# A Comprehensive Family of Biunivalent Functions Defined byk-Fibonacci Numbers

Authors: Basem Aref Frasin; Sondekola Rudra Swamy; Ibtisam Aldawish
Journal: Journal of Function Spaces
 (2021)

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## Abstract

By usingk-Fibonacci numbers, we present a comprehensive family of regular and biunivalent functions of the type gz=z+∑j=2∞djzj in the open unit disc D. We estimate the upper bounds on initial coefficients and also the functional of Fekete-Szegö for functions in this family. We also discuss few interesting observations and provide relevant connections of the result investigated.


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## Body

## 1. Introduction and Notations

Letℂ be the set of all complex numbers and the disc z∈ℂ:z<1 be symbolized by D. Let ℕ=ℕ0\0≔1,2,3,⋯ and ℝ be the collection of all real numbers. We denote the set of all normalized regular functions in D that have the series of the form(1)gz=z+∑j=2∞djzj,by A and the symbol S stands for set of all functions of A that are univalent (or Schlicht) in D. As per the popular Koebe theorem (see [1]), every function g∈S has an inverse function given by(2)g−1ω=fω=ω−d2ω2+2d22−d3ω3−5d23−5d2d3+d4ω4+⋯,such that z=g−1gz and ω=gg−1ω,ω<r0g,r0g≥1/4,z,ω∈D.A functiong of A is called biunivalent (or bi-Schlicht) in D if both g and g−1 are univalent (or Schlicht) in D. Let Σ stands for the set of biunivalent (or bi-Schlicht) functions having the form (1). Historically, investigations of the family Σ begun five decades ago by Lewin [2] and Brannan and Clunie [3]. Later, Tan [4] found some coefficient estimates for biunivalent functions. In 1986 [5], Brannan and Taha introduced certain well-known subfamilies of Σ in D. Many interesting results related to initial bounds for some special families of Σ have appeared in [6–8].In 2007, the concept ofk-Fibonacci number sequence Fk,jj=0∞, k∈ℝ+ was examined by Falcón and Plaza [9] and is given by(3)Fk,0=0,Fk,1=1,Fk,j+1=kFk,j+Fk,j−1,where j∈ℕ and(4)Fk,j=k−tkj−tkjk2+4withtk=k−k2+42.F1,j=Fj is the well-known Fibonacci number sequence.Özgür and Sokól in 2015 [10] proved that if(5)p~kz=1+tk2z21−ktkz−tk2z2,then,(6)p~kz=1+Fk,0+Fk,2tkz+Fk,1+Fk,3tk2z2+⋯=1+ktkz+k2+2tk2z2+⋯,where tk is as in (4) and z∈D. Further, if p~kz=1+Σp~k,jzj, then, we have(7)p~k,j=tkjFk,j−1+Fk,j+1,j∈ℕ.Fibonacci polynomials, Pell-Lucas polynomials, Gegenbauer polynomials, Chebyshev polynomials, Horadam polynomials, Fermat-Lucas polynomials, and generalizations of them are potentially important in many branches such as architecture, physics, combinatorics, number theory, statistics, and engineering. Additional information is associated with these polynomials one can go through [11–13]. More details about the very popular functional of Fekete-Szegö for biunivalent functions based on k-Fibonacci numbers can be found in [14–20].The recent research trends are the outcomes of the study of functions inΣ based on any one of the above-mentioned polynomials, which can be seen in the recent papers [21–28]. Generally, interest was shown to estimate the first two coefficient bounds and the functional of Fekete-Szegö for some subfamilies of Σ.For functionsg and f regular 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⇐g0=f0 and gD⊂fD.Inspired by the recent articles and the new trends on functions inΣ, we present a comprehensive family of Σ defined by using k-Fibonacci numbers as given by (3) with Fk,j as in (4).Throughout this paper,g−1ω=fω is as in (2), Tk=k−k2+2tk, tk is as in (4), and p~k is as in (5).Definition 1.
A functiong∈Σ having the power series (1) is said to be in the family SRSΣτγ,μ,p~k, if(8)zg′zτ+μz2g′′zγgz+1−γz≺p~kz,z∈Dωg′ωτ+μz2g′′ωγgω+1−γω≺p~kω,ω∈D,where τ≥1,0≤γ≤1,andμ≥0.Remark 2.
The function familiesSRSΣτγ,0,p~k and SRSΣ1γ,μ,p~k were investigated by Frasin et al. [29].
It is interesting to note that (i)γ=1, (ii) γ=0, and (iii) μ=1 lead the family SRSΣτγ,μ,p~k to various subfamilies, as illustrated in the following:(1)
SRSΣτ1,μ,p~k≡LΣτμ,p~k is the family of functions g∈Σ satisfying(9)zg′zτgz+μz2g′′zgz≺p~kzandωf′ωτfω+μω2f′′ωfω≺p~kω,z,ω∈D(2)
SRSΣτ0,μ,p~k≡KΣτμ,p~k is the set of functions g∈Σ satisfying(10)g′zτ+μzg′′z≺p~kzandf′ωτ+μωf′′ω≺p~kω,z,ω∈D(3)
SRSΣτγ,1,p~k≡MΣτγ,p~k is the collection of functions g∈Σ satisfying(11)zg′zτ+z2g′′zγgz+1−γz≺p~kz,z∈D,ωf′ωτ+ω2f′′ωγfω+1−γω≺p~kω,ω∈DRemark 3.
We note that (i)LΣτ1,p~k≡MΣτ1,p~k and (ii) KΣτ1,p~k≡MΣτ0,p~k.Remark 4.
(i)
Whenτ=1, the family KΣ1μ,p~k was introduced by Frasin et al. [30](ii)
The familyLΣ10,p~k≡SΣ∗p~k was mentioned by Güney et al. [18], when μ=0 and τ=1(iii)
Forμ=0andk=1, the class LΣτ0,p~1≡SΣp~1 was investigated by Magesh et al. [31]We now state the following lemma, which we will be using in the proof of our theorem.Lemma 5 (see [32]).
Ifp∈P, where P is the collection of regular functions p in D, satisfying Rpz>0,z∈D, with pz=1+p1z+p2z2+⋯,z∈D, then pi≤2, for each i.In the next section, we derive the estimates ford2,d3 and obtain the Fekete-Szegö [33] inequalities for functions in the class SRSΣτγ,μ,p~k.


