=1Sϕ′′′1+3ϕ′′1+ϕ′1.(4)(9)μ4′=∑n=0∞n4pn=∑n=0∞nn-1n-2n-3+6nn-1n-2+7nn-1+nanS=1Sϕiv1+6ϕ′′′1+7ϕ′′1+ϕ′1.Definition 3.
The mean of the distribution is given by(10)Mean=μ1′=ϕ′1S.Definition 4.
The variance of the distribution is given by(11)Variance=μ2′-μ1′2=1Sϕ′′1+ϕ′1-ϕ′12S.Definition 5.
The moment generating function (m.g.f.) of a random variableX is denoted by MX(t) and defined by (12)MXt=EetX.Theorem 6.
The moment generating function of generalized discrete probability distribution is given by(13)MXt=ϕetS.Proof.
One has(14)MXt=EetX=∑n=0∞etnpn=∑n=0∞etnanS=ϕetS.


## 2. Some Consequences

By specializing the values ofan, we obtain the following well-known discrete probability distributions.(1)
Letan=ρB(n,ρ+1), where ρ>0,n∈N=1,2,3,…; then it reduces to Yule–Simon Distribution [1].(2)
Letan=pn/n, where 0<p<1,n∈N=1,2,3,…; then it reduces to Logarithmic Distribution [1].(3)
Letan=mn/n! and then it reduces to Poisson distribution [1, 2].(4)
Letan=Cnkp/1-pn, n=0,1,2,…,k; then it reduces to Binomial Distribution [1, 2].(5)
Letan=CnkB(n+α,n-k+β), n=0,1,2,…,k; then it reduces to Beta-Binomial Distribution [1].(6)
Letan=1-pn, n=0,1,2,…; then it reduces to Geometric Distribution [1].(7)
Letan=1/ns, where s∈1,∞ and n∈N; then it reduces to Zeta Distribution [1].(8)
Let(15)an=p,ifn=11-p,ifn=0and then it reduces to Bernoulli Distribution [1, 2].


