F1000ResearchF1000Research2046-1402F1000 Research LimitedLondon, UK10.12688/f1000research.173513.1Research ArticleArticlesCoefficient Estimates for New Subclasses of Bi-Univalent Functions Involving Generalized Bivariate Fibonacci-like Polynomials
[version 1; peer review: 1 approved, 1 approved with reservations]
HusseinuMustafaConceptualizationFormal AnalysisInvestigationMethodologyWriting – Original Draft PreparationWriting – Review & Editinghttps://orcid.org/0009-0006-5391-4430a1H. SaloomiMohammedInvestigationSupervisionWriting – Original Draft Preparation2
University of Kerbala, Karbala, Karbala Governorate, Iraq
University of Kerbala, Karbala, Karbala Governorate, Iraqmostafa.h@s.uokerbala.edu.iq
The study of bi-univalent functions has gained considerable attention due to its relevance in geometric function theory, particularly in determining bounds for the initial coefficients of analytic functions. Previous research has introduced several subclasses of bi-univalent functions using tools such as convolution, subordination, differential subordination, and various families of special polynomials.
Methods
Motivated by these developments, this work constructs new subclasses of bi-univalent functions by employing generalized Fibonacci polynomials. The analytic properties of these polynomials are used to define the classes and derive the corresponding functional relations needed for coefficient estimation.
Results
For the newly introduced subclasses, bounds for the initial coefficients are established. Furthermore, Fekete–Szegö inequalities are obtained for each subclass, extending previously known results and demonstrating the influence of the generalized Fibonacci structure on coefficient behavior.
Conclusions
The results contribute to the ongoing development of bi-univalent function theory by providing broader subclasses and sharper estimates. The use of generalized Fibonacci polynomials offers a flexible analytical framework that can be further applied to related classes in future studies.
Regular functionunivalent functionsubordinationbi-univalent functionFibonacci-like polynomials functionsFekete- SzegöThe author(s) declared that no grants were involved in supporting this work.1. Introduction
Many polynomials including Fibonacci and Bell Many polynomials are widely used in many researches in the context of this topic in addition to Lucas polynomials and Horadam polynomials. Other special polynomials also exist. Most of these polynomials are employed in some branches comprising geometry, physics, and theoretical analysis such as.
1–
4 In addition, they are utilized in the theory of geometric functions and number theory, for example, Refs.
5–
7, and this is what interests us in these polynomials. In the context of the topic, the principle of subordination and quasi -subordination has been used in complex analysis, especially in the geometric functions theory. After forming new subclasses of regular functions, many researchers have used polynomials and estimated the initial coefficients and the Fekete-Szegö problem. Based on the relationship between polynomials and the classes of regular functions, the two new classes are defined.
After defining these two subclasses and calculating the upper bounds of the coefficients, the researchers concluded many important results and observations, after using the basic rules of polynomials.
Through the
recurrence relation:
℣n=℣n−1+℣n−2,℣0=0,℣1=1,(n≥2).The Fibonacci polynomials are a
polynomial sequence that may be regarded as a generalization of the
Fibonacci numbers. In Ref.
8, a new generalization of the Fibonacci bolynomials called generalized bivariate Fibonacci-like is introduced.
Concerning n ≥ 2, the recurrence relation:
℣n(x,y)=px℣n−1(x,y)+qy℣n−2(x,y)is possible to define the generalized bivariate Fibonacci-like polynomials where
p,
q represent positive integers while “
x,y∈ℝ,
℣0(x,y)=a,℣1(x,y)=b,px,qy≠0and(px)2+4qy≠0”.
For the Fibonacci-like polynomials,
generating function as denoted in Ref.
8, has the form:
℣(x,y)(v)=∑℣n(x,y)vn=a+(b−apx)va−pxz−qyv2
The polynomials for the special cases can be obtained as presented in
Table 1 for and of
p,q,a,bandy.
Values of Special cases of the generalized polynomial obtained by substituting specific values of the parameters.
By choosing particular parameter combinations, the polynomial reduces to several well-known sequences such as the bivariate Fibonacci polynomials, Fibonacci polynomials, Pell polynomials, bivariate Lucas polynomials, Chebyshev polynomials of the second kind, and the Horadam polynomials.
(x,y)
(a,b)
(p,q)
℣n(x,y)
(x,y)
(0,1)
(1,1)
Bivariate Fibonacci
℣n(x,y)
(x,1)
(0,1)
(1,1)
Fibonacci
℣n(x)
(x,1)
(0,1)
(2,1)
Pell,
Pn(x)
(x,y)
(2,x)
(1,1)
Bivariate Lucas
Ln(x,y)
(t,−1)
(1,2t)
(2,1)
Chebyshev of the second kind
Un(x)
(x,1)
(a,bx)
(p,q)
Horadam
Hn+1(x)
Consider
ℓ to be the function of the form
ℓ(v)=v+∑m=2∞ɛmvm,that is part of Å where Å symbolizes the class of regular functions recognized on the disk
Ų={v∈ℂ:|v|<1},regarding
ℓ(0)=ℓ´(0)−1=0. Also, consider S to be the subclass of Å containing the form
(1,1). They are univalent in
Ų. For each
ℏ∈S, the image
t=ℓ(v),v∈Ų is stated by The Koebe’s Covering Theorem.
