# Consolidation of a Certain Discrete Probability Distribution with a Subclass of Bi-Univalent Functions Involving Gegenbauer Polynomials

Authors: Ala Amourah; Mohammad Alomari; Feras Yousef; Abdullah Alsoboh
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this work, we introduce and investigate a new subclass of analytic bi-univalent functions based on subordination conditions between the zero-truncated Poisson distribution and Gegenbauer polynomials. More precisely, we will estimate the first two initial Taylor–Maclaurin coefficients and solve the Fekete–Szegö functional problem for functions belonging to this new subclass.


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## Body

## 1. Introduction

LetA denote the class of all analytic functions f defined on the open unit disk U=z∈ℂ:z<1 and normalized by the conditions f0=0 and f′0=1. Thus, each f∈A has a Taylor–Maclaurin series expansion of the form(1)fz=z+∑n=2∞anzn,z∈U.Let the functionsf and g be in the class A. We say that the function f is subordinate to g, written as f≺g, if there exists a Schwarz function w, which is analytic in U with(2)w0=0,wz<1z∈U,such that(3)fz=gwz.Further, letS denote the class of all functions f∈A that are univalent in U. It is worth to mention that if the function g is univalent in U, then the following equivalence holds (see [1]):(4)fz≺gzif and only iff0=g0,and(5)fU⊂gU.It is well known [2] that every function f∈S has an inverse f−1, defined by(6)f−1fz=zz∈U,(7)f−1fw=ww<r0f;r0f≥14,where(8)f−1w=w−a2w2+2a22−a3w3−5a23−5a2a3+a4w4+⋯.A function is said to be bi-univalent inU if both fz and f−1z are univalent in U. Let Σ denote the class of bi-univalent functions in U given by (1). For interesting examples of functions in the class Σ, see [3–10].In probability theory, the zero-truncated Poisson distribution is a certain discrete probability distribution whose support is the set of positive integers, that is, the zero-truncated Poisson distribution is the same as the Poisson distribution without the zero count [11]. This distribution is also known as the conditional Poisson distribution [12] or the positive Poisson distribution [13]. The probability density function of the zero-truncated Poisson distribution is given by(9)PmX=s=msem−1s!,s=1,2,3,…,m>0.Here, let us consider a power series whose coefficients are probabilities of the zero-truncated Poisson distribution, that is,(10)ℙm,z=z+∑n=2∞mn−1em−1n−1!zn,z∈U,where m>0. By the ratio test, the radius of convergence of this series is infinity.Define the linear operatorχ:A⟶A by(11)χmfz=ℙm,z∗fz=z+∑n=2∞mn−1em−1n−1!anzn,z∈U,=z+mem−1a2z2+m22em−1a3z3+⋯,where ∗ denotes the convolution or Hadamard product of two series (see [14]).Orthogonal polynomials have been studied extensively as early as they were discovered by Legendre in 1784 [15]. In mathematical treatment of model problems, orthogonal polynomials arise often to find solutions of ordinary differential equations under certain conditions imposed by the model. The importance of the orthogonal polynomials for the contemporary mathematics, as well as for wide range of their applications in the physics and engineering, is beyond any doubt. Recently, many researchers have been exploring bi-univalent functions associated with orthogonal polynomials [16–24]. Very recently, Amourah et al. [25, 26] considered the generating function for Gegenbauer polynomials Hαx,z, which is given by(12)Hαx,z=11−2xz+z2α,where x∈−1,1, α>−1/2, and z∈U. For fixed x, the function Hα is analytic in U, so it can be expanded in a Taylor series as(13)Hαx,z=∑n=0∞Cnαxzn.where Cnαx is Gegenbauer polynomial of degree n (see [27]).Obviously,Hα generates nothing when α=0. Therefore, the generating function of the Gegenbauer polynomial is set to be(14)H0x,z=1−log1−2xz+z2=∑n=0∞Cn0xzn,for α=0. Moreover, it is worth to mention that a normalization of α to be greater than −1/2 is desirable [28, 29]. Gegenbauer polynomials can also be defined by the following recurrence relations:(15)Cnαx=1n2xn+α−1Cn−1αx−n+2α−2Cn−1αx,with the initial values(16)C0αx=1,C1αx=2αxandC2αx=2α1+αx2−α.Special cases of Gegenbauer polynomialsCnαx are Chebyshev polynomials when α=1 and Legendre polynomials when α=1/2.


## 2. The ClassζΣx,α,μ

In this section, we introduce a new subclass ofΣ involving the new constructed series (10) and Gegenbauer polynomials.Definition 2.1.
A functionf∈Σ given by (1) is said to be in the class ζΣx,α,μ if the following subordinations are satisfied:(17)1−μχmfzz+μχmfz′≺Hαx,z,(18)1−μχmgww+μχmgw′≺Hαx,w,where α>0, μ≥0, x∈1/2,1, and the function g=f−1 is given by (8).Upon specializing the parameterμ, one can get the following new subclass of Σ as illustrated in the following example.Example 2.1.
Ifμ=1, then we have ζΣx,α,1=ζΣx,α, in which ζΣx,α denotes the class of functions f∈Σ given by (1) and satisfying the following conditions.(19)χmfz′≺Hαx,z,(20)χmgw′≺Hαx,w.


