EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 6081
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Vandermonde Determinant for the Generalized
Bounded Turning Functions Associated with Gregory

Coefficients

Nur Hazwani Aqilah Abdul Wahid1,∗, Shaharuddin Cik Soh1

1 School of Mathematical Sciences, College of Computing, Informatics and Mathematics,
Universiti Teknologi MARA, 40450 Shah Alam, Selangor, Malaysia

Abstract. This paper explores the class GG (α, δ) of analytic functions, which is associated with
generalized bounded turning and the generating functions of Gregory coefficients. By using bounds
on certain coefficient functionals for functions with a positive real part, we obtain initial Taylor
coefficient bounds and logarithmic coefficient bounds of functions and inverse functions within
this class. Consequently, we establish upper bounds of the second-order for the Vandermonde
determinant, where the entries are Taylor coefficients and logarithmic coefficients of functions
and inverse functions. Additionally, we highlight several interesting implications of these results,
contributing new insights to this generalized class.

2020 Mathematics Subject Classifications: 30C45, 30C50

Key Words and Phrases: Univalent functions, inverse functions, bounded turning functions,
Gregory coefficients, Taylor coefficients, logarithmic coefficients, Vandermonde determinant

1. Introduction

Let A denote the class of analytic functions f (z) that can be expressed as a Taylor
series expansion in the open unit disk E = {z ∈ C : |z| < 1}, given by

f (z) = z +
∞∑
n=2

anz
n , z ∈ E. (1)

We denote by S the subclass of A consisting of univalent functions in E.
The inverse of a function f (z) ∈ S of the form (1) has a series expansion given by

f−1 (w) = w +
∞∑
n=2

Anw
n, |w| < r0 (f) , r0 (f) ≥

1

4
, (2)

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6081

Email addresses: hazwaniaqilah@uitm.edu.my (N. H. A. A. Wahid), haruddin@uitm.edu.my (S. Cik Soh)

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



N. H. A. A. Wahid, S. C. Soh / Eur. J. Pure Appl. Math, 18 (2) (2025), 6081 2 of 16

where, in particular, the coefficients An for n = 2, 3, 4 are expressed in terms of the Taylor
coefficients of f (z) ∈ S as follows:

A2 = −a2, (3)

A3 = −a3 + 2a2
2, (4)

and
A4 = −a4 + 5a2a3 − 5a2

3. (5)

Let P denote the class of functions with a positive real part in the open unit disk E.
A function p (z) in P has the series expansion

p (z) = 1 +
∞∑
n=1

pnz
n, z ∈ E, (6)

that is analytic in E and satisfying the condition Re (p (z)) > 0. Functions in P have often
been used to describe geometric properties of functions in A, and to define subclasses in
S. Let H denote the class of Schwarz functions υ (z) which are analytic in E, given by

υ (z) =

∞∑
k=1

bkz
k, z ∈ E

and satisfying υ (0) = 0 and |υ (z)| < 1. If p(z) ∈ P, then a Schwarz function υ(z) ∈ H
exists such that

p (z) =
1 + υ (z)

1− υ (z)
, z ∈ E. (7)

Let g1 (z) and g2 (z) be two analytic functions in E, with the symbol ≺ representing a sub-
ordination. The function g1 (z) is subordinate to function g2 (z), denoted g1 (z) ≺ g2 (z) ,
if there exists a Schwarz function υ (z) ∈ H such that g1 (z) = g2 (υ (z)). Furthermore, if
g1 (z) is univalent in E, then we have the following equivalence

g1 (z) ≺ g2 (z) ⇔ g1 (0) = g2 (0)

and
g1 (E) = g2 (E) .

