EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 5968
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Second Hankel Determinant for a Bi-univalent Function
Subclass Based Gegenbauer (Ultraspherical)

Polynomials

Abdelbaset Zeyani1, Abdulmtalb Hussen2,∗

1 Department of Mathematics and Statistics, Wichita State University, Wichita, KS, USA
2 School of Engineering, Math, and Technology, Navajo Technical University,
Crownpoint NM, USA
3 Mathematics Department, College of Education, Al Zintan University, Dirj, Libya

Abstract. In this paper, we aim to establish a new upper bound approximation for the second
Hankel determinant utilizing a certain subclass of the class of normalized analytic and bi-univalent
functions in the open unit disk U . These functions have inverses with a bi-univalent analytic con-
tinuation to U and are associated with orthogonal polynomials; namely, Gegenbauer polynomials
that satisfy subordination conditions on U. Finally, we introduce new essential results derived by
specializing the parameter τ employed in our foundational finding.

2020 Mathematics Subject Classifications: 30C45
Key Words and Phrases: Gegenbauer (Ultraspherical) Polynomials, Bi-univalent Analytic
Functions, Hankel Determinant

1. Introduction

The nth - degree Gegenbauer (or ultraspherical) polynomials (GPS), denoted here by

U (β)
k (t), with parameter β at the point t are recursively defined by

U (β)
0 (t) = 1, U (β)

1 (t) = 2βt,

U (β)
k (t) =

1

k

[
2t (k + β − 1) U (β)

k−1(t)− (k + 2β − 2) U (β)
k−2(t)

]
, k ≥ 2.

(1)

These polynomials are orthogonal on the interval I = [−1, 1] with respect to the weight

function (1 − t2)
β− 1

2 , where β > −1
2 . That is, for any two GPS, U (β)

k (t) and U (β)
l (t),

with k ̸= l, we have ∫ 1

−1

U (β)
k (t) U (β)

l (t)
(
1− t2

)β− 1
2
dt = 0,

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5968

Email addresses: abdelbaset.zeyani@wichita.edu (A. Zeyani), ahussen@navajotech.edu (A. Hussen)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



A. Zeyani, A. Hussen / Eur. J. Pure Appl. Math, 18 (2) (2025), 5968 2 of 17

and with the condition that l = k, we have∫ 1

−1

(
U (β)
k (t)

)2 (
1− t2

)β− 1
2
dt =

√
π Γ(k + 2β − 1)

21−2β k! Γ(β) Γ(k + β − 1
2)
.

For β > 0, a generating function of GPS, G
β
(t, ζ), is defined by the form

G
β
(t, ζ) =

1

(1− 2tζ + ζ2)β
=

∞∑
k=0

U (β)
k (t) ζk, (2)

where t ∈ I, ζ is in the open unit disk U = {ζ : ζ ∈ C and |ζ| < 1}, and C is, as usual,
the set of complex numbers. For a fixed t ∈ I, G

β
(t, ζ) is analytic in U that has a Taylor

series expansion given by (2). Evidently, we see that G
β
(t, ζ) produces no values when

β = 0. Therefore, the generating function of GPS is set to be of the form

G0(t, ζ) = 1− log
(
1− 2 t ζ + ζ2

)
=

∞∑
k=0

U (0)
k (t) ζk. (3)

Note that U (β)
k (t) are particular solutions of the Gegenbauer differential equation given

by

(1− t2)
d
2
y

dt2
− (2β + 1) t

dy

dt
+ k (k + 2β) y = 0, (4)

and when setting

(i) β = 1/2, equation (4) reduces to the Legendre differential equations, and
the GPS reduce to the Legendre polynomials.
(ii) β = 1, equation (4) reduces to the Chebyshev differential equations, and
the GPS reduce to the Chebyshev polynomials of the second kind.

Let A denote the class of all functions of the form

f(ζ) = ζ +

∞∑
n=2

an ζ
n, (ζ ∈ U), (5)

which are analytic in U and normalized by these two conditions f(0) = 0 and f
′
(0) = 1.

Moreover, let S be the subclass of A consisting of all normalized univalent functions
of the form (5) which are also univalent in U. Two functions, f and g, are said to be
subordinate ( f ≺ g ) if there is an analytic function h(ζ) (namely; a Schwarz function)
in U, such that f(ζ) = g(h(ζ)) with h(0) = 0 and | h(ζ) | ≤ 1. Especially, if the function
g is univalent in U, then the following equivalence is valid [1]

f(ζ) ≺ g(ζ) ⇐⇒ f(0) = g(0)

and
f(U) ⊂ g(U).