9 In the
t–plane,
Ų,covers the disk
{t:|t|<1/4}. Thus, every function
from this theoremℏ∈S has an inverse
ℏ−1, that holds
ℓ−1(ℓ(v))=v,(v∈Ų),and,
ℓ(ℓ−1(t))=t,(|t|<r0(ℓ),r0(ℓ)≥14),where
ᶂ(t)=ℓ−1(t)=t−ɛ2t2+(2ɛ22−ɛ3)t3−(5ɛ23−5ɛ2ɛ3+ɛ4)t4+···
In Ų, the function
ℓ∈Å is named bi-univalent if both
ℓ and
ℓ−1 are univalent and the set of all bi-univalent functions are signified by Σ. Lewin
10 presented this set and revealed that
|ɛ2|≤1.5 regarding the function in the set Σ. Lately, Brannan and Clunie
11 estimated that
|ɛ2|≤2 and Netanyahu in
12 showed that
|ɛ2|=43.
Many authors recently offered and explored numerous remarkable subclasses of bi-univalent functions.
2–
4,
13,
14
The regular function
ℓ is subordinate to
ᶂ, written as
ℓ≺ᶂorℓ(v)≺ᶂ(v),(ѵ∈Ų),if there is regular function
ƙ:Ų→Ų, with
ƙ(0)=0 and
|ƙ(v)|<1 such that
ℓ(v)=ᶂ(ƙ(v)),v∈Ų.
It follows the definition stating that:
ℓ(0)=ᶂ(0)andℓ(Ų)⊂ᶂ(Ų).
see Refs.
5,
6,
15.
In the current study, a new subclass of bi-univalent functions filling the subordinate settings and it is defined by Fibonacci polynomial is offered. Moreover, the coefficient estimates for |
ɛ2| and |
ɛ3| are obtained for functions of the new classes.
Definition: A function
ℓ∈∑ offered by (1.1) is stated to be in the set
MΩ,Σp,q,x,y(X(ѵ)),0≤Ω≤1, if it fits:
[(1−Ω)ℓ(z)z+Ω1+zℓ′′(z)ℓ′(z)zℓ′(z)ℓ(z)]≺X(ѵ)=℣(x,y)(ѵ)+1−a,(ѵ∈Ų)[(1−Ω)ℓ(w)w+Ω1+wℓ′′(w)ℓ′(w)wℓ′ℓ(w)]≺X(t)=℣(x,y)(t)+1−a,(t∈Ų)◾
By assigning special values to parameters
Ωand replacing polynomials, new and prominent families of bi-univalent functions are obtained. Accordingly, novel bounds for the initial coefficients are realized.
For
Ω=1, in
MΩ,Σp,q,x,y(X(ѵ)), we get the subfamilies ◾
Given a Horadam polynomial
℣(x,y)with
b =
bx and
y = 1, we obtain
MΩ,Σp,q,x,1(X(ѵ)). In this case for
Ω=1belongs to this family. Later, we have families
LΣ(x).
7
Consider
℣(x,y) to be Chebyshev polynomials with
p = 2,
q = 1,
a = 1,
b = 2
t,
x =
t,
y = -1. In Theorem 2.7 there is
MΩ,Σ2,1,x,−1(X(ѵ)). In such incident for
Ω=0,1 belongs to this family. Then, the new
familiesMΣ2,1,x,−1(X(ѵ)) and
M1,Σ2,1,x,−1(X(ѵ))respectively
are gained.
Regard
℣(x,y) as Fibonacci polynomials with
p =
q = 1,
b = 1,
y = 1,
a = 0, there is.
MΩ,Σ1,1,x,1(X(ѵ)). In this situation for
Ω=0,1 be in this family. Then, new families
MΣ1,1,x,1(X(ѵ))and
M1,Σ1,1,x,1(X(ѵ))individually are got.