## 3. Applications on Certain Classes of Univalent Functions

LetA denote the class of functions f of the following form:(16)fz=z+∑n=2∞Anzn,which are analytic in the open unit disk U={z:z∈C and |z|<1}. As usual, by S we shall represent the class of all functions in A which are univalent in U and further, we denote T be the subclass of S consisting of functions of the following form: (17)fz=z-∑n=2∞Anzn.In 1988, Altintas and Owa [3] introduced the class T(λ,α), (α(0≤α<1), λ(0≤λ<1)), being the subclass of T consisting of functions which satisfy the following condition: (18)Rzf′zλzf′z+1-λfz>α,z∈U.Also, they introduceC(λ,α), (α(0≤α<1), λ(0≤λ<1)), being the subclass of T consisting of functions which satisfy the following condition: (19)Rf′z+zf′′zf′z+λzf′′z>α,z∈U.By using (18) and (19) we have (20)fz∈Cλ,α⟺zf′z∈Tλ,α.It is easy to see that for λ=0 the classes T(λ,α) and C(λ,α) reduce to the classes of starlike functions of order α(0≤α<1), T∗(α) and the convex functions of order α(0≤α<1), C(α), respectively, studied by Silverman [4].Mostafa [5] and Porwal and Dixit [6] obtain certain conditions for hypergeometric functions and generalized Bessel functions, respectively, for these classes.Now, we introduce a power series whose coefficients are probabilities of the generalized distribution:(21)Kϕz=z+∑n=2∞an-1Szn.Further, we define the following function: (22)TKϕz=z-∑n=2∞an-1Szn.The convolution (or Hadamard product) of two seriesf(z)=∑n=0∞anzn and g(z)=∑n=0∞bnzn is defined as the power series: (23)f∗gz=∑n=0∞anbnzn.Next, we introduce the convolution operatorTKϕ(f,z) for functions f of the form (17) as follows: (24)TKϕf,z=TKϕz∗fz=z-∑n=2∞Anan-1Szn.Recently, Porwal [7] introduced a Poisson distribution series whose coefficients are probabilities of Poisson distribution and established a correlation between Statistics and Geometric Function Theory which opened up a new direction of research. After the appearance of this paper some researchers (e.g., Ahmad et al. [8], Murugusundaramoorthy [9], and Porwal and Kumar [10]) obtained some new and interesting results by using Hypergeometric Distribution, Poisson Distribution, and Confluent Hypergeometric Distribution. In the present paper motivated with the above-mentioned work, we obtain necessary and sufficient conditions for TKϕ(z) and TKϕ(f,z) in the classes T(λ,α) and C(λ,α).To prove our main theorem, we need the following lemma.Lemma 7 (see [11]).
Iff∈Rτ(A,B) is of the form (16) then (25)An≤A-Bτn,n∈N∖1.The bounds given in (25) are sharp.Lemma 8 (see [3]).
A functionf(z) defined by (17) is in class T(λ,α), if and only if (26)∑n=2∞n-λαn-α+λαan≤1-α.Lemma 9 (see [3]).
A functionf(z) defined by (17) is in class C(λ,α), if and only if (27)∑n=2∞nn-λαn-α+λαan≤1-α.Theorem 10.
IfTKϕz that is of form (22) is in class Tλ,α, if and only if (28)1S1-αλϕ′1+1-αϕ1-ϕ0≤1-α.Proof.
Since(29)TKϕz=z-∑n=2∞an-1Szn,according to Lemma 8, we have to show that (30)∑n=2∞n1-λα-α1-λan-1S≤1-α.Now(31)∑n=2∞n1-αλ-α1-λan-1S=1S1-αλ∑n=2∞n-1an-1+1-α∑n=2∞an-1=1S1-αλ∑n=1∞nan+1-α∑n=1∞an=1S1-αλϕ′1+1-αϕ1-ϕ0≤1-α.
This completes the proof of Theorem10.Theorem 11.
IfTKϕ(z) that is of form (22) is in class C(λ,α), if and only if (32)1S1-αλϕ′′1+3-2αλ-αϕ′1+1-αϕ1-ϕ0≤1-α.Proof.
Since(33)TKϕz=z-∑n=2∞an-1Szn,according to Lemma 9, we have to prove that (34)∑n=2∞nn1-λα-α1-λan-1S≤1-α.Now (35)∑n=2∞nn1-αλ-α1-λan-1S=1S1-αλ∑n=2∞n-1n-2an-1+3-2αλ-α∑n=2∞n-1an-1+1-α∑n=2∞an-1=1S1-αλϕ′′1+3-2αλ-αϕ′1+1-αϕ1-ϕ0≤1-α.
Thus the proof of Theorem11 is established.Theorem 12.
Iff∈Rτ(A,B) is of form (17) and the operator TKϕ(f,z) defined by (24) is in the class C(λ,α), if and only if (36)A-BτS∑n=2∞1-αλϕ′1+1-αϕ1-ϕ0≤1-α.Proof.
By Lemma9, it suffices to prove that (37)P1=∑n=2∞nn-λαn-α+λαan≤1-α.Since f∈Rτ(A,B) then by using Lemma 7 we have (38)an≤A-Bτn.Hence (39)P1≤A-BτS∑n=2∞n1-αλ-α1-λan-1=A-BτS∑n=2∞n-11-αλ+1-αan-1=A-BτS∑n=2∞1-αλϕ′1+1-αϕ1-ϕ0≤1-α.Thus the proof of Theorem 12 is established.


## 4. An Integral Operator

In this section, we introduce an integral operatorTGϕ(z) as follows: (40)TGϕz=∫0zTKϕttdt,and we obtain a necessary and sufficient condition for TGϕ(z) belonging to class C(λ,α)Theorem 13.
IfTKϕ(z) is defined by (22), then TGϕ(z) defined by (40) is in class C(λ,α), if and only if (28) satisfies.Proof.
Since(41)TGϕz=z-∑n=2∞an-1nSznby Lemma 9, we have to prove that (42)∑n=2∞nn1-λα-α1-λan-1nS≤1-α.Now (43)∑n=2∞nn1-λα-α1-λan-1nS=∑n=2∞n1-λα-α1-λan-1S=1S1-αλ∑n=2∞n-1an-1+1-α∑n=2∞an-1=1S1-αλ∑n=1∞nan+1-α∑n=1∞an=1S1-αλϕ′1+1-αϕ1-ϕ0≤1-α.
This completes the proof of Theorem13.



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*Source: 8654506-2018-06-03.xml*