# Fekete–Szegö Inequality for Bi-Univalent Functions Subordinate to Horadam Polynomials

Authors: Amnah E. Shammaky; Basem Aref Frasin; Sondekola Rudra Swamy
Journal: Journal of Function Spaces
 (2022)

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## Abstract

Making use of Horadam polynomials, we propose a special family of regular functions of the typegz=z+∑j=2∞djzj which are bi-univalent (or bi-schlicht) in the disc z∈ℂ:z<1. We find estimates on the coefficients d2 and d3 and the functional of Fekete–Szegö for functions in this subfamily. Relevant connections to existing results and new observations of the main result are also presented.


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## Body

## 1. Preliminaries

Letℝ and ℕ:=1,2,3,…=ℕ0\0 be the sets of real numbers and positive integers, respectively. Let ℂ be the set of all complex numbers, and let D denote the disc z∈ℂ:z<1. We denote by A, the set of all regular functions in D that has the series of the form(1)gz=z+∑j=2∞djzj,and S be the set of all members of A that are univalent in D. According to the well-known Koebe theorem (see [1]), every univalent function g has an inverse defined by(2)g−1ω=fω=ω−d2ω2+2d22−d3ω3−5d23−5d2d3+d4ω4+⋯,satisfying z=g−1gz and ω=gg−1ω,ω<r0g,r0g≥1/4,z,ω∈D.A functiong of A is said to be bi-univalent (or bi-schlicht) in D if both g and g−1 are univalent in D. Let Σ stand for the set of bi-univalent functions having form (1). Lewin [2] investigated the family Σ and proved that d2<1.51. Brannan and Clunie [3] claimed that d2<2. Later, Tan [4] obtained initial coefficient estimates for bi-univalent functions. Subsequently, Brannan and Taha [5] examined certain well-known subfamilies of Σ in D. The momentum on the study of bi-univalent function family was gained recently, which is due to the work of Srivastava et al. [6]. This article has revived the topic apparently, and many researchers have investigated several interesting special families of Σ (see [7–10]).Recently, Hörçum and Koçer [11] (see also Horadam and Mahon [12]) examined the Horadam polynomials ℋjx (or ℋjx,a,b;p,q), which is defined by the recurrence relation(3)ℋjx=pxℋj−1x+qℋj−2x,ℋ1x=a,ℋ2x=bx,where j∈ℕ1,2,x∈ℝ,p,q,a, and b are real constants. It is seen from (3) that ℋ3x=pbx2+qa. The generating function of the sequence ℋjx, j∈ℕ, is as follows (see [11]):(4)Gx,z≔∑j=1∞ℋjxzj−1=b−apxz+a1−pxz−qz2,where z∈ℂ is such that ℜz≠x, x∈ℝ.Few particular cases ofℋjx,a,b;p,q are(1)
The Fibonacci polynomials,Fjx=ℋjx,1,1;1,1(2)
The second type Chebyshev polynomials,Ujx=ℋjx,1,2;2,−1(3)
The first type Chebyshev polynomials,Tjx=ℋjx,1,1;2,−1(4)
The Lucas polynomials,Ljx=ℋjx,2,1;1,1(5)
The Pell−Lucas polynomials,Qjx=ℋjx,2,2;2,1,(6)
The Pell polynomials,Pjx=ℋjx,1,2;2,1The estimates ond2andd3 and the very popular Fekete–Szegö functional were determined for bi-univalent functions linked with certain polynomials like Lucas polynomials, Fibonacci polynomials, Chebyshev polynomials, Horadam polynomials, and Gegenbauer polynomials. It is well-known that these polynomials play a potentially important role in architecture, approximation theory, physics, statistics, mathematical, and engineering sciences.The recent research trend is the study of functions inΣ linked with any of the abovementioned polynomials. Generally, interest was shown to obtain the initial coefficient bounds and the celebrated inequality of Fekete–Szegö for the special subfamilies of Σ. Recently, the Horadam polynomial was used by Abirami et al. [13] to find