A. Zeyani, A. Hussen / Eur. J. Pure Appl. Math, 18 (2) (2025), 5968 3 of 17

The Koebe One-Quarter Theorem [2] states that the image of U under every function
f ∈ S contains a disk of radius 1

4 and center at the origin; i.e., U 1
4
(0) ∈ f(U). Therefore,

every univalent function f ∈ S has an inverse f
−1

: f(U) → U which satisfies the following
conditions:

( f
−1 ◦ f )(ζ) = ζ

(
ζ ∈ U

)
and

( f ◦ f
−1

)(η) = η

(
|η| < r0(f); r0(f) ≥

1

4

)
,

where f
−1

has the series expansion of the form

f
−1
(η) = η − a2 η

2 + (2a2
2
− a3) η

3 − (5a3
2
− 5a2a3 + a4) η

4 + · · · . (6)

A function f ∈ A is said to be bi-univalent in U if both f and f
−1

are univalent in U . Let
Ξ be denoting the class of bi-univalent functions in U given by (5). Herein, we recall the
following examples of functions in the bi-univalent function class Ξ that have apparently
revived the study of bi-univalent functions in recent years:

f1(ζ) =
ζ

1− ζ
, f2(ζ) = −log(1− ζ), and f3(ζ) =

1

2
log
(1 + ζ

1− ζ

)
,

where their inverses are respectively given by

f
−1

1
(η) =

η

1 + η
, f

−1

2
(η) =

eη − 1

eη
, and f

−1

3
(η) =

e2η − 1

e2η + 1
.

However, the familiar Koebe function, K(ζ) = ζ
(1−ζ)2

, is not a member of the bi-univalent

function class Ξ since it maps the open unit disk U ⊂ C onto the set K(U) = C\(−∞,−1
4 ],

which does not contain U (i.e., {η : η ∈ C and |η| ≤ 1
4} ⊆ K(U)) (see [3–11]). Other

common univalent functions in S that are not members of Ξ are

ϑ1(ζ) =
ζ

1− ζ2
and ϑ2(ζ) = ζ − ζ2

2
.

Historically speaking, certain subclasses of Ξ were introduced by Brannan and Taha
(see [12]) similar to the familiar subclasses S∗(ε) and K(ε) of star-like and convex func-
tions of order ε ∈ [ 0, 1) in the open unit disk U, which are respectively defined by

S∗(ε) =

{
f : f ∈ S and ℜ

{
ζ f

′
(ζ)

f(ζ)

}
> ε, ζ ∈ U

}
,

and

K(ε) =

{
f : f ∈ S and ℜ

{
1 +

ζ f
′′
(ζ)

f
′
(ζ)

}
> ε, ζ ∈ U

}
.



A. Zeyani, A. Hussen / Eur. J. Pure Appl. Math, 18 (2) (2025), 5968 4 of 17

Analogously, the bi-star-like and bi-convex function classes S∗
Ξ
(ε) and KΞ(ε) of order

ε ∈ [ 0, 1) in the open unit disk U, corresponding to S∗(ε) and K(ε), were introduced and
studied, and non-sharp upper bound estimations of the initial Taylor-Maclaurin coefficients
were obtained as well. In 1976, Noonan and Thomas defined the qth Hankel determinant
of the function f ∈ A of the form (5) for integers n, q ∈ N = {1, 2, 3, . . . } by [13]

H
f
(n, q) =

∣∣∣∣∣∣∣∣∣
an an+1 · · · an+q−1

an+1 an+2 · · · an+q
...

...
...

...
an+q−1 an+q · · · an+2q−2

∣∣∣∣∣∣∣∣∣ , a1 := 1.

In particular, it is observed that, for n = 1, 2 and q = 2, the Hankel determinants are
given by

H
f
(1, 2) =

∣∣∣∣ a1 a2

a2 a3

∣∣∣∣ = a3 − a2
2

and H
f
(2, 2) =

∣∣∣∣ a2 a3

a3 a4

∣∣∣∣ = a2 a4 − a2
3
,

which are respectively referred to as the well-known Fekete-Szegö functional and second
Hankel determinant functional. Several other authors have considered the determinant
H

f
(n, q) in their studies. In [14], for instance, Noor found that the rate of growth of

H
f
(n, q) as n → ∞ occurs when functions f ∈ A of the form (5) with bounded boundary.

The authors in [14] and [15], in particular, achieved sharp upper bounds on H
f
(2, 2)

for several types of function classes. For f ∈ S in the open unit disk U, the authors
in [16] obtained the sharp upper inequality for the functional H

f
(1, 2) that is given by

|H
f
(1, 2) | = | a3−a2

2
| ≤ 1. Several authors have recently investigated the upper bounds of

H
f
(1, 2) and Taylor-Maclaurin coefficients for various subclasses of bi-univalent functions

(see, for examples, [17–30]). Furthermore, the subclass of S consisting of all functions
whose derivatives have positive real part, introduced in [31], was considered by the authors
of [32] in order to derive the sharp bounds for the functional H

f
(2, 2) that is given by

|H
f
(2, 2) | = | a2a4 − a2

3
| ≤ 4

9 for each function belongs to that subclass. In addition, they
discovered the sharp second Hankel determinant, H

f
(2, 2), in (see [32]) for star-like and

convex function subclasses, S∗ and K, of S with bounds of |H
f
(2, 2) | = | a2a4−a2

3
| ≤ 1

8
and |H

f
(2, 2) | = | a2a4 − a2

3
| ≤ 1, respectively. In recent times, several researchers have

explored upper bounds for the coefficients and Hankel determinant of functions within
different subclasses of univalent functions (see, for examples, [33–36]).