In this section, we investigate coefficient estimates for the function class
MΩ,Σp,q,x,y(X(ѵ))
For,
0≤Ω≤1, let
ℓ∈Σ given by
(1.1) belong to
MΩ,Σp,q,x,y(X(ѵ)), then
|κ2|≤|b||b||b2(1−Ω)−(pbx+aqy)||κ3|≤b2+|b|(1+3Ω)◾
Since
ℓ∈MΩ,Σp,q,x,y(X(ѵ)), there exist regular functions
i,λ:Ê→Ê, such that
i(ѵ)=∑1∞ikѵk,λ(t)=∑1∞λktkand|ik|≤1,|λk|≤1,
we can write
[(1−Ω)ℓ(z)z+Ω1+zℓ′′(z)ℓ′(z)zℓ′(z)ℓ(z)]=X(ѵ),(ѵ∈Ų)[(1−Ω)ℓ(w)w+Ω1+wℓ′′(w)ℓ′(w)wℓ′ℓ(w)]=X(t),(t∈Ų)◾
Thus upon comparing the corresponding coefficients in
(2.8) and
(2.9), we have
κ2=℣1(x,y)i1(1+3Ω)κ3−4Ωκ22=℣1(x,y)i2+℣2(x,y)i12−κ2=℣1(x,y)λ1and
2(1+Ω)κ22−(1+3Ω)κ3=℣1(x,y)λ2+℣2(x,y)λ12◾
Now considering
(2.10) and
(2.12), we get
i1=−λ12κ22=℣21(x,y)(i12+λ12)κ22=℣21(x,y)(i12+λ12)2◾
If we add
(2.11) to
(2.13), we find that
2(1−Ω)κ22=℣1(x,y)(i2+λ2)+℣2(x,y)(i12+λ12)
Upon substituting the value of
i12+λ12 from
(2.15) in
(2.16), we deduce the next result:
2(1−Ω)κ22=℣1(x,y)(i2+λ2)+℣2(x,y)2κ22℣21(x,y)2℣21(x,y)(1−Ω)κ22=℣31(x,y)(i2+λ2)+℣2(x,y)2κ22◾
And more simplify
2(℣21(x,y)(1−Ω)−℣2(x,y))κ22=℣31(x,y)(i2+λ2)κ22=℣31(x,y)(i2+λ2)2(℣21(x,y)(1−Ω)−℣2(x,y))|κ2|2≤|℣31(x,y)||℣21(x,y)(1−Ω)−℣2(x,y)|
Using the inequality
|ik|≤1,|λk|≤1◾
|κ2|≤|b||b||b2(1−Ω)−(pbx+aqy)|.
Subtracting
(2.11) and
(2.13), and with some computation, we have
2(1+3Ω)κ3−2(1+3Ω)κ22=℣1(x,y)(i2−λ2)
Put
(2.15) in
(2.18), then we deduce
|κ3|≤|℣21(x,y)(i12+λ12)|2+|℣1(x,y)(i2−λ2)|2(1+3Ω).|κ3|≤b2+|b|(1+3Ω)◾
Consider
ℓ presented by
(1.1) be in the subfamily
M1,Σp,q,x,y(X(ѵ)). Then,
|κ2|≤|b||b||pbx+aqy|.
|κ3|≤b2+|b|4◾
Let
ℓ offered by (1.1) be in the sub family
MΣp,q,x,y(X(ѵ)). Then,
|κ2|≤|b||b||b2−(pbx+aqy)|.
|κ3|≤b2+|b|∎
Regard
ℓ introduced by
(1.1) be in the subfamily
MΩ,Σp,q,x,1(X(ѵ)) and
℣(x,y) be Horadam
Hn+1(x). Then,
|κ2|≤|bx||bx||b2x2(1−Ω)−(pbx2+aq)|.|κ3|≤b2x2+|bx|(1+3Ω)◾
Let
ℓ offered by
(1.1) and
Ω=1be in the sub family
M,Σp,q,x,1(X(ѵ)). Then,
|κ2|≤|bx||bx||b2x2−(pbx2+aq)|.|κ3|≤b2x2+|bx|◾
Let
ℓ∈Σ shown by
(1.1) in the subfamily
MΩ,Σ2,1,x,−1(X(ѵ))and
℣(x,y) be Chebyshev polynomials with
p = 2,
b
= 2
t,
x =
t,
y = -1,
q = 1, and
a = 1. Then,
|κ2|≤|2t||2t||4t4(1−Ω)−(4t3−1)|.|κ3|≤4t4+2|t2|(1+3Ω)◾
Consider
ℓ∈Σ offered by
(1.1) in the subfamily
M,Σ2,1x,−1(X(ѵ)). Then,
|κ2|≤|2t||2t||4t4−(4t3−1)|.|κ3|≤4t4+2|t2|◾
Let
ℓ∈Σ presented by
(1.1) in the subfamily
M1,Σ2,1,x,−1(X(ѵ)). Then,
|κ2|≤|2t||2t||(4t3−1)|.|κ3|≤4t4+|t2|2◾
Consider
ℓ∈Σ stated by
(1.1) in the subfamily
MΩ,Σ1,1,x,1(X(ѵ))and
℣(x,y) be Fibonacci polynomials with
p
= 1,
q
= 1,
b = 1,
y = 1, and
a = 0. Then,
|κ2|≤1|(1−Ω)−x|.|κ3|≤1+1(1+3Ω)◾
Let
ℓ∈Σ revealed by
(1.1) in the subfamily
M,Σ1,1x,1(X(ѵ)). Then,
|κ2|≤1|1−x|.|κ3|≤2◾
Let
ℓ∈Σ stated by
(1.1) in the set
M1,Σ1,1,x,1(X(ѵ))|κ2|≤1|x|.|κ3|≤54◾
For
0≤Ω≤1,a function
ℓ∈∑ offered by
(1.1) is stated to be in the set
ΓΩ,Σp,q,x,y(X(ѵ)) if the following settings are satisfied:
[(1−Ω)ℓ′(z)+Ω1+zℓ′′ℓ′zℓ′ℓ(z)]≺X(ѵ)=℣(x,y)(ѵ)+1−a,(ѵ∈Ų)[(1−Ω)ℓ′(w)+Ω1+wℓ′′ℓ′wℓ′ℓ(w)]≺X(t)=℣(x,y)(t)+1−a,(t∈Ų).
