Coefficient Estimates for New Subclasses of Bi-Univalent Functions Involving Generalized Bivariate Fibonacci-like Polynomials

Mustafa Husseinu; Mohammed H. Saloomi

doi:10.12688/f1000research.173513.2

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Research Article

Revised

Coefficient Estimates for New Subclasses of Bi-Univalent Functions Involving Generalized Bivariate Fibonacci-like Polynomials

[version 2; peer review: 2 approved]

Mustafa Husseinu ¹, Mohammed H. Saloomi²

PUBLISHED 28 Apr 2026

Author details Author details

¹ University of Kerbala, Karbala, Karbala Governorate, Iraq
² University of Kerbala, Karbala, Karbala Governorate, Iraq

Mustafa Husseinu
Roles: Conceptualization, Formal Analysis, Investigation, Methodology, Writing – Original Draft Preparation, Writing – Review & Editing

Mohammed H. Saloomi
Roles: Investigation, Supervision, Writing – Original Draft Preparation

OPEN PEER REVIEW

REVIEWER STATUS

This article is included in the Fallujah Multidisciplinary Science and Innovation gateway.

Abstract

The study of bi-univalent functions plays an important role in geometric function theory, particularly in determining coefficient bounds for analytic functions. Despite significant progress, there remains a need to develop broader subclasses that allow for more flexible analytical frameworks. Motivated by this, the present work introduces new subclasses of bi-univalent functions defined via generalized bivariate Fibonacci-like polynomials.

The proposed classes are constructed using subordination principles and suitable functional relations associated with these polynomials. Based on this approach, estimates for the initial coefficients are derived for functions belonging to the defined subclasses. In addition, Fekete–Szegö inequalities are established, extending several existing results in the literature.

The findings demonstrate that the use of generalized Fibonacci-type structures provides an effective tool for obtaining sharper coefficient bounds. These results contribute to the ongoing development of the theory and may serve as a basis for further investigations in related subclasses of analytic and bi-univalent functions.

Keywords

Regular function, univalent function, subordination, bi-univalent function, Fibonacci-like polynomials functions, Fekete- Szegö

Corresponding author: Mustafa Husseinu

Competing interests: No competing interests were disclosed.

Grant information: The author(s) declared that no grants were involved in supporting this work.

Copyright: © 2026 Husseinu M and H. Saloomi M. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite: Husseinu M and H. Saloomi M. Coefficient Estimates for New Subclasses of Bi-Univalent Functions Involving Generalized Bivariate Fibonacci-like Polynomials [version 2; peer review: 2 approved]. F1000Research 2026, 14:1463 (https://doi.org/10.12688/f1000research.173513.2) First published: 26 Dec 2025, 14:1463 (https://doi.org/10.12688/f1000research.173513.1) Latest published: 28 Apr 2026, 14:1463 (https://doi.org/10.12688/f1000research.173513.2)

Revised Amendments from Version 1

We would like to thank you for the opportunity to revise and resubmit our manuscript entitled:
“Coefficient Estimates for New Subclasses of Bi-Univalent Functions Involving Generalized Bivariate Fibonacci-like Polynomials.”
We sincerely appreciate the valuable comments and suggestions provided by the reviewers, which have helped us to significantly improve the quality and clarity of our work.
We are pleased to submit the revised version of the manuscript. All the comments raised by the reviewer have been carefully addressed. A summary of the major revisions is provided below:
Abstract: The abstract has been completely revised to clearly include the background, aim, motivation, methodology, and main findings of the study.
Introduction: The introduction has been expanded to provide a stronger background and a more comprehensive review of recent and relevant literature.
Manuscript Structure: The manuscript has been reorganized according to the reviewer’s recommendations:
The section previously titled “Method” has been changed to “Preliminaries” and now includes all necessary definitions and remarks.
The main results have been restructured under a dedicated section titled “Main Results”.
Theorems and their corresponding corollaries have been rearranged appropriately.
Language and Presentation: All suggested corrections regarding wording and phrasing (particularly in the statements of the theorems) have been carefully implemented.
Conclusion: The conclusion section has been significantly improved and expanded to better highlight the importance of the results and possible future research directions.
References: The references have been updated, reordered according to their appearance in the manuscript, and recent relevant studies have been added as recommended.
We believe that the revised manuscript has been substantially improved and now meets the standards required for publication.
Thank you for your time and consideration. We look forward to your positive response.
Sincerely,
Mustafa Husseinu
Mohammed H. Saloomi

See the authors' detailed response to the review by Matthew Olanrewaju Oluwayemi
See the authors' detailed response to the review by Alina Alb Lupas

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 \geq 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) = p x ℣_{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 \in ℝ

℣_{0} (x, y) = a, ℣_{1} (x, y) = b, p x, qy \neq 0 and {(p x)}^{2} + 4 qy \neq 0

”.

For the Fibonacci-like polynomials, generating function as denoted in Ref. 8, has the form:

℣^{(x, y)} (v) = \sum ℣_{n} (x, y) v^{n} = \frac{a + (b - ap x) v}{a - p x v - qy v^{2}}

The polynomials for the special cases can be obtained as presented in Table 1 for and of $p, q, a, b and y$ .

Table 1. 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, $P_{n} (x)$
$(x, y)$	$(2, x)$	$(1, 1)$	Bivariate Lucas $L_{n} (x, y)$
$(t, - 1)$	$(1, 2 t)$	$(2, 1)$	Chebyshev of the second kind $U_{n} (x)$
$(x, 1)$	$(a, bx)$	$(p, q)$	Horadam $H_{n + 1} (x)$

Consider $ℓ$ to be the function of the form

(1.1)

ℓ (v) = v + \sum_{m = 2}^{\infty} κ_{m} v^{m},

that is part of Å where Å symbolizes the class of regular functions recognized on the disk

Ų = {v \in 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 \in S

, the image

t = ℓ (v), v \in 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 theorem ℏ \in S

has an inverse

ℏ^{- 1}

, that holds

ℓ^{- 1} (ℓ (v)) = v, (v \in Ų),

and,

ℓ (ℓ^{- 1} (t)) = t, (| t | < r_{0} (ℓ), r_{0} (ℓ) \geq \frac{1}{4}),

where

(1.2)

ᶂ (t) = ℓ^{- 1} (t) = t - ɛ_{2} t^{2} + (2 ɛ_{2}^{2} - ɛ_{3}) t^{3} - (5 ɛ_{2}^{3} - 5 ɛ_{2} ɛ_{3} + ɛ_{4}) t^{4} + \cdot \cdot \cdot

In Ų, the function $ℓ \in Å$ is named bi-univalent if both $ℓ$ and $ℓ^{- 1}$ are univalent and the set of all bi-univalent functions are signified by Σ. Lewin¹⁰ presented this set and revealed that $| ɛ_{2} | \leq 1.5$ regarding the function in the set Σ. Lately, Brannan and Clunie¹¹ estimated that $| ɛ_{2} | \leq \sqrt{2}$ and Netanyahu in¹² showed that $| ɛ_{2} | = \frac{4}{3} .$

Many authors recently offered and explored numerous remarkable subclasses of bi-univalent functions.^{2–4,13–17}

The regular function $ℓ$ is subordinate to $ᶂ$ , written as

(1.3)

ℓ ≺ ᶂ or ℓ (v) ≺ ᶂ (v), (ѵ \in Ų),

if there is regular function

ƙ : Ų \to Ų

, with

ƙ (0) = 0

and

| ƙ (v) | < 1

such that

ℓ (v) = ᶂ (ƙ (v)), v \in Ų .