Definition 1. Let τ ∈ [0, 1] and t ∈ (12 , 1]. A function f ∈ Ξ of the form (5) is said to

be in the class Ωβ
Ξ(t, τ) with a nonzero real constant β if the following subordinations are

satisfied

τ

(
1 +

ζ f
′′
(ζ)

f ′(ζ)

)
+ (1− τ)

(
ζ f

′
(ζ)

f(ζ)

)
≺ G

β
(t, ζ) (7)

and

τ

(
1 +

η g
′′
(η)

g′(η)

)
+ (1− τ)

(
η g

′
(η)

g(η)

)
≺ G

β
(t, η), (8)



A. Zeyani, A. Hussen / Eur. J. Pure Appl. Math, 18 (2) (2025), 5968 5 of 17

where the function g = f
−1

is defined by (6) and G
β

is the GPS - generating function
given by (2).

Remark 1. [37] By setting τ = 0 in (1), we obtain the class Ωβ
Ξ(t, 0) = Σβ

Ξ(t) that
consists of functions f ∈ Ξ satisfying the conditions

ζ f
′
(ζ)

f(ζ)
≺ G

β
(t, ζ) (9)

and
η g

′
(η)

g(η)
≺ G

β
(t, η), (10)

where the function g = f
−1

is defined by (6).

Remark 2. [37] By setting τ = 1 in (1), we obtain the class Ωβ
Ξ(t, 1) = Dβ

Ξ(t) that
consists of functions f ∈ Ξ satisfying the conditions

1 +
ζ f

′′
(ζ)

f ′(ζ)
≺ G

β
(t, ζ) (11)

and

1 +
η g

′′
(η)

g′(η)
≺ G

β
(t, η), (12)

where the function g = f
−1

is defined by (6).

Let Q : U → C be the class of functions s(ζ) with positive real part consisting of all
analytic functions satisfying the conditions that s(0) = 1 and ℜ (s(ζ)) > 0. To derive our
desirable upper bounds estimation for the second Hankel determinant H

f
(2, 2) = a2a4−a2

3

associated with the class Ωβ
Ξ(t, τ) in (1), we state the necessary lemmas:

Lemma 1. [38] If the function s ∈ Q is defined by

s(ζ) = 1 +
∞∑
k=1

s
k
ζk, (13)

then
| s

k
| ≤ 2, k = 1, 2, ...

Lemma 2. [39] If the function s ∈ Q is of the form (13), then

2s2 = s2
1
+ (4− s2

1
) ξ (14)

and
4s3 = s3

1
+ 2(4− s2

1
) s1 ξ − s1(4− s2

1
) ξ2 + 2(4− s2

1
)(1− |ξ|2) ζ (15)

for some ξ and ζ with | ξ | ≤ 1 and | ζ | ≤ 1.



A. Zeyani, A. Hussen / Eur. J. Pure Appl. Math, 18 (2) (2025), 5968 6 of 17

Also by considering the class ∆ that consists of all analytic functions ω ∈ U satisfying
the conditions that ω(0) = 0 and |ω(ζ) | < 1 for all ζ ∈ U, we state the following lemma:

Lemma 3. [2] Let ω ∈ ∆ with ω(ζ) =
∑∞

k=1 ωk ζ
k, ζ ∈ U. Then

|ω1 | ≤ 1 and |ωk | ≤ 1− |ω1 |
2
for k ≥ 2.

2. Second Hankel Determinant

Theorem 1. Let the function f ∈ Ξ of the form (5) be in the class Ωβ
Ξ(t, τ) in (1). Then

∣∣ a2a4 − a2
3

∣∣ ≤



T (2−, t) E1 ≥ 0 and E2 ≥ 0;

max
{

4β2 t2

(1+2 τ)2
, T (2−, t)

}
E1 > 0 and E2 < 0;

4β2 t2

(1+2τ)2
E1 ≤ 0 and E2 ≤ 0;

max
{
T (c0 , t), T (2−, t)

}
E1 < 0 and E2 > 0,

(16)

where

T (2−, t) =

(
U (β)
1 (t)

)2
(1 + 2 τ)2

+
E1 + E2

3 (1 + 3 τ) (1 + τ)3 (1 + 2 τ)2
, (17)

T (c0 , t) =

(
U (β)
1 (t)

)2
(1 + 2 τ)2

− E2
2

12 E1 (1 + 3 τ) (1 + τ)3 (1 + 2 τ)2
; c0 =

√
−2 E2

E1
, (18)