By assigning special values to parameters
Ωand replacing polynomials, new and known families of bi-univalent functions are obtained. Thus, new bounds for the initial coefficients are gained.
For
Ω=1, in
ΓΩ,Σp,q,x,y(X(ѵ)), the subfamilies are got.
Consider
℣(x,y) to be Horadam polynomial with
b =
bx and
y = 1. Then, there is
ΓΩ,Σp,q,x,1(X(ѵ)). For
Ω=1 be in this family. Then, families
LΣ(x).
7 are got.
Let
℣(x,y) is Chebyshev polynomials with
p = 2,
b = 2
t,x =
t,
y = -1,
q = 1,
a = 1 in
ΓΩ,Σp,q,x,y(X(ѵ)). There is
ΓΩ,Σ2,1,x,−1(X(ѵ)). In such condition, for
Ω=0,1 be in this family. Accordingly, new families
ΓΣ2,1,x,−1(X(ѵ)) and
Γ1,Σ2,1,x,−1(X(ѵ))separately are obtained.
Let
℣(x,y) is Fibonacci polynomials with
p =
q = 1,
b = 1,
y = 1,
a = 0, there is.
ΓΩ,Σ1,1,x,1(X(ѵ)). In this case for
Ω=0,1 be in this family. Then, new families
ΓΣ1,1,x,1(X(ѵ)) and
Γ1,Σ1,1,x,1(X(ѵ))respectively are gained.
In this section, coefficient estimates are investigated for the function set
ΓΩ,Σp,q,x,y(X(ѵ))
Regarding,
0≤Ω≤1, let
ℓ∈Σ offered by
(1.1) belong to
ΓΩ,Σp,q,x,y(X(ѵ)). Then,
|κ2|≤|b||b||3(1−Ω)b2−(2−Ω)2(pbx+aqy)|.|κ3|≤(3−Ω)b2(3+Ω)(2−Ω)2+|b|(3+Ω).
As
ℓ∈ΓΩ,Σp,q,x,y(X(ѵ)), there exists systematic functions
i,λ:Ê→Ê, such that
i(ѵ)=∑1∞ikѵk,λ(t)=∑1∞λktkand|ik|≤1,|λk|≤1,
It is possible to write:
[(1−Ω)ℓ′(z)+Ω1+zℓ′′(z)ℓ′(z)zℓ′(z)ℓ(z)]=X(ѵ),(ѵ∈Ų)[(1−Ω)ℓ′(w)+Ω1+wℓ′′(w)ℓ′(w)wℓ′ℓ(w)]=X(t),(t∈Ų)1+(2−Ω)κ2z+[(3+Ω)κ3−4Ωκ22]z2….=1+℣1(x,y)ɍ1z+(℣1(x,y)ɍ2+℣2(x,y)ɍ12)z2+…and
1−(2−Ω)κ2w+[(6−2Ω)κ22−(3+Ω)κ3]w2..=1+℣1(x,y)ѳ1w+(℣1(x,y)ѳ2+℣2(x,y)ѳ12)w2+…
Thus upon comparing the corresponding coefficients in
(3.6) and
(3.7), we have
(2−Ω)κ2=℣1(x,y)ɍ1(3+Ω)κ3−4Ωκ22=℣1(x,y)ɍ2+℣2(x,y)ɍ12−(2−Ω)κ2=℣1(x,y)ѳ1−[(3+Ω)κ3−(6−2Ω)κ22]=℣1(x,y)ѳ2+℣2(x,y)ѳ12
Now considering
(3.8) and
(3.10), we get
ɍ1=−ѳ12(2−Ω)2κ22=℣21(x,y)(ɍ12+ѳ12)κ22=℣21(x,y)(ɍ12+ѳ12)2(2−Ω)2
In adding
(3.9) to
(3.11), we obtain:
[6(1−Ω)℣21(x,y)−2℣2(x,y)(2−Ω)2]κ22=℣31(x,y)(ɍ2+ѳ2)κ22=℣31(x,y)(ɍ2+ѳ2)6(1−Ω)℣21(x,y)−2℣2(x,y)(2−Ω)2|κ2|2≤|℣31(x,y)||3(1−Ω)℣21(x,y)−℣2(x,y)(2−Ω)2||κ2|≤|b||b||3(1−Ω)b2−(2−Ω)2(pbx+aqy)|.
