A study of bi-univalent functions associated with the modified Tremblay operator and Horadam polynomials

1 Introduction

Let ℍ denote the class of all analytic functions defined on the open unit disk

K={z∈C:|z|<1},

and normalized by the conditions

A(0)=0 and A′(0)=1.

Accordingly, every function A∈ℍ admits the Taylor series expansion [1]:

A(z)=z+∑u=2∞auzu, z∈K.(1.1)

Let E denote the subclass of ℍ consisting of functions that are univalent in K. Let A and B be analytic functions in the unit disk K. The function A is said to be subordinate to B, written as

A≺B,

if there exists a Schwarz function w, analytic in K, satisfying w(0) = 0 and |w(z)| < 1 for all z∈K, such that

A(z)=B(w)(z) (z∈K)

In addition, whenever B is univalent in K, the condition A≺B is satisfied precisely when both functions take the same value at the origin and the range of A is contained within the range of B . In particular, A(0)=B(0) along with A(K)⊂B(K) [2]. Since A(z) is assumed to be analytic and univalent in K, it admits a unique inverse function, denoted by Q(w)=A-1(z) (refer to [3]):

A-1(A(z))=z (z∈K)

whereas

A(A-1(w))=w (|w|<r0(A);r0(A)≥14).

The inverse function can be expressed as

A-1(w)=w-a2w2+(2a22-a3)w3+⋯(1.2)

The class of bi-univalent functions occupies a significant position in the theory of anlytic and univalent functions. A function belongs to this class whenever both the function itself and its inverse are one-to-one in the unit disk. We denote this family by Ϝ. Several classical functions satisfy this condition and serve as useful examples in the study of coefficient problems and geometric properties. Some representative examples are presented below along with their inverse mappings [4–7] :

The concept of bi-univalent functions originated with this pioneering work of Lewin [8], who introduced the class and obtained the first coefficient estimate. Later, Brannan and Clunie [9] emphasized the significance of coefficient problems and encouraged further investigation into the geometric properties of these functions. Subsequently, Netanyahu [10] derived important results concerning the second coefficient of univalent functions, which have influenced many later studies on bi-univalent and related function classes. The problem of determining an upper bound for the functional |a3-μa22| is known as the Fekete-Szego problem [11] . This functional emerged from the celebrated work of Fekete and Szego, who disproved the Littlewood-Paley conjecture on the coefficients of odd univalent functions [12]. Since Horadam polynomials play a key role in our investigation, we begin by recalling some of their basic definitions and properties. Horadam polynomials have received considerable attention due to their rich mathematical structure and their ability to unify several well-known polynomial sequences within a single framework. By appropriately selecting the parameters involved in their recurrence relation, one can obtain many classical polynomials, including the Fibonacci and Lucas polynomials. Because of this versatility, Horadam polynomials have found applications in various areas of mathematics, such as number theory, combinatorics and geometric function theory [13]. In 2009, Horzum and Kocer [14] investigated a class of Horadam polynomials h_*n*(x), characterized by the recurrence relation in Equations 1.3 and 1.4.

hn(x)=pxhn-1(x)+qhn-2(x), (n≥3),(1.3)

along with

h1(x)=m,h2=tx and h3(x)=ptx2+mq(1.4)

The generating function of the Horadam polynomials is given by

Θ(x,z)=∑n=1∞hn(x)zn-1=m+(t-mp)xz1-pxz-qz2.

where the parameters m, t, p and q are real constants. The Horadam polynomials serve as a generalization of several classical polynomial sequences and therefore occupy an important place in the study of recursive polynomial families; see [15] for details. A class of bi-univalent functions associated with Horadam polynomials was introduced and coefficient estimates together with the corresponding Fekete-Szego problem were investigated by [16]. These results demonstrate the usefulness of Horadam polynomial structures in coefficient problems for bi-univalent functions and provide motivation for considering them in combination with suitable fractional calculus operators. Recent studies on bi-univalent functions have investigated coefficient problems associated with various generalized polynomial and number sequences, including initial coefficient estimates for bi-univalent functions related to generalized telephone numbers [17]. Fractional calculus operators have been widely used in geometric function theory in the study of analytic and bi-univalent functions [24–27]. Among them, the Tremblay fractional derivative operator has been investigated by several researchers because of its applications in the definition and study of new subclasses of analytical functions [28]. Tremblay [29] introduced this operator using the Riemann-Liouville fractional derivative, while Ibrahim and Jahangiri [30] enhanced its study to the complex plane.