Milin [1–3] highlighted the importance of logarithmic coefficients in estimating Taylor
coefficients of univalent functions. The inequalities conjectured by Milin attracted much
attention, which led to de Branges [4] establishing the Bieberbach conjecture. Logarithmic
coefficients also play a significant role in conformal mapping, which helped Kayumov [5]
solve Brennan’s conjecture. Since then, numerous studies on logarithmic coefficients, which
play a central role in the theory of univalent functions, have continued, with examples
found in [6–9]. Here, the logarithmic coefficients γn, n ⩾ 1 of f (z) ∈ S are defined by

log
f (z)

z
= 2

∞∑
n=1

γnz
n. (8)



N. H. A. A. Wahid, S. C. Soh / Eur. J. Pure Appl. Math, 18 (2) (2025), 6081 3 of 16

Differentiating (8) and equating coefficients of zn yield expressions for the logarithmic
coefficients in terms of the Taylor coefficients for f (z) ∈ S, specifically for n = 1, 2, 3:

γ1 =
1

2
a2, (9)

γ2 =
1

2

(
a3 −

1

2
a2

2

)
, (10)

and

γ3 =
1

2

(
a4 − a2a3 +

1

3
a2

3

)
. (11)

The logarithmic coefficients of the inverse functions, denoted Γn for f (z) ∈ S were intro-
duced by Ponnusamy et al. [10]. They are expressed in the series form as

log
f−1 (w)

w
= 2

∞∑
n=1

Γnw
n, |w| < 1

4
,

where, in particular, for n = 1, 2, 3:

Γ1 = −1

2
a2, (12)

Γ2 = −1

2

(
a3 −

3

2
a2

2

)
, (13)

and

Γ3 = −1

2

(
a4 − 4a2a3 +

10

3
a2

3

)
. (14)

A typical subject in geometric function theory is the study of coefficient functionals, which
are equations derived from various combinations of Taylor coefficients for subclasses in S.
This comprises the Taylor coefficients of inverse functions, the logarithmic coefficients of
functions and inverse functions, as well as the Vandermonde determinant.

The Vandermonde determinant, also known as a discriminant, has many applications
in a range of domains. It is used in digital signal processing to compute the discrete
Fourier transform (DFT) and the inverse discrete Fourier transform (IDFT), as well as in
approximation problems [11]. It is also an important tool in linear algebra, for example,
in determining the number of roots of polynomials (see [12]). Vijayalakshmi et al. [11]
studied the Vandermonde determinant Vq,n (f), where n, q ≥ 1 and an, n ≥ 2 are the
Taylor series coefficients in (1):

Vq,n (f) =

∣∣∣∣∣∣∣∣∣
1
1
...
1

an
an+1
...

an+q−1

· · ·
· · ·
...
· · ·

an
q−1

an+1
q−1

...
an+q−1

q−1

∣∣∣∣∣∣∣∣∣ , a1 = 1. (15)



N. H. A. A. Wahid, S. C. Soh / Eur. J. Pure Appl. Math, 18 (2) (2025), 6081 4 of 16

It is noted that with one as the first element, this determinant displays a geometric se-
quence in each row or column. Abdul Wahid et al. [13] introduced the Vandermonde
determinant Vq,n (γf ), where n, q ≥ 1 by looking at the logarithmic series coefficients
γn, n ≥ 1 in (8), because they were inspired by previous studies on Hankel and Toeplitz
determinants, which involved both Taylor coefficients and logarithmic coefficients. This
determinant is given by [13]

Vq,n (γf ) =

∣∣∣∣∣∣∣∣∣
1
1
...
1

γn
γn+1
...

γn+q−1

· · ·
· · ·
...
· · ·

γn
q−1

γn+1
q−1

...
γn+q−1

q−1

∣∣∣∣∣∣∣∣∣ . (16)

Given the importance of the Vandermonde determinant and inspired by the works of Abdul
Wahid et al. [14], Shi et al. [15], Obradovi and Tuneski [16], and Hadi et al. [17], which
deal with solving determinant and coefficient functional problems for the inverse of analytic
functions, it is natural to explore the Vandermonde determinant with An and Γn replacing
an and γn, respectively. Using this idea, we define the Vandermonde determinant of Taylor
coefficients and logarithmic coefficients of inverse functions for f (z) ∈ S, respectively, as
follows:

Vq,n

(
f−1

)
=

∣∣∣∣∣∣∣∣∣
1
1
...
1

An

An+1
...

