# Coefficient Bounds for a Certain Family of Biunivalent Functions Defined by Gegenbauer Polynomials

Authors: Isra Al-Shbeil; Abbas Kareem Wanas; Abdelkader Benali; Adriana Cătaş
Journal: Journal of Mathematics
 (2022)

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## Abstract

In the present work, by making use of Gegenbauer polynomials, we introduce and study a certain family ofλ-pseudo bistarlike and λ-pseudo biconvex functions with respect to symmetrical points defined in the open unit disk. We obtain estimates for initial coefficients and solve the Fekete–Szego¨ problem for functions that belong to this family. Furthermore, we give connections to some of the earlier known results.


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## Body

## 1. Introduction

In [1], Legendre studied orthogonal polynomials comprehensively. The importance of orthogonal polynomials for contemporary mathematics as well as for a wide range of their applications in physics and engineering is beyond any doubt. It is well known that these polynomials play an essential role in problems of approximation theory. They occur in the theory of differential and integral equations as well as in mathematical statistics. Their applications in quantum mechanics, scattering theory, automatic control, signal analysis, and axially symmetric potential theory are also known [2, 3]. In practice, Gegenbauer polynomials are a special case of orthogonal polynomials. They are representatively related to typically real functions TR as discovered in [4]. Typically, real functions play an important role in geometric function theory because of the relation TR=co¯SR and its role in estimating coefficient bounds, where SR indicates the family of univalent functions in the unit disk with real coefficients and co¯SR denotes the closed convex hull of SR.On this subject in geometric function theory, so-called Fekete–Szegö type inequalities (or problems) estimate some upper bounds fora3−μa22 for holomorphic univalent functions. Its origin was in the disproof by Fekete and Szego of the 1933 conjecture of Littlewood and Paley that the coefficients of odd univalent functions are bounded by unity (see [5])We considerA, the set of functions f, which are holomorphic in the open unit disk U=z∈ℂ:z<1, having the following form:(1)fz=z+∑n=2∞anzn.LetS be denoted as the subfamily of A consisting of the functions which are univalent in U.According to the Koebe one-quarter theorem [6], each function f∈S has an inverse f−1, which fulfills(2)f−1fz=z,z∈U,and(3)ff−1w=w,w<r0f,r0f≥14,where(4)gw=f−1w=w−a2w2+2a22−a3w3−5a23−5a2a3+a4w4+⋯.We say that a functionf∈A is biunivalent in U if both f and f−1 are univalent in U. Let Σ indicate the class of biunivalent functions in U given by (1). Starting with Srivastava et al. pioneering work [7] on the subject, a large number of works related to the subject have been presented (see, for example [4, 8–24]). We notice that the family Σ is not empty. Some examples of functions in the class Σ are(5)z1−z,12log1+z1−zand−log1−z,with the corresponding inverse functions(6)w1+w,e2w−1e2w+1,ew−1ew,respectively. We recall here other common examples of functions that are not members of Σ, namely,(7)z−z22,andz1−z2.So far, the coefficient estimate problem for each of the following Taylor–Maclaurin coefficients:(8)an,n=3,4,…,for functions f∈Σ is still an open problem.We say that a functionf∈S is starlike with respect to symmetrical points if (see [25])(9)Rezf′zfz−f−z>0,z∈U.The subset of all such functions is denoted bySs∗.The family of starlike functions with respect to symmetrical points obviously includes the family of convex functions with respect to symmetrical pointsCs, satisfying the following condition:(10)Rezf′z′fz−f−z′>0,z∈U.We say that a functionf∈A is λ-pseudostarlike function in U if (see [26])(11)Rezf′zλfz>0,z∈U,λ≥1.We consider two functionsf and g that are holomorphic in U. We say that the function f is subordinate to g if there exists a Schwarz function w holomorphic in U with w0=0 and wz<1z∈U such that fz=gwz. This subordination is denoted by f≺g or fz≺gzz∈U. It is well known that (see [27]) if the function g is univalent in U, then f≺g if and only if f0=g0 and fU⊂gU.Recently, Amourah et al. [28] have studied Gegenbauer polynomials Hδz,t, which are given by the following recurrence relation.For a nonzero real constantδ, a generating function of Gegenbauer polynomials is defined by(12)Hδz,t=11−2tz+z2δ,where t∈−1,1 and z∈U. For fixed t, the function Hδ is holomorphic in U, so it can be expanded in a Taylor series, and note that if t=cosβ, where β∈−π/3,π/3, then(13)Hδz,t=11−2tz+z2δ=∑n=0∞Gnδtzn,where Gnδt is the Gegenbauer polynomial of the degree n.Obviously,Hδ generates nothing when δ=0. Thus, the generating function of the Gegenbauer polynomial is set to be(14)H0z,t=1−log1−2tz+z2=∑n=0∞Gn0tzn.Furthermore, it is worth to mention that a normalization ofδ to be greater than −1/2 is desirable [3, 29]. Gegenbauer polynomials can also be defined by the following recurrence relations:(15)Gnδt=122tn+δ−1Gn−1δt−n+2δ−2Gn−1δt.The initial values are expressed as(16)G0δt=1,G1δt=2δt,G2δt=2δδ+1t2−δ.Remark 1.
By choosing the particular values ofδ, the Gegenbauer polynomial Gnδt reduces to well-known polynomials. These special cases are as follows:(1)
Takingδ=1, we obtain Chebyshev polynomials(2)
Takingδ=1/2, we obtain Legendre polynomials