## 2. Coefficient Bounds and Fekete-Szegö Functional

In this section, we offer to get the upper bounds on initial coefficients and find the functional of Fekete-Szegö for functions∈SRSΣτγ,μ,p~k.Theorem 6.
Letτ≥1,0≤γ≤1, and μ≥0. If gz of the form (1) in the family SRSΣτγ,μ,p~k, then(12)d2≤kktk∣γ2+τ−γ2τ+1+2μ3−γk2tk+2μ+τ−γ2Tk∣,(13)d3≤ktk32μ+τ−γ+k3tk2γ2+τ−γ2τ+1+2μ3−γk2tk+2μ+τ−γ2Tk,and for δ∈ℝ,(14)d3−δd22≤k∣tk∣32μ+τ−γ;1−δ≤J,k3tk21−δγ2+τ−γ2τ+1+2μ3−γk2tk+2μ+τ−γ2Tk;1−δ≥J,where(15)J=132μ+τ−γγ2+τ−γ2τ+1+2μ3−γ+2μ+τ−γ2Tkk2tk.Proof.
Let the functiong∈SRSΣτγ,μ,p~k. Then, from Definition 1, we have(16)zg′zτ+μz2g′′zγgz+1−γz≺p~kuz,z∈D,(17)ωf′ωτ+μω2f′′ωγfω+1−γω≺p~kvω,ω∈D.
Letpz=1+p1z+p2z2+⋯, and p≺p~k. Then, there exists a regular function u with uz<1 in D and pz=p~kuz. Therefore, the function mz is in the class P, where(18)mz=1+uz1−uz=1+u1z+u2z2+⋯.
So it follows that(19)uz=mz−1mz+1=u12z+u2−u122z22+u3−u1u2+u134z32+⋯,(20)p~kuz=1+p~k,1u1z2+u2−u122z22+⋯+p~k,2u1z2+u2−u122z22+⋯2+⋯=1+p~k,1u1z2+12u2−u122p~k,1+u124p~k,2z2+⋯.
Similarly, it follows that(21)p~kvω=1+p~k,1v1ω2+12v2−v122p~k,1+v124p~k,2ω2+⋯,where v is a regular function such that ∣vω∣<1 in D such that pω=p~kvω and the function lω is in the class P, where(22)lω=1+vω1−vω=1+v1ω+v2ω2+⋯.
By virtue of (14), (15), (18), and (19), we obtain(23)2μ+τ−γd2=u1ktk2,(24)32μ+τ−γd3+γ2−2γμ+τ+2ττ−1d22=12u2−u122ktk+u124k2+2tk2,(25)−2μ+τ−γd2=v1ktk2,(26)32μ+τ−γ2d22−d3+γ2−2γμ+τ+2ττ−1d22=12v2−v122ktk+v124k2+2tk2.
From (21) and (23), we get(27)u1=−v1,and also,(28)22μ+τ−γ2d22=u12+v12k2tk24.
If we add (26) and (24), then we obtain(29)2γ2+τ−γ2τ+1+2μ3−γd22=12u2+v2ktk−14ktk−k2+2tk2u12+v12.
Substituting the value ofu12+v12 from (26) in (27), we get(30)d22=k3tk2u2+v24γ2+τ−γ2τ+1+2μ3−γk2tk+2μ+τ−γ2Tk,which gets (10), on using Lemma 5.
On using (25) in the subtraction of (24) from (26), we arrive at(31)d3=d22+ktku2−v2432μ+τ−γ.
Then, in view of Lemma5 and equation (28), (29) reduces to (11).
From (28) and (29), for δ∈ℝ, we can easily compute that(32)d3−δd22=ktkTδ+1432μ+τ−γu2+Tδ−1432μ+τ−γv2,where(33)Tδ=1−δk2tk4γ2+τ−γ2τ+1+2μ3−γk2tk+2μ+τ−γ2Tk.
In view of (4), we find that(34)d3−δd22≤k∣tk∣32μ+τ−γ;0≤Tδ≤1432μ+τ−γ,4ktkTδ;Tδ≥1432μ+τ−γ,which enable us to conclude (12) with J as in (13). Theorem 6 is proved.Remark 7.
By takingτ=1 in the above theorem, we obtain a result of Frasin et al. ([29], Corollary 3.4) and if we let μ=0 in the above theorem, we get another result of Frasin et al. ([29], Corollary 3.7).Remark 8.
Allowingk=γ=1 and μ=0 in the above theorem, we have Theorem 2.3 of Magesh et al. [31].Remark 9.
Lettingτ=γ=1 and μ=0 in the Theorem 6, we obtain two results of Güney et al. ([18], Corollary 10 and Corollary 23). Further, if we take k=1, we get results of Güney et al. ([17], Corollary 1 and Corollary 4).In Section3, few interesting consequences and relevant observations of the main result are mentioned.