An+q−1

· · ·
· · ·
...
· · ·

An
q−1

An+1
q−1

...
An+q−1

q−1

∣∣∣∣∣∣∣∣∣ (17)

and

Vq,n

(
Γf−1

)
=

∣∣∣∣∣∣∣∣∣
1
1
...
1

Γn

Γn+1
...

Γn+q−1

· · ·
· · ·
...
· · ·

Γn
q−1

Γn+1
q−1

...
Γn+q−1

q−1

∣∣∣∣∣∣∣∣∣ . (18)

Recently, Kazımoğlu et al. [18], Srivastava et al. [19], Tang et al. [20], Murugusun-
daramoorthy et al. [21], and Al-Hawarya et al. [22] introduced new classes of univalent
functions associated with the generating function of Gregory coefficients. The Gregory
coefficients, also known as reciprocal logarithmic numbers, second-kind Bernoulli num-
bers, or Cauchy numbers, are decreasing rational numbers that serve a function similar
to Bernoulli numbers and can be found in a wide range of problems, particularly those
involving numerical analysis and number theory. The generating function of the Gregory
coefficients Λn, for n ≥ 0, is given by

z

ln (1 + z)
=

∞∑
n=0

Λnz
n = 1 +

1

2
z − 1

12
z2 +

1

24
z3 − 19

720
z4 +

3

160
z5 − . . . . (19)

In connection with this function, we define the following class:



N. H. A. A. Wahid, S. C. Soh / Eur. J. Pure Appl. Math, 18 (2) (2025), 6081 5 of 16

Definition 1. An analytic function f (z) of the form (1) is said to be in the class GG (α, δ)
if the following condition is satisfied:

eiαf ′ (z)− i sinα− δ

ταδ
≺ Ψ(z), z ∈ E,

where Ψ(z) = z
ln(1+z) , ταδ = cosα− δ, |α| < π, and 0 ⩽ δ < 1.

Remark 1. Selecting specific values for the parameters α and δ in the class GG (α, δ)
yields the following classes, which are new and have not yet been studied by others:

(i) GG (α, 0) ≡ GG (α) =
{
f ∈ S : eiαf ′(z)−i sinα

cosα ≺ Ψ(z) , z ∈ E
}

(ii) GG (0, δ) ≡ GG (δ) :=
{
f ∈ S : f ′(z)−δ

1−δ ≺ Ψ(z) , z ∈ E
}

(iii) GG (0, 0) ≡ GG = {f ∈ S : f ′ (z) ≺ Ψ(z) , z ∈ E}
The class GG (α, δ) is inspired by the generalized class of bounded turning functions
G (α, δ) introduced by Mohamad [23], which satisfies{

f ∈ S : Re
(
eiαf ′ (z)

)
> δ, z ∈ E

}
.

Then there exists a function p (z) ∈ P such that [23]

eiαf ′ (z)− i sinα− δ

ταδ
= p (z) ∈ P,

where ταδ = cosα− δ, |α| < π, 0 ⩽ δ < 1, and cosα > δ.

Remark 2. Selecting specific values for the parameters α and δ in the class G (α, δ) results
in the following classes:

(i) G (α, 0) ≡ R (α) =
{
f ∈ S : Re

(
eiαf ′ (z)

)
> 0, z ∈ E

}
(ii) G (0, δ) ≡ R (δ) = {f ∈ S : Re (f ′ (z)) > δ, z ∈ E}. The class R (δ) is called the

class of bounded turning functions of order δ.

(iii) G (0, 0) ≡ R = {f ∈ S : Re (f ′ (z)) > 0, z ∈ E}. The class R is called the class of
bounded turning functions.

Pioneering researchers like Goel and Mehrok [24], Macgregor [25], Noshiro [26], Silver-
man and Silvia [27], and Warschawski [28] explored the classes R, R (δ), and R (α), and
nonetheless, it is intriguing to examine these classes in light of the generating functions of
Gregory coefficients, leading to the geometric properties of this class and contributing to
ongoing developments in geometry function theory.