E1 = 16 (1 + 2 τ)2 U (β)
1 (t)

∣∣∣∣∣
(
U (β)
3 (t) + U (β)

2 (t) +
1

4
U (β)
1 (t)

)
(1 + τ)2 −

(
U (β)
1 (t)

)3∣∣∣∣∣
+
(
U (β)
1 (t)

)2[
3 (1 + 3 τ)(1 + τ)3 − 12 (1 + τ)2(1 + 2 τ)2

]
− 2 (1 + τ)(1 + 2 τ) U (β)

1 (t)

[
3 (1 + 3 τ)

(
U (β)
1 (t)

)2
+ 8 (1 + τ)(1 + 2 τ) U (β)

2 (t)

]
,

(19)

E2 = 12 (1 + 2 τ)2(1 + τ)2
(
U (β)
1 (t)

)2
+ 6 (1 + τ)(1 + 2 τ)(1 + 3 τ)

(
U (β)
1 (t)

)3
+ 16 (1 + τ)2(1 + 2 τ)2 U (β)

1 (t) U (β)
2 (t)− 6 (1 + 3 τ)(1 + τ)3

(
U (β)
1 (t)

)2
,

(20)

and U (β)
1 (t), U (β)

2 (t), and U (β)
3 (t) are defined by (1).



A. Zeyani, A. Hussen / Eur. J. Pure Appl. Math, 18 (2) (2025), 5968 7 of 17

Proof. Suppose f ∈ Ωβ
Ξ(t, τ) for some τ ∈ [0, 1]. Then from (7) and (8) we have

τ

(
1 +

ζ f
′′
(ζ)

f ′(ζ)

)
+ (1− τ)

(
ζ f

′
(ζ)

f(ζ)

)
≺ G

β
(t, u(ζ)) (21)

and

τ

(
1 +

η g
′′
(η)

g′(η)

)
+ (1− τ)

(
η g

′
(η)

g(η)

)
≺ G

β
(t, v(η)), (22)

where g = f
−1

and u, v ∈ ∆ are given by

u(ζ) =
∞∑
n=1

cn ζ
n and v(η) =

∞∑
n=1

dn η
n.

Then by using G
β
(t, ζ) given in (2), we can write the right hand sides of (21) and (22)

as follows:

G
β
(t, u(ζ)) = 1 + U (β)

1 (t) c1 ζ +

[
U (β)
1 (t) c2 + U (β)

2 (t) c21

]
ζ2

+

[
U (β)
1 (t) c3 + 2U (β)

2 (t) c1 c2 + U (β)
3 (t) c31

]
ζ3 + · · ·

(23)

and

G
β
(t, u(η)) = 1 + U (β)

1 (t) d1 η +

[
U (β)
1 (t) d2 + U (β)

2 (t) d21

]
η2

+

[
U (β)
1 (t) d3 + 2U (β)

2 (t) d1 d2 + U (β)
3 (t) d31

]
η3 + · · · .

(24)

Therefore, (21) and (22) become

τ

[
1 + 2 a2 ζ + (6 a3 − 4 a22) ζ

2 + 2 (4 a32 − 9 a2 a3 + 6a4)ζ
3 + · · ·

]
+ (1− τ)

[
1 + a2 ζ + (2 a3 − a22) ζ

2 + (a32 − 3 a2 a3 + 3a4)ζ
3 + · · ·

]
= 1 + U (β)

1 (t) c1 ζ +

[
U (β)
1 (t) c2 + U (β)

2 (t) c21

]
ζ2

+

[
U (β)
1 (t) c3 + 2U (β)

2 (t) c1 c2 + U (β)
3 (t) c31

]
ζ3 + · · · ,

(25)



A. Zeyani, A. Hussen / Eur. J. Pure Appl. Math, 18 (2) (2025), 5968 8 of 17

and

τ

[
1− 2 a2 η + (8 a22 − 6 a3)η

2 + (−32 a32 + 42 a2 a3 − 12 a4) η
3 + · · ·

]
+ (1− τ)

[
1− a2 η + (3 a22 − 2 a3) η

2 + (−10 a32 + 12 a2 a3 − 3 a4) η
3 + · · ·

]
= 1 + U (β)

1 (t) d1 η +

[
U (β)
1 (t) d2 + U (β)

2 (t) d21

]
η2

+

[
U (β)
1 (t) d3 + 2U (β)

2 (t) d1 d2 + U (β)
3 (t) d31

]
η3 + · · ·

(26)

At this point, the corresponding coefficients in (25) and (26) can be equated to obtain

(1 + τ) a2 = U (β)
1 (t) c1, (27)

2 (1 + 2 τ) a3 − (1 + 3τ) a22 = U (β)
1 (t) c2 + U (β)

2 (t) c21, (28)

(1+7 τ) a32− 3 (1+5 τ) a2 a3+3 (1+3 τ) a4 = U (β)
1 (t) c3+2U (β)

2 (t) c1 c2+U (β)
3 (t) c31, (29)

−(1 + τ) a2 = U (β)
1 (t) d1, (30)

(3 + 5 τ) a22 − 2 (1 + 2 τ) a3 = U (β)
1 (t) d2 + U (β)

2 (t) d21, (31)

and

−2 (5 + 11 τ) a32 + 6 (2 + 5 τ) a2 a3 − 3 (1 + 3 τ) a4 = U (β)
1 (t) d3 + 2U (β)

2 (t) d1 d2 + U (β)
3 (t) d31.