## 3. Outcome of the Main Result

By setting (i)γ=1, (ii) γ=0, and (iii) μ=1 in our main theorem, we obtain the following results, respectively.Corollary 10.
If the functiong∈LΣτμ,p~k, then(35)d2≤kktk∣τ2τ−1+4μk2tk+2μ+τ−12Tk∣,d3≤ktk32μ+τ−1+k3tk2τ2τ−1+4μk2tk+2μ+τ−12Tk.,and for δ∈ℝ,(36)d3−δd22≤ktk32μ+τ−1;1−δ≤J1,k3tk2∣1−δ∣∣τ2τ−1+4μk2tk+2μ+τ−12Tk∣;1−δ≥J1,where(37)J1=132μ+τ−1τ2τ−1+4μ+2μ+τ−12Tkk2tk.Remark 11.
(i)
By takingμ=0 and k=1 in the above corollary, we obtain Theorem 2.3 of Magesh et al. [31](ii)
By allowingμ=0 and τ=1 in the above corollary, we get two results Güney et al. ([18], Corollary 10 and Corollary 23)Corollary 12.
If the functiong∈KΣτμ,p~k, then(38)d2≤kktkτ2τ+1+6μk2tk+4μ+τ2Tk,d3≤k3tk2τ2τ+1+6μk2tk+4μ+τ2Tk+ktk32μ+τ,and for some δ∈ℝ,(39)d3−δd22≤ktk32μ+τ;1−δ≤J2,k3tk21−δτ2τ+1+6μk2tk+4μ+τ2Tk;1−δ≥J2,where(40)J2=132μ+ττ2τ+1+6μ+4μ+τ2Tkk2tk2.Remark 13.
Forτ=1, Corollary 12 reduces to a result of Frasin et al. ([30], Corollary 3.6). Further, allowing k=1, we get Corollary 10 of Altnkaya [22].Corollary 14.
If the functiong∈MΣτγ,p~k, then(41)∣d2∣≤kk∣tk∣∣1−γ2+τ−γ2τ+1+5−γk2tk+21+τ−γ2Tk∣,d3≤ktk32+τ−γ+k3tk24μ+1k2tk+2μ+12Tk,and for δ∈ℝ,(42)d3−δd22≤k∣tk∣32+τ−γ;1−δ≤J3,k3tk2∣1−δ∣1−γ2+τ−γ2τ+1+5−γk2tk+21+τ−γ2Tk;∣1−δ∣≥J3,where(43)J3=132+τ−γ1−γ2+τ−γ2τ+1+5−γ+21+τ−γ2Tkk2tk.


## 4. Conclusion

A comprehensive family of biunivalent (or bi-Schlicht) functions is introduced by usingk-Fibonacci numbers. Bounds of the first two coefficients ∣d2∣ and ∣d3∣ and the celebrated Fekete-Szegö functional have been fixed for this family. Through corollaries of our main result, we have highlighted many interesting new consequences.A comprehensive family examined in this research paper could inspire further research related to other aspects such as a comprehensive family usingq-derivative operator, a meromorphic biunivalent function family associated with Al-Oboudi differential operator, and a comprehensive family using integrodifferential operator.



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