# On Certain Properties of a Univalent Function Associated with Beta Function

Authors: Matthew Olanrewaju Oluwayemi; Sunday Olufemi Olatunji; Temitope O. Ogunlade
Journal: Abstract and Applied Analysis
 (2022)

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## Abstract

Beta function has some applications in differential equations and other areas of sciences and engineering where certain definite integrals are used. However, its applications to univalent functions have not been explored based on the available literature. In this work, therefore, the authors defined a univalent function associated with the beta functionBm,n with m,n>0. Some geometric properties of the function are discussed.


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## Body

## 1. Introduction and Preliminaries

LetS denote the class of functions of the form
(1)fz=z+∑k=2∞akzk,which are analytic in the open unit diskU=z∈ℂ:z<1 and normalised by f0=0 and f′0=1 and let T denote the subclass of S consisting of functions whose nonzero coefficients, in the second on, are negative which is defined by
(2)fz=z−∑k=2∞akzk;ak>0.See [1–4].Quite a number of life problems resulted from differential equations when formulated mathematically. These equations come in the form of partial differential equations (PDE) and ordinary differential equations (ODE). The ODE can also come in different orders such as first order or second order depending on the number of factors. ODEs could be linear equations or nonlinear equations. There are different methods of analytical solutions of the second−order differential equations and the peculiarity of the differential equations matter. For linear ODE, for instance, the most effective method of obtaining the solutions once the variable coefficients are analytic functions is the power series solution. They are called power series methods because the solutions come in certain series forms. The series solution is effective for most differential equations. Special functions such as Bessel equation, Hermite equation, Frobenius equation, hypergeometric equation, Laguerre equation, and Legendre equation are in the form of differential equations. Other special functions such as Chebyshev polynomials, Hermite polynomial, Legendre polynomial, factorial function, gamma function, and beta function are also useful techniques of solving problems in mathematics. Both beta and gamma functions are called the Euler’s integrals. A beta function can be in form of polynomial, polar, or the gamma function. It is often referred to as the Eulerian integral of the first kind while the gamma function is the Eulerian integral of the second kind.The beta function denoted byBm,n is defined by
(3)Bm,n=∫01xm−11−xn−1dx,which converges for m>0 and n>0. There are three cases of beta function which include:
(1)
Whenm is a positive integer(2)
Whenn is a positive integer(3)
When bothm and n are positive integersUsing case 3, we can express a beta function in the form of the gamma function as follows:(4)Bm,n=ΓmΓnΓm+n=m−1!n−1!m+n−1!=ΓnΓmΓn+m=Bn,m,which indicates the symmetric property of the beta function [5, 6].By ratio test, let(5)ak=m−1!n−1!m+n−1!andak+1=n!m−1!m+n!.Then,(6)ak+1ak=n!m−1!m+n!×m+n−1!m−1!n−1!=nn−1!m+n−1!m+nm+n−1!n−1!=nm+n,converges if
(7)Limk⟶∞nm+n<1.That is,m>0 and diverges for
(8)Limk⟶∞nm+n>1.That is,0>m or m<0 which implies that Bm,n converges only for case 3 and diverges for either case 1 or case 2.Supposeρ=n/m+n, then the radius of convergence 1/ρ=m+n/n=m/n+1 or 1+m/n. There are three possible cases of the radius of convergence:
(1)
Whenm=n; then, the radius of convergence is 2(2)
Whenm>n; then, the radius of convergence is greater than 2(3)
Whenm<n; then, the radius of convergence is less than 2In general, we conclude that the radius of convergence,r is such that r∈1,μ where μ≥2. Then, we consider the interval of convergence is −2,2. We consider a case of n,m∈ℕ.Supposem and n are distinct values, that is, m≠n. Then, we have that
(9)fm,nz=z+∑k=2∞ΓmΓnΓm+nzk=z+∑k=2∞m−1!n−1!m+n−1!zk;m,n>0,is a normalised univalent function which belongs to classS with beta function as its coefficients. Hence,
(10)fm,nz=z+112z2+160z3+1280z4+11260z5+15544z6+148048z7+125740z8+1437580z9+1461890z10+⋯.Ifm=n; then, equation (9) reduces to
(11)fm,nz=z+∑k=2∞Γn2Γ2nzk=z+∑k=2∞n−1!22n−1!zk;n>0.From equations (9) and (11), we have that ak=Bm,n and ak=Bn,n as the coefficients of (9) and (11), respectively. In addition,
(12)Fm,nz=2z−fm,nz=z−∑k=2∞m−1!n−1!m+n−1!zk,belongs to the class T with negative coefficients so that fm,nz∈S and Fm,nz∈T.Let(13)Fm,nz=z−∑k=2∞m−1!n−1!m+n−1!zk.Equations (9) an (13) are special case of (1) and (2), respectively.Givenm,n>0, let
(14)ak=m−1!n−1!m+n−1!;k≥1,where m≠n. Then, a1,2=0!1!/2!=1/2. But rather, we consider the case where m=n and give the coefficients as follows:
(15)a1=B1,1=0!0!1!=1,(16)a2=B2,2=1!1!3!=16,(17)a3=B3,3=2!2!5!=2.2!5.4.3.2!=15.3.2=130,(18)a4=B4,4=3!3!7!=6.3!7.6.5.4.3!=17.5.4=1140,(19)a5=B5,5=4!4!9!=24.4!9.8.7.6.5.4!=17.6.5.3=1630,(20)a6=B6,6=5!5!11!=120.5!11.10.9.8.7.6.5!=111.9.8.7.2=111088,(21)a7=B7,7=6!6!13!=720.6!13.12.11.10.9.8.7.6!=113.11.8.7.4.3=196096,(22)a8=B8,8=7!7!15!=5040.7!15.14.13.10.9.8.7!=115.13.8.3=151480,(23)a9=B9,9=8!8!17!=40320.8!17.16.15.14.13.12.11.10.9.8!=117.13.11.10.9=1240669,(24)a10=B10,10=9!9!19!=120.5!11.10.9.8.7.6.5!=111.9.8.7.2=1923780.Ifak=am,n=Bm,n, the table below shows the possible 1st−10thcoefficients of a normalised univalent function associated with the Eulerian integral of the first kind (Table 1). It is obvious from the below table that ak≤1 for all k≥1.Table 1First ten coefficients of beta functionak=am,nCase 1:m<nCase 2:m>nCase 3:m=na1=am,n11a1,1=1a2=am,na2,3=1/12a3,2=1/12a2,2=1/6a3=am,na3,4=1/60a4,3=1/60a3,3=1/30a4=am,na4,5=1/280a5,4=1/280a4,4=1/140a5=am,na5,6=1/1260a6,5=1/1260a5,5=1630a6=am,na6,7=1/5544a7,6=1/5544a6,6=111088a7=am,na7,8=1/48048a8,7=1/48048a7,7=1/96096a8=am,na8,9=1/25740a9,8=1/25740a8,8=1/51480a9=am,na9,10=1/437580a10,9=1/48048a9,9=1/218790a10=am,na10,11=1/461890a11,10=1/461890a10,10=1/923780Hence, equation (9) becomes
(25)f1,1z=z+z2+z3+⋯=z+∑k=2∞zk.Consider the special case of (9) with ak=am,n such that m=n, we have that
(26)fn,nz=z+16z2+130z3+1140z4+1630z5+111088z6+196096z7+151480z8+1218790z9+1923780z10+⋯,(27)fn,nz=z+∑k=2∞n−1!22n−1!zk;n>0,(28)Fn,nz=z−∑k=2∞n−1!22n−1!zk;n>0.In [2, 7], the authors introduced a generalised multiplier transformation as a tool to define a subclass of univalent functions. The application of the multiplier transformation was considered in [8]. One of the interests in complex analysis is that every analytic function has a power series expansion. This further encourages the use of a normalised univalent functions of the form (1); and in this work, we consider a normalised univalent function whose coefficients are in terms of the Eulerian integral of the first kind. Altunhan and Eker in [9] earlier explored subclasses of univalent functions with coefficients in Poisson distribution series while many other authors have also considered various forms of special functions, see [4, 10, 11] for examples.