While the earlier study focused on bi-univalent function classes associated with Horadam polynomials, the present work considers a different operator-polynomial framework by combining the modified Tremblay operator with the Horadam polynomial generating function. Accordingly, two new subclasses of bi-univalent functions are introduced and their initial Taylor coefficient estimates and Fekete-Szego inequalities are derived.

Recently, Yousef et al. [31] investigated two subclasses of bi-univalent functions defined through Liouville-Caputo-type fractional derivatives in conjunction with the generating function of the Gregory coefficients. They established estimates for the initial Maclaurin coefficients and the corresponding Fekete-Szego functional for these subclasses. Their work illustrates the effectiveness of combining fractional calculus operators with suitable generating functions in the study of coefficient-related problems for bi-univalent functions.

Motivated by these developments, the present work considers the modified Tremblay operator together with the Horadam polynomial generating function to define two subclasses of bi-univalent functions. The aim is to derive estimates for the initial Taylor coefficients and the corresponding Fekete-Szego inequalities, thereby extending the application of fractional calculus operators and polynomial generating functions within the framework of bi-univalent function theory. It is anticipated that these findings may stimulate further research on related subclasses of analytic and bi-univalent functions and provide a basis for future studies involving other operators and special polynomial structures in geometric function theory. For example, the works of Mahmoud et al.[32] and Abdul Ameer et al.[33] may provide potential directions for extending the present study.

Outline of the Article

The paper begins with a brief introduction to bi-univalent functions, Horadam polynomials and the modified Tremblay operator, together with some related results from the literature.

The basic definitions and supporting results used throughout this work are presented in the main results.

In Section 2 and 3, new subclasses of bi-univalent functions together with the modified Tremblay operator and the Horadam polynomial generating function are introduced and defined. The main results, including estimates for the initial Taylor coefficients and the corresponding Fekete-Szego inequalities, are established.

Several special cases and consequences of the obtained results are discussed through suitable choices of the defining parameters.

The paper concludes with a summary of the main findings.

Definition 1.1. [30] Let A∈ℍ. The Tremblay fractional derivative operator 𝕋zχ,β acting on A is defined, for all z∈K, by

𝕋zχ,βA(z)=Γ(β)Γ(χ)z1-βDzχ-βzχ-1A(z),(1.5)

(0 < β ≤ 1, 0 < χ ≤ 1, 0 ≤ χ − β < 1), It follows directly that when χ = β = 1 the Tremblay operator reduces to the identity operator,

𝕋z1,1A(z)=A(z)

Subsequently, Esa et al. [34] introduced a modified form of the classical Tremblay fractional derivative operator. Following [35], the modified Tremblay operator is defined as follows

Definition 1.2. [35] The modified Tremblay operator 𝕋^{χ, β}:ℍ → ℍ, associated with a function A∈ℍ is defined as follows

Tzχ,βA(z)=βχ𝕋zχ,βA(z)=Γ(χ+1)Γ(β+1)z1-βDzχ-βzχ-1A(z)=z+∑u=2∞Γ(β+1)Γ(χ+u)Γ(χ+1)Γ(β+u)auzu,(1.6)

where (0 < β ≤ 1, 0 < χ ≤ 1, 0 ≤ χ − β < 1),

Here, 𝕋zχ,β denotes the classical Tremblay fractional derivative operator defined in Equation 1.5, whereas Tzχ,β denotes the modified Tremblay operator defined in Equation 1.6. For further details on the classical Tremblay operator, see [26]

or

Tzχ,βA(z)=∑u=2∞Ωuauzu

where

Ωu=Γ(β+1)Γ(χ+u)Γ(χ+1)Γ(β+u)

Since the coefficients Ω_*u* are well defined for u ≥ 2 under the assumptions 0 < β ≤ 1, 0 < χ ≤ 1, and 0 ≤ χ − β < 1, and the modified Tremblay operator maps ℍ into ℍ, the above power series representation is valid for all z∈K. Further,

Tzχ,βA(z)=z+Ω2a2z2+Ω3a3z3+⋯

and if Q(w)=A-1(z)

Tzχ,βQ(w)=w-Ω2a2w2+Ω3(2a22-a3)w3+⋯

The present study focuses on deriving the initial coefficient bounds and the corresponding Fekete-Szego inequalities for two subclasses of bi-univalent functions defined by the modified Tremblay operator and the Horadam polynomial generating function.