(32)
From (27) and (30), we obtain that

c1 = − d1 (33)

and

a2 =
U (β)
1 (t)

1 + τ
c1. (34)

Upon subtracting (31) from (28), we have that

a3 = a22 +
U (β)
1 (t) (c2 − d2)

4 (1 + 2 τ)
=

(
U (β)
1 (t)

)2
(1 + τ)2

c21 +
U (β)
1 (t) (c2 − d2)

4 (1 + 2 τ)
. (35)

Furthermore, if we subtract (32) from (29), together with (27), (33) and (35), we have

a4 =
5
(
U (β)
1 (t)

)2
(c2 − d2) c1

8 (1 + τ) (1 + 2 τ)
+

U (β)
1 (t) (c3 − d3)

6(1 + 3τ)

+
U (β)
2 (t) (c2 + d2) c1

3 (1 + 3 τ)
+

[
U (β)
3 (t)

3 (1 + 3 τ)
+

2 (1 + 4 τ)
(
U (β)
1 (t)

)3
3 (1 + τ)3 (1 + 3 τ)

]
c31.

(36)



A. Zeyani, A. Hussen / Eur. J. Pure Appl. Math, 18 (2) (2025), 5968 9 of 17

Thus, when applying (27), (35), and (36), we can simply establish that

a2 a4 − a23 =

(
U (β)
1 (t)

)3
(c2 − d2) c

2
1

8 (1 + τ)2 (1 + 2 τ)
+

(
U (β)
1 (t)

)2
(c3 − d3) c1

6 (1 + τ) (1 + 3 τ)

+
U (β)
1 (t) U (β)

2 (t) (c2 + d2) c
2
1

3 (1 + τ) (1 + 3 τ)
−

(
U (β)
1 (t)

)2
(c2 − d2)

2

16 (1 + 2 τ)2

+

U (β)
1 (t)

[
U (β)
3 (t) (1 + τ)2 −

(
U (β)
1 (t)

)3]
c41

3 (1 + 3 τ) (1 + τ)3
.

(37)

Next, according to Lemma (2), we now have that

c2 − d2 =
4− c2

2
(x− y), (38)

c2 + d2 = c21 +
4− c2

2
(x+ y), (39)

and

c3 − d3 =
c31
2
+

(4− c2) c1
2

(x+ y)− (4− c21) c1
4

(x2 + y2)

+
4− c21

2

[(
1− |x|2

)
ζ −

(
1− |y|2

)
η

]
,

(40)

for some x, y, ζ, and η with |x| ≤ 1, |y| ≤ 1, |ζ| ≤ 1, and |η| ≤ 1. Then, by substituting
(38), (39), and (40) into (37), we obtain that

∣∣∣ a2 a4 − a23

∣∣∣ ≤ U (β)
1 (t)

∣∣∣(U (β)
3 (t) + U (β)

2 (t) + 1
4 U

(β)
1 (t)

)
(1 + τ)2 −

(
U (β)
1 (t)

)3∣∣∣
3 (1 + 3 τ) (1 + τ)3

c41 +

(
U (β)
1 (t)

)2
(4− c21) c1

6 (1 + τ) (1 + 3 τ)

+

[(U (β)
1 (t)

)3
(4− c21) c

2
1

16 (1 + τ)2 (1 + 2 τ)
+

(
U (β)
1 (t)

)2
(4− c21) c

2
1

12 (1 + τ) (1 + 3 τ)
+

U (β)
1 (t) U (β)

2 (t) (4− c21) c
2
1

6 (1 + τ) (1 + 3 τ)

](∣∣x∣∣+ ∣∣y∣∣)

+

[(U (β)
1 (t)

)2
(4− c21) c

2
1

24 (1 + τ) (1 + 3 τ)
−

(
U (β)
1 (t)

)2
(4− c21) c1

12 (1 + τ) (1 + 3 τ)

](∣∣x∣∣2 + ∣∣y∣∣2)

+

[(U (β)
1 (t)

)2 (
4− c21

)2
64 (1 + 2 τ)2

](∣∣x∣∣+ ∣∣y∣∣)2.
(41)

Lemma (1) allows us to assume, without any loss of generality, that c ∈ [0, 2] where
c = |c1 |. Thus, for δ1 = | x | ≤ 1 and δ2 = | y | ≤ 1, we can rewrite (41) to be in the