Subtracting
(3.9) and
(3.11), and with some computation, we get
(3+Ω)κ3−4Ωκ22=℣1(x,y)ɍ2+℣2(x,y)ɍ12−[(3+Ω)κ3−(6−2Ω)κ22]=℣1(x,y)ѳ2+℣2(x,y)ѳ122(3+Ω)κ3−(6+2Ω)κ22=℣1(x,y)(ɍ2−ѳ2)
Put
(3.13) in
(3.14)2(3+Ω)κ3−(6+2Ω)℣21(x,y)(ɍ12+ѳ12)2(2−Ω)2=℣1(x,y)(ɍ2−ѳ2)4(3+Ω)(2−Ω)2κ3−(6−2Ω)℣21(x,y)(ɍ12+ѳ12)=2(2−Ω)2℣1(x,y)(ɍ2−ѳ2)κ3=(6−2Ω)℣21(x,y)(ɍ12+ѳ12)4(3+Ω)(2−Ω)2+2(2−Ω)2℣1(x,y)(ɍ2−ѳ2)4(3+Ω)(2−Ω)2|κ3|≤(3−Ω)b2(3+Ω)(2−Ω)2+|b|(3+Ω)◾
Let
ℓ stated by
(1.1) be in the subfamily
ΓΣp,q,x,y(X(ѵ)). Then,
|κ2|≤|b||b||3b2−4(pbx+aqy)|.|κ3|≤b24+|b|3◾
Let
ℓ given by
(1.1) be in the subfamily
Γ1,Σp,q,x,y(X(ѵ)). Then,
|κ2|≤|b||b||(pbx+aqy)|.|κ3|≤|b|4◾
If
℣n is Horadam polynomial in above theorem Then, we have
|κ2|≤|bx||bx||3(1−Ω)(bx)2−(2−Ω)2(pbx2+aq)|,|κ3|≤3(1−Ω)(bx)2(3+Ω)(2−Ω)2+|bx|(3+Ω)◾
Where
℣n(x,y)=px℣n−1(x,y)−qy℣n−2(x,y),n≥2,
℣0(x,y)=a,℣1(x,y)=b
If
℣n is generalized bivariate Fibonacci-like polynomials and Ω = 0. Then,
|κ2|≤|b||b|3b2−4(pbx−aqy).
If
℣n is Horadam polynomial and Ω = 0 (remark)
H31(x,y)=(bx)3,H2(x,y)=pbx2−aq|κ2|≤|bx||bx|3b2x2−4(pbx2−aq).|κ3|≤(bx)24+|bx|3◾
Through applying the clacc
MΩ,Σp,q,x,y(X(ѵ)) and
ΓΩ,Σp,q,x,y(X(ѵ))to the Fekete-Szegö problem, the following results and notes are realized:
Let
℣(x,y) is Horadam polynomial with
b =
bx and
y = 1, we obtain
MΩ,Σp,q,x,1(X(ѵ)). In this case for
Ω=1 belongs to this family. Then, families
LΣ(x).
7 are gained.
If
b =
bx and
y = 1 and
℣(x,y) is Horadam polynomial, we obtain
M,Σp,q,x,1(X(ѵ)).
Given a Horadam polynomial
℣(x,y) with
b =
bx and
y = 1, we obtain
Γ1,Σp,q,x,1(X(ѵ)). In this case for
Ω=1 be in this family. Then, families
LΣ(x).
7 are obtained.
Let
℣(x,y) is Horadam polynomial with
b =
bx and
y = 1, we have
Γ,Σp,q,x,1(X(ѵ)).
