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

Mustafa Husseinu; Mohammed H. Saloomi

doi:10.12688/f1000research.173513.1

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Research Article

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

[version 1; peer review: awaiting peer review]

Mustafa Husseinu ¹, Mohammed H. Saloomi²

PUBLISHED 26 Dec 2025

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 AWAITING PEER REVIEW

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

Abstract

Background

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.

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: © 2025 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 1; peer review: awaiting peer review]. F1000Research 2025, 14:1463 (https://doi.org/10.12688/f1000research.173513.1) First published: 26 Dec 2025, 14:1463 (https://doi.org/10.12688/f1000research.173513.1) Latest published: 26 Dec 2025, 14:1463 (https://doi.org/10.12688/f1000research.173513.1)

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 z - 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} ɛ_{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.^2–4,13,14

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.

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. Methods

2.1 The family $M_{Ω, Σ}^{p, q, x, y} (X (ѵ))$ and the coefficients bounds

Definition: 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.

In this section, we investigate coefficient estimates for the function class $M_{Ω, Σ}^{p, q, x, y} (X (ѵ))$

Theorem 2.6:

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

(2.3)

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

(2.4)

| 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

(2.5)

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

(2.6)

[(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)

(2.7)

[(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,

(2.8)

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

(2.9)

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 (2.8) and (2.9), we have

(2.10)

κ_{2} = ℣_{1} (x, y) i_{1}

(2.11)

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

(2.12)

- κ_{2} = ℣_{1} (x, y) λ_{1}

and

(2.13)

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

Now considering (2.10) and (2.12), we get

(2.14)

i_{1} = - λ_{1}

(2.15)

\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 (2.11) to (2.13), we find that

(2.16)

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 (2.15) in (2.16), 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 $| i_{k} | \leq 1, | λ_{k} | \leq 1$ ◾

(2.17)

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

Subtracting (2.11) and (2.13), and with some computation, we have

(2.18)

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

Put (2.15) in (2.18), then we deduce

(2.19)

\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 2.7:

(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 2.8:

(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 2.9:

(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 2.10:

(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 $M_{1, Σ}^{1, 1, x, 1} (X (ѵ))$
$| κ_{2} | \leq \frac{1}{\sqrt{| x |}} .$

$| κ_{3} | \leq \frac{5}{4} ◾$

3. The family $Γ_{Ω, Σ}^{p, q, x, y} (X (ѵ))$ and the coefficients bounds

Definition 3.1.

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:

(3.1)

[(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)

(3.2)

[(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 3.2:

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

Remark 3.3:

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 3.4:

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 3.5:

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.

In this section, coefficient estimates are investigated for the function set $Γ_{Ω, Σ}^{p, q, x, y} (X (ѵ))$

Theorem 3.6:

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

(3.3)

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

(3.4)

| 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

(3.5)

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)

(3.6)

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

(3.7)

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 (3.6) and (3.7), we have

(3.8)

(2 - Ω) κ_{2} = ℣_{1} (x, y) ɍ_{1}

(3.9)

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

(3.10)

- (2 - Ω) κ_{2} = ℣_{1} (x, y) ѳ_{1}

(3.11)

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

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

(3.12)

ɍ_{1} = - ѳ_{1}

(3.13)

\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 (3.9) to (3.11), 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 (3.9) and (3.11), and with some computation, we get

(3.14)

\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 (3.13) in (3.14)

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}

Remark 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)}} .

Remark 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} ◾

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

Remark 4.1:

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 4.2:

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

Remark 4.3:

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 4.4:

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

Theorem 4.3.

Assuming that $ℓ (ѵ) = ѵ + \sum_{m = 2}^{\infty} ɛ_{m} ѵ^{m} \in M_{Ω, Σ}^{p, q, x, y} (X (ѵ))$ , 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 4.4:

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 4.5:

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 4.6:

Let 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 4.7:

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 4.8:

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.

Data availability

No data associated with this article.

Reporting guidelines

This type of study is not applicable because it is a theoretical study.

References

1. Koshy T: Fibonacci and Lucas Numbers with Applications. John Wiley and Sons Incorparation; 2001.
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. 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. 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. Duren PL: Subordination in complex analysis,Lecture Note in mathematics. Vol. 599. Berlin, Germany: Springer; 1977; pp. 22–29.
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. 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. Panwar YK, Singh M: Generalized bivariate Fibonacci-like polynomials. Internat. J. Pure Math. 2014; 1: 8–13.
9. Duren PL: Univalent Functions, Grundlehren der Mathema- tischen Wissenschaften. New York, NY, USA: Springer; 1983.
10. Lewin M: On a coefficient problem for bi-univalent functions. Proc. Am. Math. Soc. 1967; 18: 63–68. Publisher Full Text
11. Brannan D, Clunie J: Aspects of contemporary complex analysis. Academic Press; 1980.
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. Alamoush AG: On a subclass of bi-univalent functions associated to Horadam polynomials. Internat. J. Open Problems Complex Analy. 2020; 12: 58–65.
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. 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

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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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[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

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

Abstract

Background

Methods

Results

Conclusions

Keywords

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. Methods

2.1 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:

Theorem 2.6:

(2.3)

(2.4)

Proof:

(2.5)

(2.6)

(2.7)

(2.8)

(2.9)

(2.10)

(2.11)

(2.12)

(2.13)

(2.14)

(2.15)

(2.16)

(2.17)

(2.18)

(2.19)

Corollary 2.7:

Corollary 2.8:

Corollary 2.9:

Corollary 2.10:

3. The family ΓΩ,Σp,q,x,y(X(ѵ)) and the coefficients bounds

Definition 3.1.

(3.1)

(3.2)

Remark 3.2:

Remark 3.3:

Remark 3.4:

Remark 3.5:

Theorem 3.6:

(3.3)

(3.4)

Proof:

(3.5)

(3.6)

(3.7)

(3.8)

(3.9)

(3.10)

(3.11)

(3.12)

(3.13)

(3.14)

Corollary 3.7:

Corallary 3.8:

Remark 3.9:

Remark 3.10:

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

Remark 4.1:

Remark 4.2:

Remark 4.3:

Remark 4.4:

Theorem 4.3.

Proof:

Corollary 4.4:

Corollary 4.5:

Theorem 4.6:

Proof:

2.1 The family $M_{Ω, Σ}^{p, q, x, y} (X (ѵ))$ and the coefficients bounds

3. The family $Γ_{Ω, Σ}^{p, q, x, y} (X (ѵ))$ and the coefficients bounds

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