2 Coefficient analysis of the Tremblay-Horadam subfamily EϜ(Ωu,ϱ,τ,ȷ)

The focus of this section is to investigate the coefficient structure of functions belonging to the proposed Tremblay-Horadam class. Estimates for the first two Taylor coefficients and the associated Fekete-Szego inequality are determined.

Lemma 2.1. [36] If ϵ∈P, where ϵ(z)=1+a1z+a2z2+⋯, is analytic in K. Then |a_u| ≤ 2, u∈N, and the result is sharp.

Lemma 2.2. *[37] Consider x, y ∈ ℜ as well as* z1,z2∈C if |z1|,|z2|<R, therefore

Definition 2.3. [7] Assume that ϱ≥0,τ≥1,ȷ∈C *such that Re(ȷ) ≥ 0 . We say that a function* A∈Ϝ represented by Equation 1.1 lies in the subclass EϜ(Ωu,ϱ,τ,ȷ) whenever the following conditions hold, as given in Equations 2.1 and 2.2

(1-ϱ)(Tzχ,βA(z))′+ϱ((Tzχ,βA(z))′)τ(Tzχ,βA(z)z)ȷ-1≺Θ(x,z)+1-m,(2.1)

and

(1-ϱ)(Tzχ,βQ(w))′+ϱ((Tzχ,βQ(w))′)τ(Tzχ,βQ(w)w)ȷ-1≺Θ(x,w)+1-m,(2.2)

where Q(w)=A-1(z) is given in Equation 1.2.

Remark 2.4.

• By choosing ϱ = 0 in Definition 2.3, the defining condition reduces to together with the corresponding condition for the inverse function. Thus, this choice gives a simpler subclass of the proposed family.
• If we consider A(z)=z in Definition 2.3, then Since Θ(x, 0) = m, we have For this particular verification, we choose the Schwarz function ℘(z) = 0. Consequently, and hence Therefore, A(z)=z belongs to the class defined in Definition 2.3, and the class is non-empty.

Theorem 2.5. Suppose that A(z)∈EϜ(Ωu,ϱ,τ,ȷ) with ϱ≥0,τ≥1,ȷ∈C *and satisfying Re(ȷ) ≥ 0. Then the following coefficient estimates hold*:

where Φ=2Ω3(3ϱτ+ϱȷ-4ϱ+3)+2ϱΩ22(2τ2+2τ(ȷ-2)+(ȷ-1)(ȷ-2)2)-2h3(x)Ω22(2ϱτ+ϱȷ-3ϱ+2)2h22(x).

where Φ=2Ω3(3ϱτ+ϱȷ-4ϱ+3)+2ϱΩ22(2τ2+2τ(ȷ-2)+(ȷ-1)(ȷ-2)2)-2h3(x)Ω22(2ϱτ+ϱȷ-3ϱ+2)2h22(x).

where, M(ζ)=(1-ζ)h2(x)Φ

Proof. Given that A(z)∈EϜ(Ωu,ϱ,τ,ȷ). By virtue of Equations 2.2 and 2.3, suitable analytic functions exist such that ℘ and ν satisfying the condition

℘(0)=ν(0)=0, |℘(z)|<1 along with |ν(w)|<1,

where

(1-ϱ)(Tzχ,βA(z))′+ϱ((Tzχ,βA(z))′)τ(Tzχ,βA(z)z)ȷ-1≺Θ(x,℘(z))+1-m(2.3)

and

(1-ϱ)(Tzχ,βQ(w))′+ϱ((Tzχ,βQ(w))′)τ(Tzχ,βQ(w)w)ȷ-1≺Θ(x,ν(w))+1-m,(2.4)

with ℘ and ν being analytic funtions, of the form

℘(z)=c1z+c2z2+c3z3+⋯

and

ν=d1w+d2w2+⋯

from the equalities Equations 2.3 and 2.4

(1-ϱ)(Tzχ,βA(z))′+ϱ((Tzχ,βA(z))′)τ(Tzχ,βA(z)z)ȷ-1=1+h2(x)c1z+[h2(x)c2+h3(x)c12]z2+⋯(2.5)

(1-ϱ)(Tzχ,βQ(w))′+ϱ((Tzχ,βQ(w))′)τ(Tzχ,βQ(w)w)ȷ-1=1+h2d1w+[h2(x)d2+h3(x)d12]w2+⋯(2.6)