coefficient estimates for the families of bi-Bazilevic and λ-bi-starlike function, Frasin et al. [14] obtained coefficient estimates and Fekete–Szegö inequalities for certain subfamilies of Al-Oboudi-type bi-univalent functions related to k-Fibonacci numbers involving modified activation function, initial coefficient bounds for certain subsets of bi-univalent functions family subordinate to Horadam polynomials were obtained in [15, 16], Shaba and Wanas [17] obtained coefficient bounds which are sharp, for a family of bi-univalent functions using U,V-Lucas polynomials, Srivastava et al. [18] have proposed a methodology to estimate coefficient bounds and Fekete–Szegö problem for certain subsets of bi-univalent function family linked with Horadam polynomials, and Swamy [19] and Swamy et al. [20, 21] have initiated the study of some subfamilies of bi-univalent function family subordinate to Horadam polynomials involving modified activation function. Swamy and Sailaja [22] have used Horadam polynomials to investigate coefficient estimates for two families of bi-univalent functions, Swamy et al. [23] have introduced some subfamilies of Sălăgean type bi-univalent functions subordinate to m,n-Lucas polynomials and found initial coefficients, and Wanas and Alina [24] have fixed the Fekete–Szegö problem for Bazilevic bi-univalent function class linked with Horadam polynomials.For functionsg and f holomorphic in D, g is said to subordinate f, if there is a Schwarz function ψ in D, such that ψ0=0, ψz<1, and  gz=fψz,z∈D. This subordination is indicated as g≺f. In particular, if f∈S, then gz≺fz is equivalent to g0=f0 and gD⊂fD.Inspired by the article [25] and the recent trends on functions in Σ, we present a comprehensive family of Σ associated with Horadam polynomials ℋjx as in (3) having the generating function (4).Throughout this paper, the inverse functiong−1ω=fω is as in (2) and Gx,z is as in (4).Definition 1.
A functiong in Σ having the power series (1) is said to be in the set SSΣτx,γ,μ, 0≤γ≤1,μ≥0,τ≥1 and x∈ℝ, if(5)zg′zτ+μz2g″zγgz+1−γz≺1−a+Gx,z,z∈D,ωf′ωτ+μω2f″ωγfω+1−γω≺1−a+Gx,ω,ω∈D.The familySSΣτx,γ,μ is of interest as it contains many existing as well as new subfamilies of Σ for particular choices of γ,τ, and μ, as illustrated as follows:(1)
JΣτx,μ≡SSΣτx,0,μ,μ≥0 and x∈ℝ, is the collection of functions g∈Σ satisfying(6)g′zτ+μzg″z≺1−a+Gx,z,z∈D,f′ωτ+μωf″ω≺1−a+Gx,ω,ω∈D.(2)
KΣτx,μ≡SSΣτx,1,μ,μ≥0 and x∈ℝ, is the collection of functions g∈Σ satisfying(7)zg′zτgz+μz2g″zgz≺1−a+Gx,z,z∈D,ωf′ωτfω+μω2f″ωfω≺1−a+Gx,ω,ω∈D.(3)
LΣτx,γ≡SSΣτx,γ,1,0≤γ≤1 and x∈ℝ, is the collection of functions g∈Σ satisfying(8)zg′zτ+z2g″zγgz+1−γz≺1−a+Gx,z,z∈D,ωf′ωτ+ω2f″ωγfω+1−γω≺1−a+Gx,ω,ω∈D.(4)
The function classesSSΣ1x,γ,μ0≤γ≤1,μ≥0andx∈ℝ and SSΣτx,γ,00≤γ≤1andx∈ℝ were investigated by the author in [19].Remark 1.
We note that(i)
JΣτx,1≡LΣτx,0,τ≥1 and x∈ℝ(ii)
KΣτx,1≡LΣτx,1,τ≥1 and x∈ℝRemark 2.
(i)
Forτ=1, the family JΣ1x,μ was investigated by Swamy and Sailaja [22](ii)
SSΣτx,1,0≡KΣτx,0 was due to Abirami et al. [13](iii)
SSΣ1x,1,μ≡KΣx,μ was introduced by Magesh et al. [16]Remark 3.
(i)
Forμ=0 and τ=1, the class JΣ1x,0≡ℋΣx was studied by Alamoush [15](ii)
ForJΣ1x,0≡ℋΣx and τ=1, the family KΣ1x,0≡SΣ∗x was introduced by Srivastava et al. [6]
In Section2, we derive the estimates for d2 and d3 and the inequality of Fekete and Szegö [26] for functions in the class SSΣτx,γ,μ. In Section 3, relevant connections to the existing results and few interesting consequences of the main result are presented.