It follows the definition stating that:

ℓ (0) = ᶂ (0) and ℓ (Ų) \subset ᶂ (Ų) .

see Refs. 5,6,15,18,19.

Recent studies, including Refs. 16–19, have emphasized the significant role of generalized polynomials and subordination techniques in obtaining coefficient estimates for various subclasses of bi-univalent functions, which further motivates the present investigation.

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 | $ɛ_{2}$ | and | $ɛ_{3}$ | are obtained for functions of the new classes.

2. Preliminaries

The Family $M_{Ω, Σ}^{p, q, x, y} (X (ѵ))$ and the Coefficients Bounds

Definition 2.1: A function $ℓ \in \sum$ offered by (1.1) is stated to be in the set $M_{Ω, Σ}^{p, q, x, y} (X (ѵ)), 0 \leq Ω \leq 1,$ if it fits:

(2.1)

[(1 - Ω) \frac{ℓ (z)}{z} + Ω \frac{1 + \frac{z ℓ^{''} (z)}{ℓ^{'} (z)}}{\frac{z ℓ^{'} (z)}{ℓ (z)}}] ≺ X (ѵ) = ℣^{(x, y)} (ѵ) + 1 - a, (ѵ \in Ų)

(2.2)

[(1 - Ω) \frac{ℓ (w)}{w} + Ω \frac{1 + \frac{w ℓ^{''} (w)}{ℓ^{'} (w)}}{\frac{w ℓ^{'}}{ℓ (w)}}] ≺ X (t) = ℣^{(x, y)} (t) + 1 - a, (t \in Ų) ◾

By assigning special values to parameters $Ω$ and replacing polynomials, new and prominent families of bi-univalent functions are obtained. Accordingly, novel bounds for the initial coefficients are realized.

Remark 2.2:

For $Ω = 1$ , in $M_{Ω, Σ}^{p, q, x, y} (X (ѵ))$ , we get the subfamilies ◾

Remark 2.3:

Given a Horadam polynomial $℣^{(x, y)}$ with b = b $x$ and y = 1, we obtain $M_{Ω, Σ}^{p, q, x, 1} (X (ѵ))$ . In this case for $Ω = 1$ belongs to this family. Later, we have families $L_{Σ} (x)$ .⁷

Remark 2.4:

Consider $℣^{(x, y)}$ to be Chebyshev polynomials with p = 2, q = 1, $a$ = 1, b = 2t, $x$ = t, y = -1. In Theorem 2.7 there is $M_{Ω, Σ}^{2, 1, x, - 1} (X (ѵ)) .$ In such incident for $Ω = 0, 1$ belongs to this family. Then, the new families $M_{Σ}^{2, 1, x, - 1} (X (ѵ))$ and $M_{1, Σ}^{2, 1, x, - 1} (X (ѵ))$ respectively are gained.

Remark 2.5:

Regard $℣^{(x, y)}$ as Fibonacci polynomials with p = q = 1, b = 1, y = 1, $a$ = 0, there is. $M_{Ω, Σ}^{1, 1, x, 1} (X (ѵ))$ . In this situation for $Ω = 0, 1$ be in this family. Then, new families $M_{Σ}^{1, 1, x, 1} (X (ѵ))$ and $M_{1, Σ}^{1, 1, x, 1} (X (ѵ))$ individually are got.

The Family $Γ_{Ω, Σ}^{p, q, x, y} (X (ѵ))$ and the Coefficients Bounds

Definition 2.6:

For $0 \leq Ω \leq 1,$ a function $ℓ \in \sum$ offered by (1.1) is stated to be in the set $Γ_{Ω, Σ}^{p, q, x, y} (X (ѵ))$ if the following settings are satisfied:

(2.3)

[(1 - Ω) ℓ^{'} (z) + Ω \frac{1 + \frac{z ℓ^{''}}{ℓ^{'}}}{\frac{z ℓ^{'}}{ℓ (z)}}] ≺ X (ѵ) = ℣^{(x, y)} (ѵ) + 1 - a, (ѵ \in Ų)

(2.4)

[(1 - Ω) ℓ^{'} (w) + Ω \frac{1 + \frac{w ℓ^{''}}{ℓ^{'}}}{\frac{w ℓ^{'}}{ℓ (w)}}] ≺ X (t) = ℣^{(x, y)} (t) + 1 - a, (t \in Ų) .

By assigning special values to parameters $Ω$ and replacing polynomials, new and known families of bi-univalent functions are obtained. Thus, new bounds for the initial coefficients are gained.

Remark 2.7:

For $Ω = 1$ , in $Γ_{Ω, Σ}^{p, q, x, y} (X (ѵ))$ , the subfamilies are got.

Remark 2.8:

Consider $℣^{(x, y)}$ to be Horadam polynomial with b = b $x$ and y = 1. Then, there is $Γ_{Ω, Σ}^{p, q, x, 1} (X (ѵ))$ . For $Ω = 1$ be in this family. Then, families $L_{Σ} (x)$ ⁷ are got.

Remark 2.9:

Let $℣^{(x, y)}$ is Chebyshev polynomials with p = 2, b = 2t $, x$ = t, y = −1, q = 1, $a$ = 1 in $Γ_{Ω, Σ}^{p, q, x, y} (X (ѵ)) . There is$ $Γ_{Ω, Σ}^{2, 1, x, - 1} (X (ѵ))$ . In such condition, for $Ω = 0, 1$ be in this family. Accordingly, new families $Γ_{Σ}^{2, 1, x, - 1} (X (ѵ))$ and $Γ_{1, Σ}^{2, 1, x, - 1} (X (ѵ))$ separately are obtained.

Remark 2.10:

Let $℣^{(x, y)}$ is Fibonacci polynomials with p = q =1, b = 1, y = 1, $a$ = 0, there is $Γ_{Ω, Σ}^{1, 1, x, 1} (X (ѵ)) .$ In this case for $Ω = 0, 1$ be in this family. Then, new families $Γ_{Σ}^{1, 1, x, 1} (X (ѵ))$ and $Γ_{1, Σ}^{1, 1, x, 1} (X (ѵ))$ respectively are gained.