Comparing the corresponding coefficients, we obtain in Equations 2.5 and 2.6

Ω2a2[2ϱτ+ϱȷ-3ϱ+2]=h2(x)c1.(2.7)

Ω3a3(3ϱτ+ϱȷ-4ϱ+3)+ϱΩ22a22(2τ2+2τ(ȷ-2)+(ȷ-1)(ȷ-2)2)=h2(x)c2+h3(x)c12.(2.8)

-Ω2a2[2ϱτ+ϱȷ-3ϱ+2]=h2(x)d1.(2.9)

-Ω3a3(3ϱτ+ϱȷ-4ϱ+3)+2Ω3a22(3ϱτ+ϱȷ-4ϱ+3)+ϱΩ22a22(2τ2+2τ(ȷ-2)+(ȷ-1)(ȷ-2)2)=h2(x)d2+h3(x)d12.(2.10)

using Equations 2.7 and 2.9 we have

c1=-d1.(2.11)

c12+d12=2Ω22(2ϱτ+ϱȷ-3ϱ+2)2h22(x)a22.(2.12)

add Equations 2.8 and 2.10, we obtain

(2Ω3(3ϱτ+ϱȷ−4ϱ+3)+2ϱΩ22(2τ2+2τ(ȷ−2)+(ȷ−1)(ȷ−2)2))a22=h2(x)(c2+d2)+h3(x)(c12+d12).(2.13)

substituting the value of Equation 2.12 in Equation 2.13

(2Ω3(3ϱτ+ϱȷ−4ϱ+3)+2ϱΩ22 (2τ2+2τ(ȷ−2)+(ȷ−1)(ȷ−2)2)−2h3(x)Ω22(2ϱτ+ϱȷ−3ϱ+2)2h22(x))a22=h2(x)(c2+d2).(2.14)

From Equations 2.7 and 2.14, an application of the triangle inequality together with Lemma 2.1 give, respectively,

and

where,

Φ=2Ω3(3ϱτ+ϱȷ−4ϱ+3)+2ϱΩ22(2τ2+2τ(ȷ−2)+(ȷ−1)(ȷ−2)2)−2h3(x)Ω22(2ϱτ+ϱȷ−3ϱ+2)2h22(x).

subtract Equation 2.10 from Equation 2.8, we get

2(3ϱτ+ϱȷ-4ϱ+3)(Ω3a3-Ω3a22)=h2(x)(c2-d2)+h3(x)(c12-d12)(2.15)

using Equation 2.11 in Equation 2.15, we get

Ω3a3=Ω3a22+h2(x)(c2-d2)2(3ϱτ+ϱȷ-4ϱ+3)(2.16)

Ω3a3=Ω3(h22(x)c12Ω22[2ϱτ+ϱȷ-3ϱ+2]2)+h2(x)(c2-d2)2(3ϱτ+ϱȷ-4ϱ+3)(2.17)

By applying Lemma 2.1 and the triangle inequality for Equation 2.17,

And using the assertion Equation 2.14 with Equation 2.16, it follows that

where

Φ=2Ω3(3ϱτ+ϱȷ-4ϱ+3)+2ϱΩ22(2τ2+2τ(ȷ-2)+(ȷ-1)(ȷ-2)2)-2h3(x)Ω22(2ϱτ+ϱȷ-3ϱ+2)2h22(x).

from Equation 2.16

a3-ζa22=h2(x)(c2-d2)2(3ϱτ+ϱȷ-4ϱ+3)+(1-ζ)a22.

a3-ζa22=h2(x)(c2-d2)2(3ϱτ+ϱȷ-4ϱ+3)+(1-ζ)h2(x)(c2+d2)Φ,

where

Φ=2Ω3(3ϱτ+ϱȷ-4ϱ+3)+2ϱΩ22(2τ2+2τ(ȷ-2)+(ȷ-1)(ȷ-2)2)-2h3(x)Ω22(2ϱτ+ϱȷ-3ϱ+2)2h22(x).a3-ζa22=(M(ζ)+h2(x)2Ω3(3ϱτ+ϱȷ-4ϱ+3))c2+(M(ζ)-h2(x)(c2+d2)2Ω3(3ϱτ+ϱȷ-4ϱ+3))d2

where

M(ζ)=(1-ζ)h2(x)Φ

Then, using Lemma 2.1 for c₂ and d₂ and lemma 2.2, we get

Thus, the proof is complete. □

The estimates obtained in Theorem 2.5 are upper bounds derived using the triangle inequality and standard coefficient estimates. The obtained bounds are not claimed to be sharp, since the attainment of equality has not been established.