## 2. Bi-Univalent Function ClassSSΣτx,γ,μ

We determine the initial coefficients bounds and the inequality of Fekete–Szegö for functions inSSΣτx,γ,μ, in the following theorem.Theorem 1.
Let the functiongz defined by (1) be in the family SSΣτx,γ,μ and let 0≤γ≤1,μ≥0,τ≥1 and x∈ℝ. Then,(9)d2≤bxbxγ2+τ−γ2τ+1+2μ3−γbx2−2μ+τ−γ2pbx2+qa,(10)d3≤bx22μ+τ−γ2+bx2μ+τ−γ,and for δ∈ℝ,(11)d3−δd22≤bx32μ+τ−γ,1−δ≤J,bx31−δγ2+τ−γ2τ+1+2μ3−γbx2−2μ+τ−γ2pbx2+qa,1−δ≥J,where(12)J=132μ+τ−γγ2+τ−γ2τ+1+2μ3−γbx2−2μ+τ−γ2pbx2+qab2x2.Proof.
Letg∈SSΣτx,γ,μ. Then, on account of Definition 1, we get(13)zg′zτ+μz2g″zγgz+1−γz=1−a+Gx,Mz,(14)ωf′ωτ+μω2f″ωγfω+1−γω=1−a+Gx,Nω,where(15)Mz=m1z+m2z2+m3z3+⋯,(16)Nω=n1ω+n2ω2+n3ω3+⋯,are some regular functions in D with M0=0,Mz<1,N0=0, and Nω<1,z,ω∈D. It follows from (13)–(16) with (4) that(17)zg′zτ+μz2g″zγgz+1−γz=1−a+ℋ1x+ℋ2xmz+ℋ3xm2z+⋯,(18)ωf′ωτ+μω2f″ωγfω+1−γω=1−a+ℋ1x+ℋ2xnω+ℋ3xn2ω+⋯.
From (17) and (18), in view of (3), we find(19)zg′zτ+μz2g″zγgz+1−γz=1+ℋ2xm1z+ℋ2xm2+ℋ3xm12z2+⋯,(20)ωf′ωτ+μω2f″ωγfω+1−γω=1+ℋ2xn1ω+ℋ2xn2+ℋ3xn12ω2+⋯.
It is known that ifMz=m1z+m2z2+m3z3+⋯<1,z∈D, then(21)mi≤1,i∈ℕ.
Similarly, ifNω=n1ω+n2ω2+n3ω3+⋯<1,ω∈D, then(22)ni≤1,i∈ℕ.
Comparing (19) and (20), we have(23)2μ+τ−γd2=ℋ2xm1,(24)32μ+τ−γd3+γ2−2γμ+τ+2ττ−1d22=ℋ2xm2+ℋ3xm12,(25)−2μ+τ−γd2=ℋ2xn1,(26)32μ+τ−γ2d22−d3+γ2−2γμ+τ+2ττ−1d22=ℋ2xn2+ℋ3xn12.
From (23) and (25), we easily obtain(27)m1=−n1,and also(28)22μ+τ−γ2d22=m12+n12ℋ2x2.
If we add (24) and (26), then we obtain(29)2γ2+τ−γ2τ+1+2μ3−γd22=ℋ2xm2+n2+ℋ3xm12+n12.
Substituting the value ofm12+n12 from (28) in (29), we get(30)d22=ℋ23xm2+n22γ2+τ−γ2τ+1+2μ3−γℋ22x−2μ+τ−γ2ℋ3x,which yields (10) on using (21) and (22).
After subtracting (26) from (24) and then using (27), we obtain(31)d3=d22+ℋ2xm2−n2232μ+τ−γ.
Then, in view of (28), (31) becomes(32)d3=ℋ22xm12+n1222μ+τ−γ2+ℋ2xm2−n2232μ+τ−γ,which yields (10) using (21) and (22).
From (30) and (31), for δ∈ℝ, we get in view of (3) that(33)d3−δd22=ℋ2xBδ,x+1232μ+τ−γm2+Bδ,x−1232μ+τ−γn2,where(34)Bδ,x=1−δℋ22x2γ2+τ−γ2τ+1+2μ3−γℋ22x−2μ+τ−γ2H3x.
Clearly,(35)d3−δd22≤ℋ2x32μ+τ−γ,0≤Bδ,x≤1232μ+τ−γ,2ℋ2xBδ,x,Bδ,x≥1232μ+τ−γ,from which we conclude (9) with J as in (12). Thus, Theorem 1 is proved.Remark 4.
By takingτ=1 in Theorem 1, we get a result of the author (Corollary 2.4 [19]), and by letting μ=0 in Theorem 1, we obtain another result of the author (Corollary 3.3 in [19]).