Assuming that
ℓ(ѵ)=ѵ+∑m=2∞ɛmѵm∈MΩ,Σp,q,x,y(X(ѵ)), then,
|κ3−νκ22|={|b|(1+3Ω)if|1−V|≤|(1−Ω)b2−(pbx+aqy)|(1+3Ω)b2(1−V)|b|3|(1−Ω)b2−(pbx+aqy)|if|1−V|>|(1−Ω)b2−(pbx+aqy)|(1+3Ω)b2
Where
h(ν,Ω)=(1−ν)b32(b2(1−Ω)−(pbx+aqy))κ3−νκ22=(h(ν)+b2(3+Ω))ɍ2+(h(ν)−b2(3+Ω))ѳ2
Let
ℓ assumed by
(1.1) be in the subfamily
M1,Σp,q,x,1(X(ѵ)) and
℣(x,y) be Horadam
Hn+1(x). Then,
|κ3−νκ22|={|bx|4if|1−V|≤|pbx2+aq|4(bx)2(1−V)|bx|3|pbx2+aq|if|1−V|>|pbx2+aq|4(bx)2
Let
ℓ offered by
(1.1) be in the subfamily
M,Σp,q,x,1(X(ѵ)) and
℣(x,y) be Horadam
Hn+1(x). Then,
|κ3−νκ22|={|bx|if|1−V|≤|(bx)2−(pbx2+aq)|(bx)2(1−V)|bx|3|(bx)2−(pbx2+aq)|if|1−V|>|(bx)2−(pbx2+aq)|(bx)2
Let suppose that
ℓ(ѵ)=ѵ+∑m=2∞ɛkѵm∈ΓΩ,Σp,q,x,y(X(ѵ)). Then
|κ3−νκ22|={|b|(3+Ω)if|1−V|≤|3(1−Ω)b2−(pbx+aqy)(2−Ω)2|(3+Ω)b2(1−V)|b|3|3(1+Ω)b2−(pbx+aqy)(2−Ω)2|if|1−V|>|3(1−Ω)b2−(pbx+aqy)(2−Ω)2|(3+Ω)b2
Let
ℓ offered by
(1.1) be in the subfamily
Γ1,Σp,q,x,1(X(ѵ))and
℣(x,y) be Horadam
Hn+1(x). Then
|κ3−νκ22|={|bx|3if|1−V|≤|3bx2−4(pbx2+aq)|3(bx)2(1−V)|b|3|3(bx)2−4(pbx2+aq)|if|1−V|>|3bx2−4(pbx2+aq)|3(bx)2
Let
ℓ offered by
(1.1) be in the subfamily
Γ,Σp,q,x,1(X(ѵ))and
℣(x,y) be Horadam
Hn+1(x). Then
|κ3−νκ22|={|bx|4if|1−V|≤|(pbx2+aq)|4b2(1−V)|b|3|6b2−(pbx2+aq)|if|1−V|>|(pbx2+aq)|4b2
Ethical considerations
This study was not conducted on human or animal participants; therefore this field was not required.
Data availability
No data associated with this article.
Reporting guidelines
This type of study is not applicable because it is a theoretical study.
ReferencesKoshyT:
Fibonacci and Lucas Numbers with Applications.John Wiley and Sons Incorparation;2001.SaloomiMHAbdEHQahtanGA:
Coefficient bounds for biconvex and bistarlike functions associated byquasi-subordination.J. Interdiscip. Math.2021;24(3):753–764.
10.1080/09720502.2021.1884387SrivastavaHMMishraAKGochhayatP:
Certain subclasses of analytic and bi-univalent functions.Appl. Math. Lett.2010;23(10):1188–1192.
10.1016/j.aml.2010.05.009SaloomiMHAwaadBKAl-SultaniM:
An Application of generalized Fibonacci like polynomial on New Subfamilies of Regular and Bi-univalent Functions.Adv. Theory Nonlinear Anal. Appl.2023;7(4 Special Issue):181–191.DurenPL:
Subordination in complex analysis,Lecture Note in mathematics. Vol. 599.
Berlin, Germany:
Springer;1977; pp.22–29.SaloomiMHSwamySRAl-SultaniM:
quasi-subordination on pseudo and Bazilevic Bi-univalent Functions involving the Horadam Polynomial.ISMSIT 2024-8th International Symposium on Multidisciplinary Studies and Innovative Technologies.2024.OrhanHMageshNSwamySR:
Certain classes of bi-univalent functions associated with the Horadam polynomials.Acta UniSapientiae, Mathematica.2021;13(1):258–272.
10.2478/ausm-2021-0015PanwarYKSinghM:
Generalized bivariate Fibonacci-like polynomials.Internat. J. Pure Math.2014;1:8–13.DurenPL:
Univalent Functions, Grundlehren der Mathema- tischen Wissenschaften.New York, NY, USA:
Springer;1983.LewinM:
On a coefficient problem for bi-univalent functions.Proc. Am. Math. Soc.1967;18:63–68.
10.1090/S0002-9939-1967-0206255-1BrannanDClunieJ:
Aspects of contemporary complex analysis.Academic Press;1980.NetanyahuE:
The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in |z| < 1.Arch. Ration. Mech. Anal.1969;32:100–112.