Example 2.6. Consider the specialization

m=t=p=1, q=0,

for which the Horadam polynomial generating function becomes

Θ(x,z)=11-xz.

Let x=12, χ = β, ϱ = 0 *and m = 1. Then Ω_*u* = 1, and hence the modified Tremblay operator reduces to the identity operator. Choose the Schwarz function*

℘(z)=z4.

Define

A(z)=-8log(1-z8)=z+z216+z3192+⋯.

Thus, a2=116 and a3=1192, so that A is non-trivial. Moreover,

A′(z)=11-z/8=Θ(12,℘(z)),

and hence the first subordination in Definition 2.3 is satisfied. The inverse function is

Q(w)=8(1-e-w/8),

and therefore

Q′(w)=e-w/8.

Let

ν(w)=2(1-ew/8).

Then ν(0) = 0 and for |w| < 1,

Thus ν is a Schwarz function. Furthermore,

Θ(12,ν(w))=11-12ν(w)=e-w/8=Q′(w).

Hence the second subordination is also satisfied. Consequently,

A(z)∈EϜ(Ωu,ϱ,τ,ȷ).

Considering Theorem 2.5 with χ = β, ϱ = 0 and τ = ȷ = 1, for which Ω_*u* = 1, we obtain the following corollary.

Corollary 2.7. For A(z)∈EϜ(Ωu,ϱ,τ,ȷ). Then

and

3 Bounds on the coefficients for the subfamily CϜ(Ωu,ϕ)

In this section, we introduce a new subclass of bi-univalent functions by combining the modified Tremblay operator with the Horadam polynomial generating function. Using appropriate subordination conditions, we study some fundamental properties of this class. In particular, we obtain estimates for the initial coefficients and establish the corresponding Fekete-Szego inequalities.

The subsequent results concerning the proposed subfamily CϜ(Ωu,ϕ) rely on the following lemma

Lemma 3.1. [38] Suppose that ϵ(z)=1+a1z+a2z2+⋯∈P, z∈K . Then there exist parameters ϰ, ℵ satisfying |ϰ|, |ℵ| ≤ 1, such that

2s2=s12+ϰ(4-s12),and4s3=s13+2s1ϰ(4-s12)-(4-s12)s1ϰ2+2(4-s12)(1-|ϰ|2)ℵ.

Definition 3.2. A function A∈Ϝ as mentioned in Equation 1.1 is said to belong to the subfamily CϜ(Ωu,ϕ), where ϕ ∈ (−π, π], whenever it holds that

(Tzχ,βA(z))′+(eiϕ+12)z(Tzχ,βA(z))″≺Θ(x,z)+1-m,(3.1)

and

(Tzχ,βQ(w))′+(eiϕ+12)w(Tzχ,βQ(w))″≺Θ(x,w)+1-m(3.2)

where Θ(x, z) + 1 − m is a Horadam polynomials Q(w)=A-1(w) as presented in Equation 1.2

Remark 3.3.

• For ϕ = 0, we have Consequently, the defining conditions in Definition 3.2 reduce to Thus, ϕ = 0 gives a distinct special case of the proposed Tremblay-Horadam family.
• For the identity function A(z)=z, we have and hence Therefore, the left-hand side of the first defining condition in Definition 3.2 is equal to 1. Similarly, the corresponding condition for the inverse function is also satisfied. By choosing the Schwarz function s(z) ≡ 0, we have since Θ(x, 0) = m. Hence, Therefore, A(z)=z belongs to CϜ(Ωu,ϕ), and the class is non-empty.