## 2. Main Results

This section starts with defining the new familySΣλ,γ,t,δ.Definition 1.
We consider thatλ≥1, 0≤γ≤1 , t∈1/2,1 and that δ is a nonzero real constant. A function f∈Σ is said to be in the family SΣλ,γ,t,δ if both f and its inverse g=f−1 given by (4) fulfill the following subordinations:(17)2zf′zλfz−f−zγ2zf′z′λfz−f−z′1−γ≺Hδz,t=11−2tz+z2δ,(18)2wg′wλgw−g−wγ2wg′w′λgw−g−w′1−γ≺Hδw,t=11−2tw+w2δ.
In particular, if we chooseδ=1 and λ=1 in Definition 1, the family SΣλ,γ,t,δ becomes the family DΣsγ,t, which Wanas defined in [30].
If we chooseδ=1 in Definition 1, the family SΣλ,γ,t,δ reduces to the family TΣsλ,γ,t, which Wanas gave in [31]. As a consequence, we will generalize, in the following theorem. his main result given in [31, Theorem 2.1]Theorem 1.
Let f, a holomorphic function, given by (1). If f is in the family SΣλ,γ,t,δ, where λ≥1, 0≤γ≤1, t∈1/2,1, and δ≠0∈ℝ, then the following are the upper bonds for the Taylor–Maclaurin coefficients a2 and a3:(19)a2≤δt2δtδ2t2λ+2γ−3−δλ2γ−222t2−1,a3≤δ2t2λ2γ−22+2δt3λ−13−2γ.Proof.
We suppose thatf is an element of the family SΣλ,γ,t,δ, and by Definition 1, f and its inverse g are biunivalent functions of Σ from U to U, satisfying subordinations (17) and (18), and hence, we have two holomorphic functions:(20)u=∑n=1∞unzn, andv=∑n=1∞vnwn,forz,w∈Uand(21)uz<1,andvw<1,forz,w∈U.
We have(22)2zf′zλfz−f−zγ2zf′z′λfz−f−z′1−γ=∑n=0∞Gnδtunz=1+G1δtu1z+G1δtu2+G2δtu12z2+⋯,(23)2wg′wλgw−g−wγ2wg′w′λgw−g−w′1−γ=∑n=0∞Gnδtvnw=1+G1δtv1w+G1δtv2+G2δtv12w2+⋯.
Since the holomorphic functionsu and v are from U to U with u0=v0=0, uz<1 and vw<1 for z,w∈U, and hence, we have the well-known fact(24)un≤1,andvn≤1,foralln∈ℕ.
By comparing corresponding coefficients in (22) and (23), we obtain(25)−2λγ−2a2=G1δtu1,(26)2λ2γ−22+λ3γ−4a22+3λ−13−2γa3=G1δtu2+G2δtu12,(27)2λγ−2a2=G1δtv1.(28)2λ2γ−22+λ5−3γ+2γ−3a22+3λ−12γ−3a3=G1δtv2+G2δtv12.
It follows from (25) and (27) that(29)u1=−v1,(30)8λ2γ−22a22=G1δt2u12+v12.
By adding (26) to (28), we obtain(31)22λ2γ−22+λ+2γ−3a22=G1δtu2+v2+G2δtu12+v12.
By eliminatingu12+v12 from (30) and (31), we obtain(32)22λ2γ−221−2G2δtG1δt2+λ+2γ−3a22=G1δtu2+v2.
Using (16), (24) and (32), we reach the first inequality(33)a2≤δt2δtδ2t2λ+2γ−3−δλ2γ−222t2−1.
Next, by subtracting (28) from (26), we obtain(34)23λ−13−2γa3−a22=G1δtu2−v2+G2δtu12−v12.
Thus, in view of (29) and (30), (34) gives(35)a3=G1δt28λ2γ−22u12+v12+G1δt23λ−13−2γu2−v2.
Finally, by applying (16), we complete the proof by(36)a3≤δ2t2λ2γ−22+2δt3λ−13−2γ.
In particular, Theorem1 gives the following corollaries for some specific values of γ,δ, and λ.
Ifγ=1, then the family SΣλ,γ,t,δ=SΣλ,1,t,δ, the class of λ-pseudo bistarlike functions with respect to symmetrical points.Corollary 1.