## 2. Main Results

Theorem 1.
Let the functionfz be defined by (13). Then,
(29)∑k=2∞km−1!n−1!m+n−1!≤1.Proof.
Letfz∈T, then
(30)f′z=1−∑k=2∞km−1!n−1!m+n−1!zk−1,is nonnegative. That is,
(31)f′r=1−∑k=2∞km−1!n−1!m+n−1!rk−1>0;z=randk≥2.
The sequenceak=km−1!n−1!/m+n−1!k=2∞ is a monotonic decreasing sequence such that ∑k=2∞ak⟶1 as k⟶∞ and by letting z=r⟶1, we have that
(32)f′r=1−∑k=2∞km−1!n−1!m+n−1!>0.
Sincem,n>0. Then,
(33)∑k=2∞km−1!n−1!m+n−1!<1,for k≥2.Theorem 2.
Let the function(34)Fm,n=z−∑k=2∞m−1!n−1!m+n−1!zk,
belong to classT. Then, the function Gm,nz defined as
(35)Gm,nz=∑μ=1jΓμFm,nz;j∈ℕandΓμ≥0,also belongs to class T with
(36)∑μ=1jΓμ=1.Proof.
By definition,(37)Gm,nz=∑μ=1j∑k=2∞Γμm−1!n−1!m+n−1!zk.
SinceFm,n belongs to class T for every m,n>0 and k≥2. Then,
(38)∑k=2∞km−1!n−1!m+n−1!≤1,for every m,n>0 and k>2. Thus,
(39)∑μ=1jk∑k=2∞Γμm−1!n−1!m+n−1!=∑μ=1jΓμ∑k=2∞km−1!n−1!m+n−1!≤∑μ=1jΓμ=1,which completes the proof.Theorem 3.
Let the function(40)Fm,n=z−∑k=2∞m−1!n−1!m+n−1!zk,
be in the classT and let ρ be a real number such that ρ>−1. Then, the function
(41)fm,n=ρ+1zρ∫0zsρ−1fsds,also belongs to the class T.Proof.
From the definition, we can expressFm,n as follows
(42)Fm,n=∑k=2∞ckzk,where
(43)ck=ρ+1ρ+km−1!n−1!m+n−1!.
Thus,(44)∑k=2∞kck=∑k=2∞kρ+1ρ+km−1!n−1!m+n−1!≤∑k=2∞km−1!n−1!m+n−1!≤1,which completes the proof.Theorem 4.
Letρ be a real number such that ρ>−1. If Fm,n∈T, then, the function fm,n defined by (41) is univalent in z<R where
(45)R=infkρ+1ρ+k1/k−1k>2.Proof.
Letfm,n be defined by (13). Then, from (41), it follows that
(46)fm,n=z1−ρzρfm,n′ρ+1=z−∑k=2∞ρ+kρ+1m−1!n−1!m+n−1!zk;ρ>−1.
We now need to show thatfm,n−1<1 in z<R where
(47)fm,n−1≤∑k=2∞kρ+kρ+1m−1!n−1!m+n−1!zk−1.
Hence,fm,n−1<1 provided
(48)∑k=2∞kρ+kρ+1m−1!n−1!m+n−1!zk−1<1.
From (29), we have that (48) is satisfied if
(49)kρ+kρ+1zk−1≤1,which implies that
(50)z≤infkρ+1ρ+k1/k−1k≥2,as required. Thus, the function fm,mz is univalent in z<R with the best estimate
(51)fm,n=z−ρ+1kρ+kzkk>2.


## 3. Application of Fractional Calculus

We now consider an application of fractional calculus earlier introduced by [12, 13] to the univalent function Fm,nm,n>0 involving beta function Bm,n as follows.Definition 5.
The fractional integral of orderσ>0 for a function fm,nz is defined as
(52)Dz−σfm,nz=1Γσ∫0zfsdsz−s1−σ,
where0≤σ<1, fm,nz is an analytic function in a simply connected region of the z− plane containing the origin and the multiplicity of z−sσ−1 is removed by requiring logz−η to be real when z−s>0.Definition 6.
The fractional integral of orderσ>0 for a function fm,nz is defined by
(53)Dzσfm,nz=1Γ1−σddz∫0zfsdsz−sσ,where σ>0, fm,nz is an analytic function in a simply connected region of the z− plane containing the origin and the multiplicity of z−s−σ is removed by requiring logz−η to be real when z−s>0.Definition 7.
Under the hypotheses of Definition6., the fractional derivative of order k+σ is defined by
(54)Dzk+σfm,nz=dkdzkDzσfm,nz,where 0<σ<1 and k∈ℕ0.Theorem 8.
Letfm,nz∈T. Then,
(55)z1+σΓ2+σ1−Γ3Γσ+12Γ3+σz≤Dz−σfm,nz≤z1+σΓ2+σ1+Γ3Γσ+12Γ3+σz.Proof.
Using the above definition and upon simple calculation, we have that(56)Γ2+σz−σDz−σfm,nz=z−∑k=2∞Γk+1Γσ+1Γk+σ+1m−1!n−1!m+n−1!zk=z−∑k=2∞m−1!n−1!m+n−1!zk,setting
(57)Γk+1Γσ+1Γk+σ+1=Φk,(58)am,n=m−1!n−1!m+n−1!=ak,so that km,n is a function of m and n such that m,n≥0. It is obvious that Φk is a strictly monotonic decreasing sequence since Γk+1σ+1<Γk+σ+1 for all values of k≥2. It thus follows that
(59)0<Φk≤Φ2=Γ3Γσ+1Γ3+σ.
Thus,(60)Γ2+σz−σDz−σfm,nz≥Φ2z2∑k=2∞ak≥z−12Φ2z2=z−Γ3Γσ+12Γ3+σz2,(61)Γ2+σz−σDz−σfm,nz≤z+Γ3Γσ+12Γ3+σz2.
Hence,(62)z1+σΓ2+σ1−Γ3Γσ+12Γ3+σz≤Dz−σfm,nz≤z1+σΓ2+σ1+Γ3Γσ+12Γ3+σz,as required.


## 4. Conclusion

In this study, a certain univalent function associated with the Eulerian integral of the first kind, otherwise known as beta function, is established and some of the geometric properties of the functions discussed. The study opens up a new direction of investigation in geometric function theory (GFT).



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