10.1007/BF00247676AlamoushAG:
On a subclass of bi-univalent functions associated to Horadam polynomials.Internat. J. Open Problems Complex Analy.2020;12:58–65.AldawishIAl-HawaryTFrasinBA:
Subclasses of bi-univalent functions defined by Frasin differential operator.Mathematics.2020;8:783.
10.3390/math8050783MishraAKPanigrahiTMishraRK:
Subordination and inclusion theorems for subclasses of meromorphic functions with applications to electromagnetic cloaking.Math. Comput. Model.2013;57:945–962.
10.1016/j.mcm.2012.10.00510.5256/f1000research.191340.r459891Reviewer response for version 1OluwayemiMatthew Olanrewaju1Refereehttps://orcid.org/0000-0003-3170-6818
Nile University of Nigeria, Abuja, Nigeria
Competing interests: No competing interests were disclosed.
Title: Coefficient Estimates for New Subclasses of Bi-Univalent Functions Involving Generalized Bivariate Fibonacci-like Polynomials
Author: Husseinu M and H. Saloomi M
Journal: F1000 Research
In Review:
Title: The title “Coefficient Estimates for New Subclasses of Bi-Univalent Functions Involving Generalized Bivariate Fibonacci-like Polynomials” agrees with the contents of the work.
Abstract: The abstract provides some fundamental information about the work. For instance, a brief background to the study in one sentence or two sentences and the knowledge gap. However, the purpose of the study, aim and objectives of the work, motivation, purpose and justification of the study, methodology and findings are missing. I therefore recommend that the authors should review or modify the abstract.
Introduction: The authors considered coefficient estimates for certain classes of univalent functions (Bi-Univalent Functions) associated Generalized Bivariate Fibonacci-like Polynomials in the work using differential subordination. New and interesting results were presented in the study.
Structural adjustment is also recommended in terms of the presentation. In particular, the following sections are necessary:
1
Introduction: Introduction provided but requires some modifications and corrections as pointed out above. The authors need to provide sufficient background to the study and adequate review of relevant and current literature.
2
Preliminaries: Section 2, titled “Method” should be the preliminaries. More so, Definition 3.1 and Remark 3.2 - Remark 3.5 should also come under the second section, titled “Preliminaries”.
3
Main Results: The authors presented six (6) results in the work as follows, The section 3 titled “The family Γp,q,x,yΩ,Σ(X(ѵ)) and the coefficients bounds” and section 4, titled “Fekete-Szegö inequality for the class Mp,q,x,yΩ,Σ(X(ѵ)) and Γp,q,x,yΩ,Σ(X(ѵ))” should come under “Main Results”. However, Definition 3.1 and Remark 3.2 - Remark 3.5 should come under “Preliminaries”. The following results, Theorem 3.6, Theorem 4.3 and Theorem 4.6 with all associated corollaries should come under this section, section 3, titled “Main Results”.
Theorem 3.6: The word “Regarding” should be replaced with “if” or “For”
Theorem 4.3: The phrase “Assuming that” should be replaced with “Let” or “Suppose”. The proof of the Theorem should begin with an assumption.
Theorem 4.6: The statement of the Theorem should be reviewed or modified. In particular, the phrase “Let suppose that” is tautology and rather be written as “Let” or “Suppose that” and not both.
4
Conclusion: The conclusion in the work is not well presented by the authors. It should be well detailed. Hence, I recommend that it is modified.
References: The authors should follow the required reference style by arranging the references in the way and manner that they were cited in the body of the work. That is, [1], [2], [3], …and [14], [15] and [16]. The authors should also ensure that all cited papers are referenced and in the same manner, they also need to ensure that all reference papers are adequately cited in the body of the work. The references are current and relevant to some extent but need to be improved. Six (6) out of the fifteen (15) references representing 40% of the references are in the last ten (10) years of 2016 – 2026. I recommend that the authors update the references using the following:
https://doi.org/10.3390/axioms12040360 (Ref 1);
https://doi.org/10.1016/j.sciaf.2023.e01876 (Ref 2); https://
doi.org/10.3390/math12020169 (Ref 3);
https://doi.org/10.3390/math14020292 (Ref 4); and any other relevant and current references.
General comments: The results are correct and scientific enough for indexing in the journal, F1000 journal. However, more review of background to the study and literature review is suggested. The references also need to be updated. The authors should consider the following references:
1.
https://doi.org/10.3390/ axioms12040360;
2.
https://doi.org/10.1016/j.sciaf.2023.e01876;
3. https://
doi.org/10.3390/math12020169;
4.
https://doi.org/10.3390/math14020292; and
any other relevant and current papers
Recommendation: Recommended for indexing after necessary corrections.
Is the work clearly and accurately presented and does it cite the current literature?
Yes
If applicable, is the statistical analysis and its interpretation appropriate?
Not applicable
Are all the source data underlying the results available to ensure full reproducibility?