Forλ≥1, t∈1/2,1, and δ is a nonzero real constant, let f∈A be in the family SΣλ,1,t,δ. Then,(37)a2≤δt2δtδ2t2λ−1−δλ22t2−1.
And we obtain(38)a3≤δ2t2λ2+2δt3λ−1.Ifγ=0, then the family SΣλ,γ,t,δ=SΣλ,t,δ, the class of λ-pseudo biconvex functions with respect to symmetrical points.Corollary 2.
Forλ≥1, t∈1/2,1, and δ is a nonzero real constant, let f∈A be in the family SΣλ,t,δ. Then,(39)a2≤δt2δtδ2t2λ−3−4δλ22t2−1.
And we obtain(40)a3≤δ2t24λ2+2δt33λ−1.Ifγ=0, Theorem 1 gives the following corollary by Wanas [31] for the well-known family TΣsλ,γ,t.Corollary 3 (see [31]).
Forλ≥1, 0≤γ≤1, and t∈1/2,1, let f∈A be in the family TΣsλ,γ,t. Then,(41)a2≤t2tλ+2γ−3t2−λ2γ−222t2−1.And we obtain(42)a3≤t2λ2γ−22+2t3λ−13−2γ.Ifδ=λ=1, Theorem 1 the following corollary by Wanas [30] for the well-known family DΣsγ,t.Corollary 4 (see [30]).
For0≤γ≤1 and t∈1/2,1, let f∈A be in the family DΣsγ,t. Then,(43)a2≤t2t2γ−1t2−γ−222t2−1.And we obtain(44)a3≤t2γ−22+t3−2γ.Furthermore, ifδ=λ=1, Corollary 2 gives the following corollary by Wanas and Majeed [32] for the well-known family FΣsct.Corollary 5 (see [32]).
Fort∈1/2,1, let f∈A be in the family FΣsct. Then,(45)a2≤tt2−5t2.And we obtain(46)a3≤t3t+412.In the next theorem, we present the “Fekete–Szegö problem” for the familySΣλ,γ,t,δ.Theorem 2.
Forλ≥1, 0≤γ≤1, and t∈1/2,1, μ∈ℝ, and δ is a nonzero real constant, let f∈A be in the family SΣλ,γ,t,δ. Then,(47)a3−μa22≤2tδ3λ−13−2γ;forμ−1≤δ2t2λ+2γ−3−δλ2γ−222t2−1δ2t23λ−13−2γ4t3δ3μ−1δ2t2λ+2γ−3−δλ2γ−222t2−1;forμ−1≥δ2t2λ+2γ−3−δλ2γ−222t2−1δ2t23λ−13−2γ.Proof.
In the light of (32) and (34), we deduce that(48)a3−μa22=1−μG1δt3u2+v222λ2γ−22+λ+2γ−3G1δt2−4λ2γ−22G2δt+G1δtu2−v223λ−13−2γ=G1δtψμ+123λ−13−2γu2+ψμ−123λ−13−2γv2,where(49)ψμ=G1δt21−μ22λ2γ−22+λ+2γ−3G1δt2−4λ2γ−22G2δt.
According to (16), we deduce that(50)a3−μa22≤2tδ3λ−13−2γ,0≤ψμ≤123λ−13−2γ,4tδψμ,ψμ≥123λ−13−2γ.
After some computations, we obtain(51)a3−μa22≤2tδ3λ−13−2γ;forμ−1≤δ2t2λ+2γ−3−δλ2γ−222t2−1δ2t23λ−13−2γ4t3δ3μ−1δ2t2λ+2γ−3−δλ2γ−222t2−1;forμ−1≥δ2t2λ+2γ−3−δλ2γ−222t2−1δ2t23λ−13−2γ
In particular, puttingμ=1 in Theorem 2, we conclude the following result.Corollary 6.
Forλ≥1, 0≤γ≤1, and t∈1/2,1, and δ is a nonzero real constant, let f∈A be in the family SΣλ,γ,t,δ. Then,(52)a3−a22≤2tδ3λ−13−2γ.


## 3. Conclusion

In the present work, we obtain a familySΣλ,γ,t,δ of λ-pseudo bistarlike and λ-pseudo biconvex functions with respect to symmetrical points defined by Gegenbauer polynomials. We generated Taylor–Maclaurin coefficient inequalities of functions belonging to this family and viewed the famous Fekete–Szegö problem. Furthermore, by specifying parameters, consequences of this family are mentioned.



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