Therefore, this paper aims to estimate the upper bounds of the Taylor coefficients and
logarithmic coefficients of functions and inverse functions belonging to the classGG (α, δ) of
analytic functions, which is associated with generalized bounded turning and the generat-
ing functions of Gregory coefficients. For example, |an| (n = 2, 3, 4, 5), |An| (n = 2, 3, 4, 5),
|γn| (n = 1, 2, 3), and |Γn| (n = 1, 2, 3). As a result, we focus on estimating the upper
bounds of the second-order Vandermonde determinant, whose entries are Taylor coeffi-
cients and logarithmic coefficients of functions and inverse functions in GG (α, δ).



N. H. A. A. Wahid, S. C. Soh / Eur. J. Pure Appl. Math, 18 (2) (2025), 6081 6 of 16

2. Preliminary results

This section gives a few sharp bounds on coefficient functionals for functions with a positive
real part, in the form of the following lemmas, to verify our main findings:

Lemma 1. ([29]) For a function p (z) ∈ P of the form (6), the sharp inequality |pn| ⩽ 2
holds for each n ⩾ 1. The equality holds for the function p (z) = 1+z

1−z .

Lemma 2. ([30]) Let p (z) ∈ P be a function of the form (6) and µ∗ ∈ C. Then

|pn − µ∗pkpn−k| ⩽ 2max {1, |2µ∗ − 1|} , 1 ⩽ k ⩽ n− 1.

If |2µ∗ − 1| ⩾ 1, then the inequality is sharp for the function p (z) = 1+z
1−z or its rotations.

If |2µ∗ − 1| < 1, then the inequality is sharp for the function p (z) = 1+zn

1−zn or its rotations.

Lemma 3. ([31]) Let p (z) ∈ P be a function of the form (6) and α∗, β∗, γ∗ ∈ ℜ. Then∣∣α∗p1
3 − β∗p1p2 + γ∗p3

∣∣ ⩽ 2 |α∗|+ 2 |β∗ − 2α∗|+ 2 |α∗ − β∗ + γ∗| .

Lemma 4. ([32]) Let p (z) ∈ P be a function of the form (6) and 0 < β < 1, 0 < µ < 1,
and
8β (1− β)

[
(µη − 2α)2 + (µ (β + µ)− η)2

]
+ µ (1− µ) (η − 2βµ)2 ≤ 4µ2β(1− µ)2 (1− β).

Then ∣∣∣∣αp14 + βp2
2 + 2µp1p3 −

3

2
ηp1

2p2 − p4

∣∣∣∣ ≤ 2.

3. Main results

This section presents the proof of our main findings, focusing primarily on the upper
bounds of Taylor coefficients, logarithmic coefficients, and Vandermonde determinant of
second-order of functions and inverse functions belonging to the class GG (α, δ).

3.1. Taylor coefficients

We now estimate the upper bounds of the Taylor coefficients of functions and inverse
functions in GG (α, δ).

Theorem 1. Let f (z) ∈ GG (α, δ) . Then

|an| ≤
ταδ
2n

, n = 2, 3, 4, 5,

where ταδ = cosα− δ.

Proof. Let a function f (z) ∈ GG (α, δ) given by (1). Then there exists a Schwarz
function υ (z) with υ (0) = 0 and |υ (z)| < 1 in E such that

eiαf ′ (z)− i sinα− δ

ταδ
= Ψ(υ (z)) , z ∈ E, (20)



N. H. A. A. Wahid, S. C. Soh / Eur. J. Pure Appl. Math, 18 (2) (2025), 6081 7 of 16

where ταδ = cosα− δ.
Define the function p (z) by

p (z) =
1 + υ (z)

1− υ (z)
, z ∈ E,

or equivalently

υ (z) = −1+p(z)
1+p(z)

= 1
2p1z +

1
2

(
p2 − 1

2p1
2
)
z2 + 1

2

(
1
4p1

3 − p1p2 + p3
)
z3

+1
2

(
−1

8p1
4 + 3

4p1
2p2 − p1p3 − 1

2p2
2 + p4

)
z4 + · · · .