Fekete-Szegö inequality for the class $M_{Ω, Σ}^{p, q, x, y} (X (ѵ))$ and $Γ_{Ω, Σ}^{p, q, x, y} (X (ѵ))$

Through applying the class $M_{Ω, Σ}^{p, q, x, y} (X (ѵ))$ and $Γ_{Ω, Σ}^{p, q, x, y} (X (ѵ))$ to the Fekete-Szegö problem, the following results and notes are realized:

Remark 2.11:

Let $℣^{(x, y)}$ is Horadam polynomial with b = b $x$ and y = 1, we obtain $M_{Ω, Σ}^{p, q, x, 1} (X (ѵ))$ . In this case for $Ω = 1$ belongs to this family. Then, families $L_{Σ} (x)$ ⁷ are gained.

Remark 2.12:

If b = b $x$ and y = 1 and $℣^{(x, y)}$ is Horadam polynomial, we obtain $M_{, Σ}^{p, q, x, 1} (X (ѵ))$ .

Remark 2.13:

Given a Horadam polynomial $℣^{(x, y)}$ with b = b $x$ and y = 1, we obtain $Γ_{1, Σ}^{p, q, x, 1} (X (ѵ))$ . In this case for $Ω = 1$ be in this family. Then, families $L_{Σ} (x)$ ⁷ are obtained.

Remark 2.14:

Let $℣^{(x, y)}$ is Horadam polynomial with b = bx and y = 1, we have $Γ_{, Σ}^{p, q, x, 1} (X (ѵ))$ .

3. Main Results

Theorem 3.1:

For, $0 \leq Ω \leq 1,$ let $ℓ \in Σ$ given by (1.1) belong to $M_{Ω, Σ}^{p, q, x, y} (X (ѵ))$ , then

(3.1)

| κ_{2} | \leq \frac{| b | \sqrt{| b |}}{\sqrt{| b^{2} (1 - Ω) - (pbx + aqy) |}}

(3.2)

| κ_{3} | \leq b^{2} + \frac{| b |}{(1 + 3 Ω)} ◾

Proof:

Since $ℓ \in M_{Ω, Σ}^{p, q, x, y} (X (ѵ))$ , there exist regular functions $i, λ : \hat{E} \to \hat{E}$ , such that

(3.3)

i (ѵ) = \sum_{1}^{\infty} i_{k} ѵ^{k}, λ (t) = \sum_{1}^{\infty} λ_{k} t^{k} and | i_{k} | \leq 1, | λ_{k} | \leq 1,

we can write

(3.4)

[(1 - Ω) \frac{ℓ (z)}{z} + Ω \frac{1 + \frac{z ℓ^{''} (z)}{ℓ^{'} (z)}}{\frac{z ℓ^{'} (z)}{ℓ (z)}}] = X (ѵ), (ѵ \in Ų)

(3.5)

[(1 - Ω) \frac{ℓ (w)}{w} + Ω \frac{1 + \frac{w ℓ^{''} (w)}{ℓ^{'} (w)}}{\frac{w ℓ^{'}}{ℓ (w)}}] = X (t), (t \in Ų) ◾

Equivalently,

(3.6)

1 + κ_{2} z + {(1 + 3 Ω) κ_{3} - 4 Ω {κ_{2}}^{2}} z^{2} + . . . = 1 + ℣_{1} (x, y) i_{1} z + (℣_{1} (x, y) i_{2} + ℣_{2} (x, y) {i_{1}}^{2}) z^{2} + \dots

and

(3.7)

1 - κ_{2} w + {2 (1 + Ω) {κ_{2}}^{2} - (1 + 3 Ω) κ_{3}} w^{2} - . . . = 1 + ℣_{1} (x, y) λ_{1} w + (℣_{1} (x, y) λ_{2} + ℣_{2} (x, y) {λ_{1}}^{2}) w^{2} + \dots

Thus upon comparing the corresponding coefficients in (3.6) and (3.7), we have

(3.8)

κ_{2} = ℣_{1} (x, y) i_{1}

(3.9)

(1 + 3 Ω) κ_{3} - 4 Ω {κ_{2}}^{2} = ℣_{1} (x, y) i_{2} + ℣_{2} (x, y) {i_{1}}^{2}

(3.10)

- κ_{2} = ℣_{1} (x, y) λ_{1}

and

(3.11)

2 (1 + Ω) {κ_{2}}^{2} - (1 + 3 Ω) κ_{3} = ℣_{1} (x, y) λ_{2} + ℣_{2} (x, y) {λ_{1}}^{2} ◾

Now considering (3.8) and (3.10), we get

(3.12)

i_{1} = - λ_{1}

(3.13)

\begin{matrix} 2 {κ_{2}}^{2} = {℣^{2}}_{1} (x, y) ({i_{1}}^{2} + {λ_{1}}^{2}) \\ {κ_{2}}^{2} = \frac{{℣^{2}}_{1} (x, y) ({i_{1}}^{2} + {λ_{1}}^{2})}{2} ◾ \end{matrix}

If we add (3.9) to (3.11), we find that

(3.14)

2 (1 - Ω) {κ_{2}}^{2} = ℣_{1} (x, y) (i_{2} + λ_{2}) + ℣_{2} (x, y) ({i_{1}}^{2} + {λ_{1}}^{2})

$Upon substituting$ the value of ${i_{1}}^{2} + {λ_{1}}^{2}$ from (3.13) in (3.14), we deduce the next result:

2 (1 - Ω) {κ_{2}}^{2} = ℣_{1} (x, y) (i_{2} + λ_{2}) + ℣_{2} (x, y) \frac{2 {κ_{2}}^{2}}{{℣^{2}}_{1} (x, y)}

2 {℣^{2}}_{1} (x, y) (1 - Ω) {κ_{2}}^{2} = {℣^{3}}_{1} (x, y) (i_{2} + λ_{2}) + ℣_{2} (x, y) 2 {κ_{2}}^{2} ◾

And more simplify

2 ({℣^{2}}_{1} (x, y) (1 - Ω) - ℣_{2} (x, y)) {κ_{2}}^{2} = {℣^{3}}_{1} (x, y) (i_{2} + λ_{2})

{κ_{2}}^{2} = \frac{{℣^{3}}_{1} (x, y) (i_{2} + λ_{2})}{2 ({℣^{2}}_{1} (x, y) (1 - Ω) - ℣_{2} (x, y))}

{| κ_{2} |}^{2} \leq \frac{| {℣^{3}}_{1} (x, y) |}{| {℣^{2}}_{1} (x, y) (1 - Ω) - ℣_{2} (x, y) |}

Using the inequality $| i_{k} | \leq 1, | λ_{k} | \leq 1$ ◾

(3.15)

| κ_{2} | \leq \frac{| b | \sqrt{| b |}}{\sqrt{| b^{2} (1 - Ω) - (pbx + aqy) |}} .