## 3. Estimates of the ClassζΣx,α,μ

First, we give the coefficient estimates for the classζΣx,α,μ given in Definition 2.1.Theorem 3.1.
Letf∈Σgiven by (1) belong to the classζΣx,α,μ.Then,(21)a2≤2αxem−12αxm2α21+2μem−1−2α1+μ21+αx2+α1+μ2,(22)a3≤4α2x2em−12m21+μ2+4αxem−1m21+2μ.Proof.
Letf∈ζΣx,α,μ. From Definition 2.1, for some analytic functions w, v such that w0=v0=0 and wz<1, vw<1 for all z,w∈U, then we can write(23)1−μχmfzz+μχmfz′=Hαx,wz,(24)1−μχmgww+μχmgw′=Hαx,vw.
From the equalities (23) and (24), we obtain that(25)1−μχmfzz+μχmfz′=1+C1αxc1z+C1αxc2+C2αxc12z2+⋯,(26)1−μχmgww+μχmgw′=1+C1αxd1w+C1αxd2+C2αxd12w2+⋯.
It is fairly well known that if(27)wz=c1z+c2z2+c3z3+⋯<1,z∈U,(28)vw=d1w+d2w2+d3w3+⋯<1,w∈U,then (see [30])(29)|cj|≤1and|dj|≤1for allj∈ℕ.
Thus, upon comparing the corresponding coefficients in (25) and (26), we have(30)1+μmem−1a2=C1αxc1,(31)1+2μm22em−1a3=C1αxc2+C2αxc12,(32)−1+μmem−1a2=C1αxd1,(33)1+2μm22em−12a22−a3=C1αxd2+C2αxd12.
It follows from (30) and (32) that(34)c1=−d1,(35)21+μ2m2em−12a22=C1αx2c12+d12.
If we add (31) and (33), we get(36)1+2μm2em−1a22=C1αxc2+d2+C2αxc12+d12.
Substituting the value ofc12+d12 from (35) in the right hand side of (36), we deduce that(37)1+2μ−21+μ2em−1C2αxC1αx2m2em−1a22=C1αxc2+d2.
Moreover, using (16), (29), and (37), we find that(38)a2≤2αxem−12αxm2α21+2μem−1−2α1+μ21+αx2+α1+μ2.
Now, if we subtract (33) from (31), we obtain(39)1+2μm2em−1a3−a22=C1αxc2−d2+C2αxc12−d12.
Then, in view of (16) and (35), (39) becomes(40)a3=em−12C1αx22m21+μ2c12+d12+em−1C1αxm21+2μc2−d2.
Thus, applying (16) and (29), we conclude that(41)a3≤4α2x2em−12m21+μ2+4αxem−1m21+2μ.
This completes the proof of Theorem3.1.Making use of the values ofa22 and a3, we prove the following Fekete–Szegö inequality for functions in the class ζΣx,α,μ.Theorem 3.2.
Letf∈Σgiven by (1) belong to the classζΣx,α,μ. Then,a3−ηa22≤4αxem−1/m21+2μ,8α2x3em−121−η/m22α1+2μem−1−21+μ21+αx2+1+μ2,η−1≤δη−1≥δ, where(42)δ=1−1+μ221+αx2−12αx2em−11+2μ.Proof.
From (37) and (39),(43)a3−ηa22=1−ηem−12C1αx3c2+d2m2em−11+2μC1αx2−21+μ2C2αx+em−1C1αxm21+2μc2−d2=C1αxhη+em−1m21+2μc2+C1αxhη−em−1m21+2μd2,where(44)hη=em−12C1αx21−ηm2em−11+2μC1αx2−21+μ2C2αx.
Then, in view of (16), we conclude that(45)a3−ηa22≤2em−1C1αxm21+2μ2C1αxhη0≤hη≤em−1m21+2μ,hη≥em−1m21+2μ,which completes the proof of Theorem 3.2.Corresponding essentially to Example2.1, Theorems 3.1 and 3.2 yield the following consequence.Corollary 3.3.
Letf∈Σgiven by (1) belong to the classζΣx,α,0. Then,(46)a2≤2αxem−12αxm2α2em−1−2α1+αx2+α,a3≤4α2x2em−12m2+4αxem−1m2,(47)a3−ηa22≤4αxem−1m2,8α2x3em−121−ηm22αem−1−21+αx2+1,η−1≤1−21+αx2−12αx2em−1η−1≥1−21+αx2−12αx2em−1.


## 4. Conclusions

In our present investigation, we have introduced a new subclassζΣx,α,μ of normalized analytic and bi-univalent functions associated with the zero-truncated Poisson distribution and Gegenbauer polynomials. For functions belonging to this class, we have derived the estimates of the Taylor–Maclaurin coefficients a2 and a3 and the Fekete–Szegö functional problem. Furthermore, by suitably specializing the parameter μ, one can deduce the result for the subclass ζΣx,α which is defined in Example 2.1.The results presented in this paper would lead to various other new results for the classesζΣx,1,μ for Chebyshev polynomials and ζΣx,1/2,μ for Legendre polynomials.The special families examined in this research paper and linked with zero-truncated Poisson distribution and Gegenbauer polynomials could inspire further research related to other aspects, such as families usingq−derivative operator [31, 32] and bi-univalent function families associated with differential operators [33].



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