(21)

Using (21) along with the expression Ψ (υ (z)) = υ(z)
ln(1+υ(z)) , we obtain

Ψ (υ (z)) = 1 + 1
4p1z +

1
48

(
12p2 − 7p1

2
)
z2 + 1

192

(
17p1

3 − 56p1p2 + 48p3
)
z3

+ 1
11520

(
−649p1

4 + 3060p1
2p2 − 3360p1p3 − 1680p2

2 + 2880p4
)
z4 + · · · .

(22)
Thus, by applying (22) to (20), we have

eiα
[(
1 + 2a2z + 3a3z

2 + 4a4z
3 + 5a5z

4 + · · ·
)
− 1

]
= ταδ

[
1
4p1z +

1
48

(
12p2 − 7p1

2
)
z2 + 1

192

(
17p1

3 − 56p1p2 + 48p3
)
z3

+ 1
11520

(
−649p1

4 + 3060p1
2p2 − 3360p1p3 − 1680p2

2 + 2880p4
)
z4 + · · ·

]
.
(23)

Comparing the coefficients of zn for n = 1, 2, 3, 4 on both sides of (23) gives

a2 =
ταδe

−iα

8 p1,

a3 =
ταδe

−iα

144

(
12p2 − 7p1

2
)
,

a4 =
ταδe

−iα

768

(
17p1

3 − 56p1p2 + 48p3
)
,

a5 =
ταδe

−iα

57600

(
−649p1

4 + 3060p1
2p2 − 3360p1p3 − 1680p2

2 + 2880p4
)
.

 (24)

Based on Lemmas 1-4, the equations in (24) can be expressed as follows:

|a2| =
∣∣∣∣ταδe−iα

8
p1

∣∣∣∣ , (25)

|a3| =
∣∣∣∣ταδe−iα

144

[
12

(
p2 −

7

12
p1

2

)]∣∣∣∣ , (26)

|a4| =
∣∣∣∣ταδe−iα

768

[
17p1

3 − 56p1p2 + 48p3
]∣∣∣∣ , (27)

|a5| =
∣∣∣∣ταδe−iα

20

[
649

2880
p1

4 +
1680

2880
p2

2 + 2

(
1680

2880

)
p1p3 −

3

2

(
2040

2880

)
p1

2p2 − p4

]∣∣∣∣ . (28)



N. H. A. A. Wahid, S. C. Soh / Eur. J. Pure Appl. Math, 18 (2) (2025), 6081 8 of 16

It is observed that

|p1| ≤ 2,∣∣p2 − 7
12p1

2
∣∣ ≤ 2max

{
1,
∣∣2 ( 7

12

)
− 1

∣∣} = 2,∣∣17p13 − 56p1p2 + 48p3
∣∣ ≤ 2 |17|+ 2 |56− 2 (17)|+ 2 |17− 56 + 48| = 96,∣∣ 649

2880p1
4 + 1680

2880p2
2 + 2

(
1680
2880

)
p1p3 − 3

2

(
2040
2880

)
p1

2p2 − p4
∣∣ ≤ 2.

The upper bounds for |a2|, |a3|, |a4|, and |a5| result from applying Lemmas 1-4, respec-
tively:

|a2| ≤ ταδ
8 (2) = ταδ

4 ,
|a3| ≤ ταδ

12 (2) = ταδ
6 ,

|a4| ≤ ταδ
768 (96) =

ταδ
8 ,

|a5| ≤ ταδ
20 (2) = ταδ

10 .

 (29)

Thus, we get the desired bound. This completes the proof of Theorem 1.