Subtracting (3.9) and (3.11), and with some computation, we have

(3.16)

2 (1 + 3 Ω) κ_{3} - 2 (1 + 3 Ω) {κ_{2}}^{2} = ℣_{1} (x, y) (i_{2} - λ_{2})

Put (3.13) in (3.16), then we deduce

(3.17)

\begin{matrix} | κ_{3} | \leq \frac{| {℣^{2}}_{1} (x, y) ({i_{1}}^{2} + {λ_{1}}^{2}) |}{2} + \frac{| ℣_{1} (x, y) (i_{2} - λ_{2}) |}{2 (1 + 3 Ω)} . \\ | κ_{3} | \leq b^{2} + \frac{| b |}{(1 + 3 Ω)} ◾ \end{matrix}

Corollary 3.2:

(i) Consider $ℓ$ presented by (1.1) be in the subfamily $M_{1, Σ}^{p, q, x, y} (X (ѵ))$ . Then,
$| κ_{2} | \leq \frac{| b | \sqrt{| b |}}{\sqrt{| pb x + aqy |}} .$

| κ_{3} | \leq b^{2} + \frac{| b |}{4} ◾

(ii) Let $ℓ$ offered by (1.1) be in the sub family $M_{Σ}^{p, q, x, y} (X (ѵ))$ . Then,
$| κ_{2} | \leq \frac{| b | \sqrt{| b |}}{\sqrt{| b^{2} - (pbx + aqy) |}} .$

| κ_{3} | \leq b^{2} + | b | ∎

Corollary 3.3:

(i) Regard $ℓ$ introduced by (1.1) be in the subfamily $M_{Ω, Σ}^{p, q, x, 1} (X (ѵ))$ and $℣ (x, y)$ be Horadam $H_{n + 1} (x)$ . Then,
$| κ_{2} | \leq \frac{| bx | \sqrt{| b x |}}{\sqrt{| b^{2} x^{2} (1 - Ω) - (pb x^{2} + aq) |}} .$

$| κ_{3} | \leq b^{2} x^{2} + \frac{| bx |}{(1 + 3 Ω)} ◾$
(ii) Let $ℓ$ offered by (1.1) and $Ω = 1$ be in the sub family $M_{, Σ}^{p, q, x, 1} (X (ѵ))$ . Then,
$| κ_{2} | \leq \frac{| bx | \sqrt{| b x |}}{\sqrt{| b^{2} x^{2} - (pb x^{2} + aq) |}} .$

$| κ_{3} | \leq b^{2} x^{2} + | b x | ◾$

Corollary 3.4:

(i) Let $ℓ \in Σ$ shown by (1.1) in the subfamily $M_{Ω, Σ}^{2, 1, x, - 1} (X (ѵ))$ and $℣^{(x, y)}$ be Chebyshev polynomials with p = 2, b = 2t, $x$ = t, y = −1, q = 1, and $a$ = 1. Then,
$| κ_{2} | \leq \frac{| 2 t | \sqrt{| 2 t |}}{\sqrt{| 4 t^{4} (1 - Ω) - (4 t^{3} - 1) |}} .$

$| κ_{3} | \leq 4 t^{4} + \frac{2 | t^{2} |}{(1 + 3 Ω)} ◾$
(ii) Consider $ℓ \in Σ$ offered by (1.1) in the subfamily $M_{, Σ}^{2, 1 x, - 1} (X (ѵ))$ . Then,
$| κ_{2} | \leq \frac{| 2 t | \sqrt{| 2 t |}}{\sqrt{| 4 t^{4} - (4 t^{3} - 1) |}} .$

$| κ_{3} | \leq 4 t^{4} + 2 | t^{2} | ◾$
(iii) Let $ℓ \in Σ$ presented by (1.1) in the subfamily $M_{1, Σ}^{2, 1, x, - 1} (X (ѵ))$ . Then,
$| κ_{2} | \leq \frac{| 2 t | \sqrt{| 2 t |}}{\sqrt{| (4 t^{3} - 1) |}} .$

$| κ_{3} | \leq 4 t^{4} + \frac{| t^{2} |}{2} ◾$

Corollary 3.5:

(i) Consider $ℓ \in Σ$ stated by (1.1) in the subfamily $M_{Ω, Σ}^{1, 1, x, 1} (X (ѵ))$ and $℣^{(x, y)}$ be Fibonacci polynomials with p = 1, q = 1, b = 1, y = 1, and $a$ = 0. Then,
$| κ_{2} | \leq \frac{1}{\sqrt{| (1 - Ω) - x |}} .$

$| κ_{3} | \leq 1 + \frac{1}{(1 + 3 Ω)} ◾$
(ii) Let $ℓ \in Σ$ revealed by (1.1) in the subfamily $M_{, Σ}^{1, 1 x, 1} (X (ѵ))$ . Then,
$| κ_{2} | \leq \frac{1}{\sqrt{| 1 - x |}} .$

$| κ_{3} | \leq 2 ◾$
(iii) Let $ℓ \in Σ$ stated by (1.1) in the set $M_{1, Σ}^{1, 1, x, 1} (X (ѵ))$
$| κ_{2} | \leq \frac{1}{\sqrt{| x |}} .$

$| κ_{3} | \leq \frac{5}{4} ◾$

Theorem 3.6:

For $0 \leq Ω \leq 1,$ let $ℓ \in Σ$ offered by (1.1) belong to $Γ_{Ω, Σ}^{p, q, x, y} (X (ѵ))$ . Then,

(3.18)

| κ_{2} | \leq \frac{| b | \sqrt{| b |}}{\sqrt{| 3 (1 - Ω) b^{2} - {(2 - Ω)}^{2} (pbx + aqy) |}} .

(3.19)

| κ_{3} | \leq \frac{(3 - Ω) b^{2}}{(3 + Ω) {(2 - Ω)}^{2}} + \frac{| b |}{(3 + Ω)} .

Proof:

As $ℓ \in Γ_{Ω, Σ}^{p, q, x, y} (X (ѵ))$ , there exists systematic functions $i, λ : \hat{E} \to \hat{E}$ , such that

(3.20)

i (ѵ) = \sum_{1}^{\infty} i_{k} ѵ^{k}, λ (t) = \sum_{1}^{\infty} λ_{k} t^{k} and | i_{k} | \leq 1, | λ_{k} | \leq 1,

It is possible to write:

[(1 - Ω) ℓ^{'} (z) + Ω \frac{1 + \frac{z ℓ^{''} (z)}{ℓ^{'} (z)}}{\frac{z ℓ^{'} (z)}{ℓ (z)}}] = X (ѵ), (ѵ \in Ų)

[(1 - Ω) ℓ^{'} (w) + Ω \frac{1 + \frac{w ℓ^{''} (w)}{ℓ^{'} (w)}}{\frac{w ℓ^{'}}{ℓ (w)}}] = X (t), (t \in Ų)

(3.21)