Theorem 3.4. [7] Suppose that ϕ ∈ (−π, π] and A∈CϜ(Ωu,ϕ) it follows that

and

Proof: Since A∈CϜ(Ωu,ϕ), the subordinations (Equations 3.1 and 3.2) imply the existence of Schwarz functions s and b satisfying

s(0)=b(0)=0,|s(z)| including |b(w)|≤1

such that

(Tzχ,βA(z))′+(eiϕ+12)z(Tzχ,βA(z))″≺Θ(x,s(z))+1-m,z∈K(3.3)

and

(Tzχ,βQ(w))′+(eiϕ+12)w(Tzχ,βQ(w))″≺Θ(x,b(w))+1-m w∈K(3.4)

This gives us

(Tzχ,βA(z))′+(eiϕ+12)z(Tzχ,βA(z))″=1+h2(x)s1z+[h2(x)s2+h3(x)s12]z2+⋯, z∈K(3.5)

and

(Tzχ,βQ(w))′+(eiϕ+12)w(Tzχ,βQ(w))″=1+h2(x)b1w+[h2(x)b2+h3(x)b12]w2+⋯ w∈K(3.6)

Examining the coefficients in Equations 3.5 and 3.6, we get Equations 3.7–3.10.

Ω2(3+eiϕ)a2=h2(x)s1(3.7)

3Ω3(2+eiϕ)a3=h2(x)s2+h3(x)s12(3.8)

-Ω2(3+eiϕ)a2=h2(x)b1(3.9)

3Ω3(2+eiϕ)(2a22-a3)=h2(x)b2+h3(x)b12(3.10)

From the equations we get

s1=-b1(3.11)

Also

2Ω22(3+eiϕ)2a22=h22(x)(s12+b12)(3.12)

By adding Equations 3.8–3.10, we obtain

6Ω3(2+eiϕ)a22=h2(x)(s2+b2)+h3(x)(s12+b12)(3.13)

On replacing s12+b12 in Equation 3.13 with its value from Equation 3.12, the result is

(6Ω3(2+eiϕ)-h3(x)(2Ω22(3+eiϕ)2)h22(x))a22=h2(x)(s2+b2)(3.14)

Applying Lemma 2.1 in combination with the triangle inequality, we can derive the corresponding outcomes for Equations 3.12 and 3.14

Subtracting Equation 3.10 from Equation 3.8, we obtain

6Ω3(2+eiϕ)(a3-a22)=h2(x)(s2-b2)+h3(x)(s12-b12)(3.15)

Considering Equation 3.11, we obtain the following version of Equation 3.15

Ω3a3=Ω3a22+h2(x)(s2-b2)6(2+eiϕ)(3.16)

From the relation between Equations 3.16 and 3.12, we get

a3=h22(x)(s12+b12)2Ω22(3+eiϕ)2+h2(x)(s2-b2)6Ω3(2+eiϕ)

An application of the triangle inequality and Lemma 2.1, Consequently, we obtain the following relation

In the same way, applying Equations 3.14 within 3.16 leads to

By making use of Equations 3.11 and 3.12, the following relation is obtained

a3-ϖa22=h2(x)(s2-b2)6Ω3(2+eiϕ)+(1-ϖ)a22

a3-ϖa22=h2(x)(s2-b2)6Ω3(2+eiϕ)+(1-ϖ)h22(x)s12Ω22(3+eiϕ)2

Based on 3.1, it follows that

2s2=s12+ϰ(4-s12) and

2b2=b12+ℵ(4-b12),|ϰ|, |ℵ| ≤ 1 Equation 3.11 yields

s2-b2=4-s122(ϰ-ℵ)

Consequently,

a3-ϖa22=h2(x)(4-s12)(ϰ-ℵ)12Ω3(2+eiϕ)+(1-ϖ)h22(x)s12Ω22(3+eiϕ)2

Applying the triangle inequality and setting |ϰ| = λ, |ℵ| = y, λ, |ℵ| ∈ [0, 1] considering s₁ ∈ ℜ, s₁ = ♭ ∈ [0, 2]; Hence, we obtain the following Equation 3.17.

Since

we maximize the right-hand side with respect to λ and y. Let

O(♭)=|1-ϖ||h2(x)|2♭2Ω22|3+eiϕ|2≥0

and

B(♭)=(4-♭2)|h2(x)|12Ω3|2+eiϕ|≥0.

Then

Since λ, y ∈ [0, 1] and B(♭)≥0, we have

λ+y≤2.

Therefore,

maxλ,y∈[0,1]𝕃(λ,y)=𝕃(1,1)=O(♭)+2B(♭)=:P(♭).

Consequently,

It remains to determine the maximum of P(♭) on the interval [0, 2].