A. Zeyani, A. Hussen / Eur. J. Pure Appl. Math, 18 (2) (2025), 5968 10 of 17

following form:∣∣∣ a2 a4 − a23

∣∣∣ ≤ Υ1 +Υ2 (δ1 + δ2) + Υ3

(
δ21 + δ22

)
+Υ4

(
δ1 + δ2

)2
= φ(δ1, δ2), (42)

where

Υ1 =
U (β)
1 (t)

∣∣∣(U (β)
3 (t) + U (β)

2 (t) + 1
4 U

(β)
1 (t)

)
(1 + τ)2 −

(
U (β)
1 (t)

)3∣∣∣
3 (1 + 3 τ) (1 + τ)3

c4+

(
U (β)
1 (t)

)2
(4− c2) c

6 (1 + τ) (1 + 3 τ)
≥ 0,

(43)

Υ2 =

[(U (β)
1 (t)

)3
(4− c2) c2

16 (1 + τ)2 (1 + 2 τ)
+

(
U (β)
1 (t)

)2
(4− c2) c2

12 (1 + τ) (1 + 3 τ)
+

U (β)
1 (t) U (β)

2 (t) (4− c2) c2

6 (1 + τ) (1 + 3 τ)

]
≥ 0,

(44)

Υ3 =

[(U (β)
1 (t)

)2
(4− c2) (c− 2) c

24 (1 + τ) (1 + 3 τ)

]
≤ 0, (45)

and

Υ4 =

[(U (β)
1 (t)

)2 (
4− c2

)2
64 (1 + 2 τ)2

]
≥ 0. (46)

Now, we have to maximize the function φ(δ1, δ2) in (42) on the closed square S = [0, 1]×
[0, 1] by investigating the maximum values of φ(δ1, δ2) in accordance with 0 < c < 2,
c = 0, and c = 2. For the case that 0 < c < 2, since Υ3 < 0 and Υ3 + 2Υ4 > 0 for all
t ∈ (12 , 1), we deduce that

φ
δ1δ1

φ
δ2δ2

− φ2
δ1δ2

< 0, for all δ1, δ2 ∈ S.

Therefore, as a result of this, the function φ cannot have a local maximum in the interior
of the square S. Now, we will explore the maximum value of φ on the boundary of S.
(1) for δ1 = 0 and 0 ≤ δ2 ≤ 1 (similarly, for δ2 = 0 and 0 ≤ δ1 ≤ 1), φ(δ1, δ2) takes
the form

ψ1(δ2) := φ(0, δ2) = Υ1 +Υ2 δ2 + (Υ3 +Υ4) δ
2

2.

Next, we will separately discuss the following two cases.

Case (i) : When Υ3 +Υ4 ≥ 0, for 0 < δ2 < 1, for any fixed c ∈ (0, 2), and
for all t ∈ (12 , 1), it is obvious that ψ

′

1
(δ2) = Υ2 + 2 (Υ3 +Υ4) δ2 > 0.

Case (ii) : When Υ3 + Υ4 < 0 and since Υ2 + 2 (Υ3 + Υ4) ≥ 0, for
0 < δ2 < 1, for any fixed c ∈ (0, 2), and for all t ∈ (12 , 1), it is obvi-

ous that Υ2 + 2 (Υ3 + Υ4) < Υ2 + 2 (Υ3 + Υ4) δ2 < Υ2 and thus ψ
′

1
(δ2) =

Υ2 + 2 (Υ3 +Υ4) δ2 > 0 .



A. Zeyani, A. Hussen / Eur. J. Pure Appl. Math, 18 (2) (2025), 5968 11 of 17

In both cases, ψ1(δ2) is an increasing function and; therefore, for any fixed
c ∈ (0, 2) and t ∈ (12 , 1), the maximum value of ψ1(δ2) occurs at δ2 = 1 and

max
δ2

{
ψ1(δ2)

}
= ψ1(1) = Υ1 +Υ2 +Υ3 +Υ4. (47)

For c = 0 and c = 2, we respectively obtain that

φ(δ1, δ2) = Υ4

∣∣∣
c=0

=

(
U (β)
1 (t)

)2
4 (1 + 2 τ)2

(
δ1 + δ2

)2
and

φ(δ1, δ2) = Υ1

∣∣∣
c=2

=
16 U (β)

1 (t)
∣∣∣(U (β)

3 (t) + U (β)
2 (t) + 1

4 U
(β)
1 (t)

)
(1 + τ)2 −

(
U (β)
1 (t)

)3∣∣∣
3 (1 + 3 τ) (1 + τ)3

.

(48)

By taking equation (48) and the two mentioned cases in account, for 0 ≤ δ2 < 1, for any
fixed c ∈ [0, 2], and for all t ∈ (12 , 1), the maximum value of ψ1(δ2) is

max
δ2

{
ψ1(δ2)

}
= ψ1(1) = Υ1 +Υ2 +Υ3 +Υ4.