1 + (2 - Ω) κ_{2} z + [(3 + Ω) κ_{3} - 4 Ω {κ_{2}}^{2}] z^{2} \dots . = 1 + ℣_{1} (x, y) ɍ_{1} z + (℣_{1} (x, y) ɍ_{2} + ℣_{2} (x, y) {ɍ_{1}}^{2}) z^{2} + \dots

and

(3.22)

1 - (2 - Ω) κ_{2} w + [(6 - 2 Ω) {κ_{2}}^{2} - (3 + Ω) κ_{3}] w^{2} . . . = 1 + ℣_{1} (x, y) ѳ_{1} w + (℣_{1} (x, y) ѳ_{2} + ℣_{2} (x, y) {ѳ_{1}}^{2}) w^{2} + \dots

Thus upon comparing the corresponding coefficients in (3.21) and (3.22), we have

(3.23)

(2 - Ω) κ_{2} = ℣_{1} (x, y) ɍ_{1}

(3.24)

(3 + Ω) κ_{3} - 4 Ω {κ_{2}}^{2} = ℣_{1} (x, y) ɍ_{2} + ℣_{2} (x, y) {ɍ_{1}}^{2}

(3.25)

- (2 - Ω) κ_{2} = ℣_{1} (x, y) ѳ_{1}

(3.26)

- [(3 + Ω) κ_{3} - (6 - 2 Ω) {κ_{2}}^{2}] = ℣_{1} (x, y) ѳ_{2} + ℣_{2} (x, y) {ѳ_{1}}^{2}

Now considering (3.23) and (3.25), we get

(3.27)

ɍ_{1} = - ѳ_{1}

(3.28)

\begin{matrix} 2 {(2 - Ω)}^{2} {κ_{2}}^{2} = {℣^{2}}_{1} (x, y) ({ɍ_{1}}^{2} + {ѳ_{1}}^{2}) \\ {κ_{2}}^{2} = \frac{{℣^{2}}_{1} (x, y) ({ɍ_{1}}^{2} + {ѳ_{1}}^{2})}{2 {(2 - Ω)}^{2}} \end{matrix}

In adding (3.24) to (3.26), we obtain:

[6 (1 - Ω) {℣^{2}}_{1} (x, y) - 2 ℣_{2} (x, y) {(2 - Ω)}^{2}] {κ_{2}}^{2} = {℣^{3}}_{1} (x, y) (ɍ_{2} + ѳ_{2})

{κ_{2}}^{2} = \frac{{℣^{3}}_{1} (x, y) (ɍ_{2} + ѳ_{2})}{6 (1 - Ω) {℣^{2}}_{1} (x, y) - 2 ℣_{2} (x, y) {(2 - Ω)}^{2}}

{| κ_{2} |}^{2} \leq \frac{| {℣^{3}}_{1} (x, y) |}{| 3 (1 - Ω) {℣^{2}}_{1} (x, y) - ℣_{2} (x, y) {(2 - Ω)}^{2} |}

| κ_{2} | \leq \frac{| b | \sqrt{| b |}}{\sqrt{| 3 (1 - Ω) b^{2} - {(2 - Ω)}^{2} (pbx + aqy) |}} .

Subtracting (3.24) and (3.26), and with some computation, we get

(3.29)

\begin{matrix} (3 + Ω) κ_{3} - 4 Ω {κ_{2}}^{2} = ℣_{1} (x, y) ɍ_{2} + ℣_{2} (x, y) {ɍ_{1}}^{2} \\ - [(3 + Ω) κ_{3} - (6 - 2 Ω) {κ_{2}}^{2}] = ℣_{1} (x, y) ѳ_{2} + ℣_{2} (x, y) {ѳ_{1}}^{2} \\ 2 (3 + Ω) κ_{3} - (6 + 2 Ω) {κ_{2}}^{2} = ℣_{1} (x, y) (ɍ_{2} - ѳ_{2}) \end{matrix}

Put (3.28) in (3.29)

2 (3 + Ω) κ_{3} - (6 + 2 Ω) \frac{{℣^{2}}_{1} (x, y) ({ɍ_{1}}^{2} + {ѳ_{1}}^{2})}{2 {(2 - Ω)}^{2}} = ℣_{1} (x, y) (ɍ_{2} - ѳ_{2})

4 (3 + Ω) {(2 - Ω)}^{2} κ_{3} - (6 - 2 Ω) {℣^{2}}_{1} (x, y) ({ɍ_{1}}^{2} + {ѳ_{1}}^{2}) = 2 {(2 - Ω)}^{2} ℣_{1} (x, y) (ɍ_{2} - ѳ_{2})

κ_{3} = \frac{(6 - 2 Ω) {℣^{2}}_{1} (x, y) ({ɍ_{1}}^{2} + {ѳ_{1}}^{2})}{4 (3 + Ω) {(2 - Ω)}^{2}} + \frac{2 {(2 - Ω)}^{2} ℣_{1} (x, y) (ɍ_{2} - ѳ_{2})}{4 (3 + Ω) {(2 - Ω)}^{2}}

| κ_{3} | \leq \frac{(3 - Ω) b^{2}}{(3 + Ω) {(2 - Ω)}^{2}} + \frac{| b |}{(3 + Ω)} ◾

Corollary 3.7:

(i) Let $ℓ$ stated by (1.1) be in the subfamily $Γ_{Σ}^{p, q, x, y} (X (ѵ))$ . Then,
$| κ_{2} | \leq \frac{| b | \sqrt{| b |}}{\sqrt{| 3 b^{2} - 4 (pbx + aqy) |}} .$

$| κ_{3} | \leq \frac{b^{2}}{4} + \frac{| b |}{3} ◾$
(ii) Let $ℓ$ given by (1.1) be in the subfamily $Γ_{1, Σ}^{p, q, x, y} (X (ѵ))$ . Then,
$| κ_{2} | \leq \frac{| b | \sqrt{| b |}}{\sqrt{| (pb x + aqy) |}} .$

$| κ_{3} | \leq \frac{| b |}{4} ◾$

Corallary 3.8:

If $℣_{n}$ is Horadam polynomial in above theorem Then, we have

\begin{matrix} | κ_{2} | \leq \frac{| bx | \sqrt{| bx |}}{\sqrt{| 3 (1 - Ω) {(bx)}^{2} - {(2 - Ω)}^{2} (pb x^{2} + aq) |}}, \\ | κ_{3} | \leq \frac{3 (1 - Ω) {(bx)}^{2}}{(3 + Ω) {(2 - Ω)}^{2}} + \frac{| bx |}{(3 + Ω)} ◾ \end{matrix}

Corallary 3.9:

Where $℣_{n} (x, y) = p x ℣_{n - 1} (x, y) - qy ℣_{n - 2} (x, y), n \geq 2$ ,

℣_{0} (x, y) = a, ℣_{1} (x, y) = b

If $℣_{n}$ is generalized bivariate Fibonacci-like polynomials and Ω = 0. Then,

| κ_{2} | \leq \frac{| b | \sqrt{| b |}}{\sqrt{3 b^{2} - 4 (pbx - aqy)}} .