Theorem 2. Let f (z) ∈ GG (α, δ) . Then

|A2| ≤
ταδ
4

,

|A3| ≤
ταδ
6

,

|A4| ≤
ταδ
24

[∣∣5ταδe−iα + 7
∣∣+ 3

]
,

where ταδ = cosα− δ.

Proof. By substituting (24) into (3)-(5), we get

A2 = − ταδe
−iα

8 p1,

A3 = − ταδe
−iα

288

[
24p2 −

(
14 + 9ταδe

−iα
)
p1

2
]
,

A4 = − ταδe
−iα

16

[(
360ταδ

2e−2iα + 1120ταδe
−iα + 816

) p13

2304 + p3 −
(
5ταδe

−iα+7
6

)
p1p2

]
.


(30)

Taking the modulus on both sides of the equations in (30) and applying Lemmas 1-3, we
can express the equations in (30) as follows:

|A2| =
∣∣∣∣−ταδe

−iα

8
p1

∣∣∣∣ , (31)

|A3| =
∣∣∣∣−ταδe

−iα

288

[
24

(
p2 −

(
14 + 9ταδe

−iα

24

)
p1

2

)]∣∣∣∣ , (32)

|A4| =
∣∣∣∣ταδe−iα

16

{
p1

(
5ταδe

−iα + 7

6

)[
p2 −

(
45ταδ

2e−2iα + 140ταδe
−iα + 102

240ταδe−iα + 336

)
p1

2

]
− p3

}∣∣∣∣ .
(33)



N. H. A. A. Wahid, S. C. Soh / Eur. J. Pure Appl. Math, 18 (2) (2025), 6081 9 of 16

It is observed that

|p1| ≤ 2,
|p3| ≤ 2,∣∣∣p2 − (

14+9ταδe
−iα

24

)
p1

2
∣∣∣ ≤ 2max

{
1,
∣∣∣2(14+9ταδe

−iα

24

)
− 1

∣∣∣} = 2,∣∣∣p2 − (
45ταδ

2e−2iα+140ταδe
−iα+102

240ταδe−iα+336

)
p1

2
∣∣∣ ≤ 2max

{
1,
∣∣∣2(45ταδ

2e−2iα+140ταδe
−iα+102

240ταδe−iα+336

)
− 1

∣∣∣} = 2.

Thus, the upper bounds for |A2| and |A3| are obtained by applying Lemma 1 and Lemma
2, respectively. Meanwhile, the bound for |A4| follows from the combined application of
Lemmas 1 and 2, along with the triangle inequality. This completes the proof of Theorem
2.

3.2. Logarithmic coefficients

Next, we estimate the upper bounds of the logarithmic coefficients for functions and their
inverse functions in the class GG (α, δ).

Theorem 3. Let f (z) ∈ GG (α, δ) . Then

|γ1| ≤
ταδ
8

,

|γ2| ≤
ταδ
12

,

and
|γ3| ≤

ταδ
48

[
3 +

∣∣ταδe−iα + 7
∣∣] ,

where ταδ = cosα− δ.

Proof. Substituting (24) into (9)-(11) and simplifying, we obtain

γ1 =
ταδe

−iα

16
p1, (34)

γ2 =
ταδe

−iα

2304

[
96p2 −

(
57 + 9ταδe

−iα
)
p1

2
]
, (35)

γ3 =
ταδe

−iα

9216

[
288p3 −

(
48ταδe

−iα + 336
)
p1p2 +

(
3ταδ

2e−2iα + 28ταδe
−iα + 102

)
p1

3
]
.

(36)
Using Lemma 1 on (34) yields

|γ1| ≤
ταδ
16

(2) =
ταδ
8

.

Applying Lemma 2 to (35) implies that

|γ2| =
ταδ
24

∣∣∣∣p2 − (
57 + 9ταδe

−iα

96

)
p1

2

∣∣∣∣ ≤ ταδ
24

(2) =
ταδ
12

.