(2) for δ1 = 1 and 0 ≤ δ2 ≤ 1 (similarly, for δ2 = 1 and 0 ≤ δ1 ≤ 1), φ(δ1, δ2) takes
the form

ψ2(δ2) := φ(1, δ2) = (Υ3 +Υ4) δ
2
2 + (Υ2 + 2Υ4) δ2 +Υ1 +Υ2 +Υ3 +Υ4.

Analogous to the previously mentioned cases of Υ3 +Υ4, we conclude that

max
δ2

{
ψ2(δ2)

}
= ψ2(1) = Υ1 + 2Υ2 + 2Υ3 + 4Υ4. (49)

Since ψ1(1) ≤ ψ2(1) for c ∈ (0, 2) and t ∈ (12 , 1), we see that

max
δ1 , δ2

{
φ(δ1, δ2)

}
= φ(1, 1) (50)

on the boundary of S. Therefore, the maximum value of φ(δ1, δ2) occurs at δ1 = 1 and
δ2 = 1 in the closed square S. Now, for a fixed value of t, let T : [0, 2] → R be the
function defined by

T (c, t) = max
δ1 , δ2

(
φ(δ1, δ2)

)
= φ(1, 1) = Υ1 + 2Υ2 + 2Υ3 + 4Υ4 (51)

Upon substituting the expressions of Υ1, Υ2, Υ3, and Υ4 into (51), we obtain that

T (c, t) =

(
U (β)
1 (t)

)2
(1 + 2 τ)2

+
E1 c4 + 4 E2 c2

48 (1 + 3 τ) (1 + τ)3 (1 + 2 τ)2
, (52)



A. Zeyani, A. Hussen / Eur. J. Pure Appl. Math, 18 (2) (2025), 5968 12 of 17

where

E1 = 16 (1 + 2 τ)2 U (β)
1 (t)

∣∣∣∣∣(U (β)
3 (t) + U (β)

2 (t) +
1

4
U (β)
1 (t)

)
(1 + τ)2 −

(
U (β)
1 (t)

)3∣∣∣∣∣
+
(
U (β)
1 (t)

)2 [
3 (1 + 3 τ) (1 + τ)3 − 12 (1 + τ)2 (1 + 2 τ)2

]

− 2 (1 + τ) (1 + 2 τ) U (β)
1 (t)

[
3 (1 + 3 τ)

(
U (β)
1 (t)

)2
+ 8 (1 + τ) (1 + 2 τ)U (β)

2 (t)

]
(53)

and

E2 = 12 (1 + 2 τ)2 (1 + τ)2
(
U (β)
1 (t)

)2
+ 6 (1 + τ) (1 + 2 τ) (1 + 3 τ)

(
U (β)
1 (t)

)3
+ 16 (1 + τ)2 (1 + 2 τ)2 U (β)

1 (t) U (β)
2 (t)− 6 (1 + 3 τ) (1 + τ)3

(
U (β)
1 (t)

)2
.

(54)

By assuming that the function T (c, t) has a maximum value at an interior point 0 < c < 2,
we obtain

d T
d c

=
E1 c3 + 2 E2 c

12 (1 + 3 τ) (1 + τ)3 (1 + 2 τ)2
. (55)

With some calculations, we can examine the sign of d T
d c taking into account the following

four cases.

(i) Suppose that E1 ≥ 0 and E2 ≥ 0, then d T
d c ≥ 0; indicating that T (c, t) is an

increasing function. Therefore, we get that

max
0< c<2

{
T (c, t)

}
= T (2−, t) =

(
U (β)
1 (t)

)2
(1 + 2 τ)2

+
E1 + E2

3 (1 + 3 τ) (1 + τ)3 (1 + 2 τ)2
, (56)

which means:

max
0< c<2

{
max
S

{
φ(δ1, δ2)

}}
= T (2−, t).

(ii) Suppose that E1 > 0 and E2 < 0, then c0 =
√

−2 E2
E1 is a critical value of the

function T (c, t). By assuming c0 ∈ (0, 2), we get that d2 T
d c2

∣∣∣
c=c0

> 0, that is, c = c0

is a local minimum value of T (c, t). Thus, the function T (c, t) can not possess a
local maximum.



A. Zeyani, A. Hussen / Eur. J. Pure Appl. Math, 18 (2) (2025), 5968 13 of 17

(iii) Suppose that E1 ≤ 0 and E2 ≤ 0, then d T
d c ≤ 0; indicating that T (c, t) is a

decreasing function. Thus,

max
0< c<2

{
T (c, t)

}
= T (0+, t) = 4Υ4 =

(
U (β)
1 (t)

)2
(1 + 2 τ)2

. (57)

(iv) Suppose that E1 < 0 and E2 > 0, then c0 is a critical value of the function T (c, t).