Corallary 3.10:

If $℣_{n}$ is Horadam polynomial and Ω = 0 (remark)

{H^{3}}_{1} (x, y) = {(bx)}^{3}, H_{2} (x, y) = pb x^{2} - aq

| κ_{2} | \leq \frac{| bx | \sqrt{| bx |}}{\sqrt{3 b^{2} x^{2} - 4 (pb x^{2} - aq)}} .

| κ_{3} | \leq \frac{{(bx)}^{2}}{4} + \frac{| bx |}{3} ◾

Theorem 3.11:

Suppose $ℓ (ѵ) = ѵ + \sum_{m = 2}^{\infty} ɛ_{m} ѵ^{m} \in M_{Ω, Σ}^{p, q, x, y} (X (ѵ))$ , then,

| κ_{3} - ν {κ_{2}}^{2} | = {\begin{cases} \frac{| b |}{(1 + 3 Ω)} & if & | 1 - V | \leq \frac{| (1 - Ω) b^{2} - (pbx + aqy) |}{(1 + 3 Ω) b^{2}} \\ \frac{(1 - V) {| b |}^{3}}{| (1 - Ω) b^{2} - (pbx + a qy) |} & if & | 1 - V | > \frac{| (1 - Ω) b^{2} - (pbx + aqy) |}{(1 + 3 Ω) b^{2}} \end{cases}

Proof:

κ_{3} - ν {κ_{2}}^{2} = \frac{℣_{1} (x, y) (ɍ_{2} - ѳ_{2})}{2 (1 + 3 Ω)} + {κ_{2}}^{2} - ν {κ_{2}}^{2}

κ_{3} - {νκ}_{2}^{2} = \frac{b (ɍ_{2} - ѳ_{2})}{2 (1 + 3 Ω)} + \frac{b^{3} (ɍ_{2} + ѳ_{2})}{2 (b^{2} (1 - Ω) - (pbx + aqy))} (1 - ν)

κ_{3} - {νκ}_{2}^{2} = {\frac{(1 - ν) b^{3}}{2 (b^{2} (1 - Ω) - (pbx + aqy))} + \frac{b}{2 (1 + 3 Ω)}} ɍ_{2} + {\frac{(1 - ν) b^{3}}{2 (b^{2} (1 - Ω) - (pbx + aqy))} - \frac{b}{2 (1 + 3 Ω)}} ѳ_{2}

κ_{3} - {νκ}_{2}^{2} = {h (ν, Ω) + \frac{b}{2 (1 + 3 Ω)}} ɍ_{2} + {h (ν, Ω) - \frac{b}{2 (1 + 3 Ω)}} ѳ_{2}

Where

h (ν, Ω) = \frac{(1 - ν) b^{3}}{2 (b^{2} (1 - Ω) - (pbx + aqy))}

κ_{3} - {νκ}_{2}^{2} = (h (ν) + \frac{b}{2 (3 + Ω)}) ɍ_{2} + (h (ν) - \frac{b}{2 (3 + Ω)}) ѳ_{2}

Corollary 3.12:

Let $ℓ$ assumed by (1.1) be in the subfamily $M_{1, Σ}^{p, q, x, 1} (X (ѵ))$ and $℣ (x, y)$ be Horadam $H_{n + 1} (x)$ . Then,

| κ_{3} - ν {κ_{2}}^{2} | = {\begin{cases} \frac{| bx |}{4} & if & | 1 - V | \leq \frac{| p {bx}^{2} + aq |}{4 {(bx)}^{2}} \\ \frac{(1 - V) {| bx |}^{3}}{| p {bx}^{2} + aq |} & if & | 1 - V | > \frac{| p {bx}^{2} + aq |}{4 {(bx)}^{2}} \end{cases}

Corollary 3.13:

Let $ℓ$ offered by (1.1) be in the subfamily $M_{, Σ}^{p, q, x, 1} (X (ѵ))$ and $℣ (x, y)$ be Horadam $H_{n + 1} (x)$ . Then,

| κ_{3} - ν {κ_{2}}^{2} | = {\begin{cases} | b x | & if & | 1 - V | \leq \frac{| {(b x)}^{2} - (pb x^{2} + aq) |}{{(b x)}^{2}} \\ \frac{(1 - V) {| b x |}^{3}}{| {(b x)}^{2} - (pb x^{2} + aq) |} & if & | 1 - V | > \frac{| {(b x)}^{2} - (pb x^{2} + aq) |}{{(b x)}^{2}} \end{cases}

Theorem 3.14:

Suppose that $ℓ (ѵ) = ѵ + \sum_{m = 2}^{\infty} ɛ_{k} ѵ^{m} \in Γ_{Ω, Σ}^{p, q, x, y} (X (ѵ))$ . Then

| κ_{3} - ν {κ_{2}}^{2} | = {\begin{cases} \frac{| b |}{(3 + Ω)} & if & | 1 - V | \leq \frac{| 3 (1 - Ω) b^{2} - (pbx + aqy) {(2 - Ω)}^{2} |}{(3 + Ω) b^{2}} \\ \frac{(1 - V) {| b |}^{3}}{| 3 (1 + Ω) b^{2} - (pbx + aqy) {(2 - Ω)}^{2} |} & if & | 1 - V | > \frac{| 3 (1 - Ω) b^{2} - (pbx + aqy) {(2 - Ω)}^{2} |}{(3 + Ω) b^{2}} \end{cases}

Proof:

κ_{3} - {vκ}_{2}^{2} = \frac{℣_{1} (x, y) (ɍ_{2} - ѳ_{2})}{2 (3 + Ω)} + \frac{(6 - 2 Ω)}{2 (3 + Ω)} {κ_{2}}^{2} - ν {κ_{2}}^{2}

κ_{3} - {vκ}_{2}^{2} = \frac{℣_{1} (x, y) (ɍ_{2} - ѳ_{2})}{2 (3 + Ω)} + (1 - ν) \frac{{℣^{3}}_{1} (x, y) (ɍ_{2} + ѳ_{2})}{6 (1 - Ω) {℣^{2}}_{1} (x, y) - 2 ℣_{2} (x, y) {(2 - Ω)}^{2}}