## 3. Outcome of the Main Result

Theorem1 would yield the following outcome when γ=0.Corollary 1.
Let the functiongz defined by (1) be in the family JΣτx,μ≡SSΣτx,0,μ, and let μ≥0,τ≥1 and x∈ℝ. Then,(36)d2≤bxbxτ2τ+1+6μbx2−4μ+τ2pbx2+qa,d3≤b2x24μ+τ2+bx32μ+τ,and for some δ∈ℝ,(37)d3−δd22≤bx32μ+τ,1−δ≤J1,bx31−δτ2τ+1+6μbx2−4μ+τ2pbx2+qa,1−δ≥J1,where(38)J1=132μ+ττ2τ+1+6μ−4μ+τ2pbx2+qab2x2.Remark 5.
Forτ=1, Corollary 1 reduces to Corollary 2.1 of Swamy and Sailaja [22]. Also, we obtain Theorem 2.2 of Alamoush [15] from Corollary 1 when μ=0 and τ=1.Allowingγ=1 in Theorem 1, we obtain the following.Corollary 2.
Let the functiongz defined by (1) be in the set KΣτx,μ≡SSΣτx,1,μ, and let μ≥0,τ≥1 and x∈ℝ. Then,(39)d2≤bxbxτ2τ−1+4μbx2−2μ+τ−12pbx2+qa,d3≤bx22μ+τ−12+bx32μ+τ−1,and for δ∈ℝ,(40)d3−δd22≤bx32μ+τ−1,1−δ≤J2,bx31−δτ2τ+1+4μbx2−2μ+τ−12pbx2+qa,1−δ≥J2,where(41)J2=132μ+τ−1τ2τ−1+4μ−2μ+τ−12pbx2+qab2x2.Remark 6.
(i)
Corollary2 coincides with Theorem 2.1 of Abirami et al. [13] when μ=0(ii)
Corollary2 further coincides with Theorem 2.1 of Magesh et al. [16] when τ=1(iii)
We obtain Corollary 1 and Corollary 3 of Srivastava et al. [18] from Corollary 1 when μ=0 and τ=1
Settingμ=1  in Theorem 1, we have the following.Corollary 3.
Let the functiongz defined by (1) be in the class LΣτx,γ≡SSΣτx,γ,1 and let 0≤γ≤1,τ≥1, and x∈ℝ. Then,(42)d2≤bxbx1−γ2+τ−γ2τ+1+5−γbx2−21+τ−γ2pbx2+qa,d3≤bx221+τ−γ2+bx32+τ−γ,and for δ∈ℝ,(43)d3−δd22≤bx32μ+τ−γ,1−δ≤J3,bx31−δ1−γ2+τ−γ2τ+1+5−γbx2−21+τ−γ2pbx2+qa,1−δ≥J3,where(44)J3=132+τ−γ1−γ2+τ−γ2τ+1+5−γ−21+τ−γ2pbx2+qab2x2.By choosing appropriate values for parametersγ,τ, and μ in Theorem 1 and above corollaries, we have effectively shown the similarity between many subfamilies of bi-univalent functions.


## 4. Conclusion

A subfamily of bi-univalent (or bi-schlicht) functions is examined by using Horadam polynomial. Bounds of the first two coefficientsd2andd3 and the Fekete–Szegö functional have been fixed for this subfamily. We have presented relevant connections to the existing results and few interesting consequences of the main result.A subfamily examined in this article could inspire researchers to focus on other aspects such as a family bi-univalent functions usingq-derivative operator [27, 28], a family using q-integral operator [29], meromorphic bi-univalent function family based on Al-Oboudi differential operator [30], regular bi-univalent function family based on Frasin operator [31], and a family using integro-differential operator [32].



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