P(♭)=(|1-ϖ||h2(x)|2Ω22|3+eiϕ|2-|h2(x)|6Ω3|2+eiϕ|)♭2+2|h2(x)|3Ω3|2+eiϕ|

As

P′(♭)=2(|1-ϖ||h2(x)|2Ω22|3+eiϕ|2-|h2(x)|6Ω3|2+eiϕ|)♭

If

then P′(♭)≤0 for every ♭ ∈ [0, 2]. Hence P is decreasing on [0, 2]. Therefore,

max0≤♭≤2P(♭)=P(0)=2|h2(x)|3Ω3|2+eiϕ|.

If

then P′(♭)≥0 for every ♭ ∈ [0, 2]. Hence P is increasing on [0, 2]. Therefore,

max0≤♭≤2P(♭)=P(2)=4|1-ϖ∥h2(x)|2Ω22|3+eiϕ|2.

Therefore, combining the above two cases with

we obtain exactly the Fekete-Szego estimate stated in Theorem 3.4. Hence, the proof is complete. □

The results established in Theorem 2.5 and Theorem 3.4 provides coefficient estimates and fekete-szego inequalities for a broad family of bi-univalent functions associated with the Horadam polynomial generating function and the Tremblay fractional derivative operator.

Example 3.5. Consider the particular choice

m=t=p=1, q=0, x=12, χ=β, ϕ=π.

Then the Horadam polynomial generating function becomes

Θ(x,z)=11-xz,

and

Θ(12,z)=11-z/2.

Since χ = β, we have Ω_*u* = 1 for u ≥ 2 and hence the modified Tremblay operator reduces to the identity operator. Moreover,

eiπ+12=0.

Choose

s(z)=z4

and define

A(z)=-8log(1-z8)=z+z216+z3192+⋯.

Thus,

a2=116, a3=1192.

Since

A′(z)=11-z/8=Θ(12,s(z))+1-m,

the first subordination condition in Equation 3.3 is satisfied. The inverse function of A is

Q(w)=8(1-e-w/8),

so that

Q′(w)=e-w/8.

Let

b(w)=2(1-ew/8).

Then b(0) = 0 and

Furthermore,

Θ(12,b(w))+1-m=11-12b(w)=e-w/8=Q′(w).

Hence the second subordination condition in Equation 3.4 is also satisfied. Consequently,

A(z)∈CϜ(Ωu,π).

This example demonstrates that the subclass contains a non-trivial function with non-zero second and third Taylor coefficients.

Remark 3.6. Setting ϕ = π in Theorem 3.4, we obtain a special subclass of the family.

Since e^*iπ* = −1, it follows that 3 + e^*iϕ* = 2, 2 + e^*iϕ* = 1 . *Consequently, the coefficient estimates for a₂ and a₃, together with the corresponding Fekete-Szego inequality, reduce to simpler forms. Moreover, if the Horadam polynomial generating function satisfies*

ℜ{Θ(x,z)}>0,

then the condition

(Tzχ,βA(z))′≺Θ(x,z)

represents a Tremblay-Horadam analog of the bounded turning class.

Remark 3.7. Taking χ = β, where 0 < β ≤ 1, in Theorem 3.4, yields the following result

Ωu=1

Consequently,

Tzχ,βA(z)=A(z),

*and the Tremblay operator reduces to the identity operator. Therefore, Theorem 3.4 reduces to the corresponding Horadam polynomial class without the Tremblay operator. Hence, the coefficient estimates for a₂, a₃ and the associated Fekete-Szego inequality reduce to those of the corresponding classical Horadam polynomial family*.

4 Conclusion

New subclasses of bi-univalent functions associated with the modified Tremblay operator and the Horadam polynomial generating function have been introduced and investigated. By means of suitable subordination conditions, estimates for the initial Taylor coefficients and the corresponding Fekete-Szego inequalities were obtained. Furthermore, several special cases were discussed by appropriately choosing the parameters involved in the defining conditions. The results obtained enrich the existing literature on bi-univalent functions and their connections with fractional calculus operators and special polynomial families. The combination of the modified Tremblay operator and the Horadam polynomial generating function provides an interesting setting for the study of coefficient-related problems in geometric function theory. It is anticipated that these findings may stimulate further research on related subclasses of analytic and bi-univalent functions and provide a basis for future studies involving other operators and special polynomial structures in geometric function theory.

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Footnotes

Edited by: Firdous A. Shah, University of Kashmir, India

Reviewed by: Raheam Al-Saphory, Mustansiriyah University, Iraq

Shyamsunder, SRM University (Delhi-NCR), India

Sibel Yalçln, Bursa Uludag Universitesi, Türkiye