κ_{3} - {vκ}_{2}^{2} = (\frac{(1 - ν) b^{3}}{6 (1 - Ω) b^{2} - 2 (pbx + aqy) {(2 - Ω)}^{2}} + \frac{b}{2 (3 + Ω)}) ɍ_{2} + (\frac{(1 - ν) b^{3}}{6 (1 - Ω) b^{2} - 2 (pbx + aqy) {(2 - Ω)}^{2}} - \frac{b}{2 (3 + Ω)}) ѳ_{2}

put

h (ν) = \frac{(1 - ν) b^{3}}{6 (1 - Ω) b^{2} - 2 (pbx + aqy) {(2 - Ω)}^{2}}

κ_{3} - {vκ}_{2}^{2} = (h (ν) + \frac{b}{2 (3 + Ω)}) ɍ_{2} + (h (ν) - \frac{b}{2 (3 + Ω)}) ѳ_{2}

| κ_{3} - ν {κ_{2}}^{2} | = {\begin{cases} \frac{| b |}{(3 + Ω)} & if & | 1 - V | \leq \frac{| 3 (1 - Ω) b^{2} - (pbx + aqy) {(2 - Ω)}^{2} |}{(3 + Ω) b^{2}} \\ \frac{(1 - V) {| b |}^{3}}{| 3 (1 + Ω) b^{2} - (pbx + aqy) {(2 - Ω)}^{2} |} & if & | 1 - V | > \frac{| 3 (1 - Ω) b^{2} - (pbx + aqy) {(2 - Ω)}^{2} |}{(3 + Ω) b^{2}} \end{cases}

Corollary 3.15:

Let $ℓ$ offered by (1.1) be in the subfamily $Γ_{1, Σ}^{p, q, x, 1} (X (ѵ))$ and $℣ (x, y)$ be Horadam $H_{n + 1} (x)$ . Then

| κ_{3} - ν {κ_{2}}^{2} | = {\begin{cases} \frac{| b x |}{3} & if & | 1 - V | \leq \frac{| 3 {bx}^{2} - 4 (pb x^{2} + aq) |}{3 {(bx)}^{2}} \\ \frac{(1 - V) {| b |}^{3}}{| 3 {(b x)}^{2} - 4 (pb x^{2} + a q) |} & if & | 1 - V | > \frac{| 3 {b x}^{2} - 4 (pb x^{2} + aq) |}{3 {(b x)}^{2}} \end{cases}

Corollary 3.16:

Let $ℓ$ offered by (1.1) be in the subfamily $Γ_{, Σ}^{p, q, x, 1} (X (ѵ))$ and $℣ (x, y)$ be Horadam $H_{n + 1} (x)$ . Then

| κ_{3} - ν {κ_{2}}^{2} | = {\begin{cases} \frac{| b x |}{4} & if & | 1 - V | \leq \frac{| (pb x^{2} + aq) |}{4 b^{2}} \\ \frac{(1 - V) {| b |}^{3}}{| 6 b^{2} - (pb x^{2} + a q) |} & if & | 1 - V | > \frac{| (pb x^{2} + aq) |}{4 b^{2}} \end{cases}

Ethical considerations

This study was not conducted on human or animal participants; therefore this field was not required.

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Reporting guidelines

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¹ University of Kerbala, Karbala, Karbala Governorate, Iraq
² University of Kerbala, Karbala, Karbala Governorate, Iraq

Mustafa Husseinu
Roles: Conceptualization, Formal Analysis, Investigation, Methodology, Writing – Original Draft Preparation, Writing – Review & Editing

Mohammed H. Saloomi
Roles: Investigation, Supervision, Writing – Original Draft Preparation

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No competing interests were disclosed.

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The author(s) declared that no grants were involved in supporting this work.

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Husseinu M and H. Saloomi M. Coefficient Estimates for New Subclasses of Bi-Univalent Functions Involving Generalized Bivariate Fibonacci-like Polynomials [version 2; peer review: 2 approved]. F1000Research 2026, 14:1463 (https://doi.org/10.12688/f1000research.173513.2)

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REVIEWER’S REPORT MANUSCRIPT ID:
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
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

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
Are the conclusions drawn adequately supported by the results?

Partly

References

1. Olatunji S, Oluwayemi M, Oros G: Coefficient Results concerning a New Class of Functions Associated with Gegenbauer Polynomials and Convolution in Terms of Subordination. Axioms. 2023; 12 (4). Publisher Full Text
2. Frasin B, Oluwayemi M, Porwal S, Murugusundaramoorthy G: Harmonic functions associated with Pascal distribution series. Scientific African. 2023; 21. Publisher Full Text
3. Olatunji S, Sakar F, Breaz N, Aydoǧan S, et al.: Bi-Univalency of m-Fold Symmetric Functions Associated with a Generalized Distribution. Mathematics. 2024; 12 (2). Publisher Full Text
4. Aldawish I, Srivastava H, El-Deeb S, Seoudy T: Comprehensive Subfamilies of Bi-Univalent Functions Involving a Certain Operator Subordinate to Generalized Bivariate Fibonacci Polynomials. Mathematics. 2026; 14 (2). Publisher Full Text

Competing Interests: No competing interests were disclosed.

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.

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Author Response 28 Apr 2026

Mustafa Husseinu, University of Kerbala, Karbala, Iraq

28 Apr 2026

Author Response

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 ... Continue reading 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
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
Competing Interests: No competing interests were disclosed. Close
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COMMENTS ON THIS REPORT

Author Response 28 Apr 2026

Mustafa Husseinu, University of Kerbala, Karbala, Iraq

28 Apr 2026

Author Response

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 ... Continue reading 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
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
Competing Interests: No competing interests were disclosed. Close
Report a concern

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Reviewer Report 24 Feb 2026

Alina Alb Lupas, University of Oradea, Oradea, Romania

Approved

https://doi.org/10.5256/f1000research.191340.r459885

REVIEWER’s REPORT
On the paper
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
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

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
Are the conclusions drawn adequately supported by the results?

Yes

Competing Interests: No competing interests were disclosed.

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.

CITE

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Author Response 28 Apr 2026

Mustafa Husseinu, University of Kerbala, Karbala, Iraq

28 Apr 2026

Author Response

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 ... Continue reading 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.
Sincerely,
The Authors
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.
Sincerely,
The Authors
Competing Interests: No competing interests were disclosed. Close
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COMMENTS ON THIS REPORT

Author Response 28 Apr 2026

Mustafa Husseinu, University of Kerbala, Karbala, Iraq

28 Apr 2026

Author Response

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 ... Continue reading 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.
Sincerely,
The Authors
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.
Sincerely,
The Authors
Competing Interests: No competing interests were disclosed. Close
Report a concern

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Version 2

VERSION 2 PUBLISHED 26 Dec 2025

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Alina Alb Lupas, University of Oradea, Oradea, Romania
Matthew Olanrewaju Oluwayemi, Nile University of Nigeria, Abuja, Nigeria

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27 May 2026 | for Version 2

Matthew Olanrewaju Oluwayemi, Nile University of Nigeria, Abuja, Nigeria

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Approved

The authors have implemented the necessary corrections. Hence, the manuscript is recommended for indexing in its current form.

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09 Mar 2026 | for Version 1

Matthew Olanrewaju Oluwayemi, Nile University of Nigeria, Abuja, Nigeria

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Approved With Reservations

Is the work clearly and accurately presented and does it cite the current literature?