By assuming c0 ∈ (0, 2), we obtain that d2 T
d c2

∣∣∣
c=c0

< 0, which means that the

function T (c, t) has a local maximum occurring at c = c0 . Thus,

max
0< c<2

{
T (c, t)

}
= T (c0 , t), (58)

where

T (c0 , t) =

(
U (β)
1 (t)

)2
(1 + 2 τ)2

− E2
2

12 E1 (1 + 3 τ) (1 + τ)3 (1 + 2 τ)2
.

Therefore, the proof of the above Theorem is evidently completed.

Ultimately, we introduce two essential corollaries that obtained from the classes Σβ
Ξ(t)

and Λβ
Ξ(t).

Corollary 1. Let f ∈ Ξ of the form (5) be in the class Ωβ
Ξ(t, 0) = Λβ

Ξ(t). Then

∣∣ a2a4 − a2
3

∣∣ ≤



T (2−, t) E∗
1 ≥ 0 and E∗

2 ≥ 0;

max
t

{
4β2 t2, T (2−, t)

}
E∗

1 > 0 and E∗
2 < 0;

4β2 t2 E∗
1 ≤ 0 and E∗

2 ≤ 0;

max
t

{
T (c0 , t), T (2−, t)

}
E∗

1 < 0 and E∗
2 > 0,

(59)

where

T (2−, t) = 4β2 t2 +
E∗

1 + E∗
2

3
, (60)

T (c0 , t) = 4β2 t2 − E∗2
2

12 E∗
1
; c0 =

√
−2 E∗

2

E∗
1
, (61)

E∗
1 = 16 U (β)

1 (t)

∣∣∣∣∣U (β)
3 (t) + U (β)

2 (t) +
1

4
U (β)
1 (t)−

(
U (β)
1 (t)

)3∣∣∣∣∣
− 9

(
U (β)
1 (t)

)2
− 2 U (β)

1 (t)

[
3
(
U (β)
1 (t)

)2
+ 8 U (β)

2 (t)

]
,

(62)



A. Zeyani, A. Hussen / Eur. J. Pure Appl. Math, 18 (2) (2025), 5968 14 of 17

E∗
2 = 2 U (β)

1 (t)

[
3 U (β)

1 (t) + 3
(
U (β)
1 (t)

)2
+ 8 U (β)

2 (t)

]
, (63)

and U (β)
1 (t), U (β)

2 (t), and U (β)
3 (t) are defined by (1).

Corollary 2. Let f ∈ Ξ of the form (5) be in the class Ωβ
Ξ(t, 1) = Σβ

Ξ(t). Then

∣∣ a2a4 − a2
3

∣∣ ≤



T (2−, t) D∗
1 ≥ 0 and D∗

2 ≥ 0;

max
t

{
4β2 t2

9 , T (2−, t)
}

D∗
1 > 0 and D∗

2 < 0;

4β2 t2

9 D∗
1 ≤ 0 and D∗

2 ≤ 0;

max
t

{
T (c0 , t), T (2−, t)

}
D∗

1 < 0 and D∗
2 > 0,

(64)

where

T (2−, t) =
4β2 t2

9
+

D∗
1 +D∗

2

864
, (65)

T (c0 , t) =
4β2 t2

9
− D∗2

2

3456D∗
1
, c0 =

√
−2D∗

2

D∗
1
, (66)

D∗
1 = 144 U (β)

1 (t)

∣∣∣∣∣4(U (β)
3 (t) + U (β)

2 (t) +
1

4
U (β)
1 (t)

)
−
(
U (β)
1 (t)

)3∣∣∣∣∣
− 336

(
U (β)
1 (t)

)2
− 144 U (β)

1 (t)

[(
U (β)
1 (t)

)2
+ 4 U (β)

2 (t)

]
,

(67)

D∗
2 = 48 U (β)

1 (t)

[
5 U (β)

1 (t) + 3
(
U (β)
1 (t)

)2
+ 12 U (β)

2 (t)

]
, (68)

and U (β)
1 (t), U (β)

2 (t), and U (β)
3 (t) are defined by (1).

3. Conclusion

In our present study, we have derived the new upper bound estimates and inequal-
ities for the second Hankel determinant, H

f
(2, 2), of a certain subclass of normalized

bi-univalent functions in the open unit disk U. The upper bound estimates are deter-
mined by using orthogonal ultraspherical polynomials, which provide information about
the properties and characteristics of these functions in the context of H

f
(2, 2). Further-

more, we provide new findings acquired by specializing the parameter τ utilized in our
analysis.



A. Zeyani, A. Hussen / Eur. J. Pure Appl. Math, 18 (2) (2025), 5968 15 of 17

Acknowledgements

The authors would like to express their sincere gratitude to the editor and the anony-
mous reviewers for their insightful and constructive feedback. Their valuable comments
and suggestions have significantly contributed to enhancing the quality of this work.

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