No source data required
Is the study design appropriate and is the work technically sound?
Yes
Are the conclusions drawn adequately supported by the results?
Partly
Are sufficient details of methods and analysis provided to allow replication by others?
Yes
Reviewer Expertise:
Complex analysis (Geometric Function Theory) and Numerical analysis (or Computational Mathematics)
I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.
References:
Coefficient Results concerning a New Class of Functions Associated with Gegenbauer Polynomials and Convolution in Terms of Subordination.
Axioms.2023;12(4) :
10.3390/axioms1204036010.3390/axioms12040360:
Harmonic functions associated with Pascal distribution series.
Scientific African.2023;21:
10.1016/j.sciaf.2023.e0187610.1016/j.sciaf.2023.e01876:
Bi-Univalency of m-Fold Symmetric Functions Associated with a Generalized Distribution.
Mathematics.2024;12(2) :
10.3390/math1202016910.3390/math12020169:
Comprehensive Subfamilies of Bi-Univalent Functions Involving a Certain Operator Subordinate to Generalized Bivariate Fibonacci Polynomials.
Mathematics.2026;14(2) :
10.3390/math1402029210.3390/math14020292HusseinuMustafaDepartment of Mathematics, University of Kerbala, Karbala, Karbala Governorate, Iraq
Competing interests: No competing interests were disclosed.
1342026
Dear Reviewers,
We would like to express our sincere gratitude for your careful review of our manuscript entitled:
“Coefficient Estimates for New Subclasses of Bi-Univalent Functions Involving Generalized Bivariate Fibonacci-like Polynomials.”
We truly appreciate your insightful comments and constructive suggestions, which have been invaluable in improving the quality, clarity, and overall presentation of our work.
We have carefully revised the manuscript in accordance with all your comments. A summary of the major revisions is provided below:
Abstract: The abstract has been completely revised to clearly present the background, aim, motivation, methodology, and main findings of the study.
Introduction: The introduction has been expanded to include a stronger background and a more comprehensive review of recent and relevant literature.
Manuscript Structure: The manuscript has been reorganized following your recommendations:
The section previously titled “Method” has been renamed to “Preliminaries” and now includes all necessary definitions and remarks.
The main results are now presented in a dedicated section titled “Main Results”.
Theorems and their corresponding corollaries have been properly arranged.
Language and Presentation: All suggested corrections related to wording and phrasing, especially in the statements of the theorems, have been carefully implemented.
Conclusion: The conclusion has been significantly improved and expanded to better emphasize the importance of the results and highlight potential directions for future research.
References: The references have been updated, reordered according to their appearance in the manuscript, and recent relevant studies have been added as recommended.
We hope that the revisions adequately address all your comments and that the manuscript is now suitable for publication.
Thank you once again for your valuable time and thoughtful feedback.
Sincerely,
Mustafa Husseinu
Mohammed H. Saloomi
10.5256/f1000research.191340.r459885Reviewer response for version 1LupasAlina Alb1Refereehttps://orcid.org/0000-0002-2855-7535
University of Oradea, Oradea, Romania
Competing interests: No competing interests were disclosed.
Coefficient Estimates for New Subclasses of Bi-Univalent Functions Involving Generalized Bivariate Fibonacci-like Polynomials
by Mustafa Husseinu, Mohammed H. Saloomi
In this paper the authors introduce new subclasses of bi-univalent functions involving generalized Fibonacci polynomials and establish coefficient estimation and Fekete–Szegö inequalities for the functions belong to the defined subclasses.
The results are new, correct and detailed. The paper is original and doesn’t contradict to ethical or policy issues, the question posed by authors is new and well defined, the methods used by authors are appropriate and well described, the data are sound and well controlled, the discussion and conclusions are well balanced, the title and abstract convey the obtained results, the writing is acceptable, the paper contains good scientific results.
The paper doesn’t require a revision.
Taking the above into consideration, I recommend the paper for Indexing.
Is the work clearly and accurately presented and does it cite the current literature?
Yes
If applicable, is the statistical analysis and its interpretation appropriate?
Not applicable
Are all the source data underlying the results available to ensure full reproducibility?
Yes
Is the study design appropriate and is the work technically sound?
Yes
Are the conclusions drawn adequately supported by the results?
Yes
Are sufficient details of methods and analysis provided to allow replication by others?
Yes
Reviewer Expertise:
Complex analysis
I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.
HusseinuMustafaDepartment of Mathematics, University of Kerbala, Karbala, Karbala Governorate, Iraq
Competing interests: No competing interests were disclosed.
1342026
Dear Reviewer,
We sincerely thank you for your careful evaluation of our manuscript and for your positive and encouraging comments. We greatly appreciate your recognition of the originality, correctness, and scientific contribution of our work.
Your valuable feedback is highly appreciated and encourages us to continue our research efforts.