Yes
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

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
Are the conclusions drawn adequately supported by the results?

Partly

References

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Complex analysis (Geometric Function Theory) and Numerical analysis (or Computational Mathematics)

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Author Response

28 Apr 2026

Mustafa Husseinu, University of Kerbala, Karbala, Iraq

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

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Competing Interests

No competing interests were disclosed.

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Reviewer Report

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24 Feb 2026 | for Version 1

Alina Alb Lupas, University of Oradea, Oradea, Romania

23 Views Cite this report Responses(1)

Approved

Is the work clearly and accurately presented and does it cite the current literature?

Yes
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

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
Are the conclusions drawn adequately supported by the results?

Yes

Competing Interests

No competing interests were disclosed.

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.

Respond to this report

Responses (1)

Alongside their report, reviewers assign a status to the article:

Approved - the paper is scientifically sound in its current form and only minor, if any, improvements are suggested

Approved with reservations - A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit.

Not approved - fundamental flaws in the paper seriously undermine the findings and conclusions

[1] 1. Koshy T: Fibonacci and Lucas Numbers with Applications. John Wiley and Sons Incorparation; 2001.

[2] 2. Saloomi MH, Abd EH, Qahtan GA: Coefficient bounds for biconvex and bistarlike functions associated byquasi-subordination. J. Interdiscip. Math. 2021; 24(3): 753–764. Publisher Full Text

[3] 3. Srivastava HM, Mishra AK, Gochhayat P: Certain subclasses of analytic and bi-univalent functions. Appl. Math. Lett. 2010; 23(10): 1188–1192. Publisher Full Text

[4] 4. Saloomi MH, Awaad BK, Al-Sultani M: 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.

[5] 5. Duren PL: Subordination in complex analysis,Lecture Note in mathematics. Vol. 599. Berlin, Germany: Springer; 1977; pp. 22–29.

[6] 6. Saloomi MH, Swamy SR, Al-Sultani M, et al.: 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.

[7] 7. Orhan H, Magesh N, Swamy SR, et al.: Certain classes of bi-univalent functions associated with the Horadam polynomials. Acta UniSapientiae, Mathematica. 2021; 13(1): 258–272. Publisher Full Text

[8] 8. Panwar YK, Singh M: Generalized bivariate Fibonacci-like polynomials. Internat. J. Pure Math. 2014; 1: 8–13.

[9] 9. Duren PL: Univalent Functions, Grundlehren der Mathema- tischen Wissenschaften. New York, NY, USA: Springer; 1983.

[10] 10. Lewin M: On a coefficient problem for bi-univalent functions. Proc. Am. Math. Soc. 1967; 18: 63–68. Publisher Full Text

[11] 11. Brannan D, Clunie J: Aspects of contemporary complex analysis. Academic Press; 1980.

[12] 12. Netanyahu E: 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. Publisher Full Text

[13] 13. Alamoush AG: On a subclass of bi-univalent functions associated to Horadam polynomials. Internat. J. Open Problems Complex Analy. 2020; 12: 58–65.

[14] 14. Aldawish I, Al-Hawary T, Frasin BA: Subclasses of bi-univalent functions defined by Frasin differential operator. Mathematics. 2020; 8: 783. Publisher Full Text

[15] 15. Mishra AK, Panigrahi T, Mishra RK: Subordination and inclusion theorems for subclasses of meromorphic functions with applications to electromagnetic cloaking. Math. Comput. Model. 2013; 57: 945–962. Publisher Full Text

[16] 16. Olatunji SO, Oluwayemi MO, Oros GI:Coefficient Results concerning a New Class of Functions Associated with Gegenbauer Polynomials and Convolution in Terms of Subordination.Axioms2023; 12: 360. Publisher Full Text

[17] 17. Frasin BA, Oluwayemi MO, Porwal S, et al.:Harmonic functions associated with Pascal distribution series.Sci. Afr.2023; 21: e01876. Publisher Full Text

[18] 18. Olatunji SO, Sakar FM, Breaz N, et al.:Bi-Univalency of m-Fold Symmetric Functions Associated with a Generalized Distribution.Mathematics2024; 12: 169. Publisher Full Text

[19] 19. Aldawish I, Srivastava HM, El-Deeb SM, et al.:Comprehensive Subfamilies of Bi-Univalent Functions Involving a Certain Operator Subordinate to Generalized Bivariate Fibonacci Polynomials.Mathematics.2026; 14: 292. Publisher Full Text

Coefficient Estimates for New Subclasses of Bi-Univalent Functions Involving Generalized Bivariate Fibonacci-like Polynomials

Abstract

Keywords

Revised Amendments from Version 1

1. Introduction

Table 1. Values of Special cases of the generalized polynomial obtained by substituting specific values of the parameters.

(1.1)

(1.2)

(1.3)

2. Preliminaries

The Family MΩ,Σp,q,x,y(X(ѵ)) and the Coefficients Bounds

(2.1)

(2.2)

Remark 2.2:

Remark 2.3:

Remark 2.4:

Remark 2.5:

The Family ΓΩ,Σp,q,x,y(X(ѵ)) and the Coefficients Bounds

Definition 2.6:

(2.3)

(2.4)

Remark 2.7:

Remark 2.8:

Remark 2.9:

Remark 2.10:

Fekete-Szegö inequality for the class MΩ,Σp,q,x,y(X(ѵ)) and ΓΩ,Σp,q,x,y(X(ѵ))

Remark 2.11:

Remark 2.12:

Remark 2.13:

Remark 2.14:

3. Main Results

Theorem 3.1:

(3.1)

(3.2)

Proof:

(3.3)

(3.4)

(3.5)

(3.6)

(3.7)

(3.8)

(3.9)

(3.10)

(3.11)

(3.12)

(3.13)

(3.14)

(3.15)

(3.16)

(3.17)

Corollary 3.2:

Corollary 3.3:

Corollary 3.4:

Corollary 3.5:

Theorem 3.6:

(3.18)

(3.19)

Proof:

(3.20)

(3.21)

(3.22)

(3.23)

(3.24)

(3.25)

(3.26)

(3.27)

(3.28)

(3.29)

Corollary 3.7:

Corallary 3.8:

Corallary 3.9:

Corallary 3.10:

Theorem 3.11:

Proof:

Corollary 3.12:

Corollary 3.13:

Theorem 3.14:

Proof:

Corollary 3.15:

Corollary 3.16:

The Family $M_{Ω, Σ}^{p, q, x, y} (X (ѵ))$ and the Coefficients Bounds

The Family $Γ_{Ω, Σ}^{p, q, x, y} (X (ѵ))$ and the Coefficients Bounds

Fekete-Szegö inequality for the class $M_{Ω, Σ}^{p, q, x, y} (X (ѵ))$ and $Γ_{Ω, Σ}^{p, q, x, y} (X (ѵ))$