N. H. A. A. Wahid, S. C. Soh / Eur. J. Pure Appl. Math, 18 (2) (2025), 6081 10 of 16

Now, by rearranging the terms in (36), we can rewrite it as

|γ3| =
∣∣∣∣ταδe−iα

9216

{
288p3 −

(
48ταδe

−iα + 336
)
p1

[
p2 −

(
3ταδ

2e−2iα + 28ταδe
−iα + 102

48ταδe−iα + 336

)
p1

2

]}∣∣∣∣ .
Consequently, by applying Lemmas 1 and 2, along with the triangle inequality, and sim-
plifying, we obtain

|γ3| ≤
ταδ
48

[
3 +

∣∣ταδe−iα + 7
∣∣] .

This completes the proof of Theorem 3.

Theorem 4. Let f (z) ∈ GG (α, δ) . Then

|Γ1| ≤
ταδ
8

,

|Γ2| ≤
ταδ
12

,

and
|Γ3| ≤

ταδ
48

[
3 +

∣∣4ταδe−iα + 7
∣∣] ,

where ταδ = cosα− δ.

Proof. Substituting (24) into (12)-(14), we have

Γ1 = −ταδe
−iα

16
p1, (37)

Γ2 = − ταδe
−iα

2304

[
96p2 −

(
27ταδe

−iα + 56
)
p1

2
]

= − ταδe
−iα

24

[
p2 −

(
27ταδe

−iα+56
96

)
p1

2
]
,

(38)

Γ3 = − ταδe
−iα

4608

[(
51 + 56ταδe

−iα + 15ταδ
2e−2iα

)
p1

3 −
(
168 + 96ταδe

−iα
)
p1p2 + 144p3

]
= − ταδe

−iα

4608

{
−p1

(
168 + 96ταδe

−iα
) [

p2 −
(
51+56ταδe

−iα+15ταδ
2e−2iα

168+96ταδe−iα

)
p1

2
]
+ 144p3

}
.

(39)
The bounds for |Γ1| and |Γ2| follow from Lemma 1 and Lemma 2, respectively. Meanwhile,
the bound for |Γ3| results from applying both Lemmas 1 and 2, along with the triangle
inequality. This completes the proof of Theorem 4.

3.3. Vandermonde determinant of Taylor coefficients

In this subsection, we use the results of Theorems 1 and 2 to estimate the upper bounds
of the Vandermonde determinant of second-order, where the entries are Taylor coefficients
of functions and inverse functions in GG (α, δ), that is, |V2,2 (f)| and

∣∣V2,2

(
f−1

)∣∣.



N. H. A. A. Wahid, S. C. Soh / Eur. J. Pure Appl. Math, 18 (2) (2025), 6081 11 of 16

Theorem 5. Let f (z) ∈ GG (α, δ) . Then

|V2,2 (f)| ≤
5ταδ
12

,

where ταδ = cosα− δ.

Proof. In view of (15), we can establish

|V2,2 (f)| = |a3 − a2| ≤ |a3|+ |a2| . (40)

Using |a2| ≤ ταδ
4 and |a3| ≤ ταδ

6 from Theorem 1, we obtain

|V2,2 (f)| ≤
ταδ
6

+
ταδ
4

=
5ταδ
12

. (41)

Thus, we get the desired inequality, thereby completing the proof of Theorem 5.

Theorem 6. Let f (z) ∈ GG (α, δ) . Then∣∣V2,2

(
f−1

)∣∣ ≤ 5ταδ
12

,

where ταδ = cosα− δ.

Proof. From (17), we can establish∣∣V2,2

(
f−1

)∣∣ = |A3 −A2| ≤ |A3|+ |A2| . (42)

Making use of |A2| ≤ ταδ
4 and |A3| ≤ ταδ

6 from Theorem 2, we get∣∣V2,2

(
f−1

)∣∣ ≤ ταδ
6

+
ταδ
4

=
5ταδ
12

, (43)

thereby concluding the proof of Theorem 6.

3.4. Vandermonde determinant of logarithmic coefficients

In this subsection, we use the results of Theorems 3 and 4 to estimate the upper bounds
of the Vandermonde determinant of second-order, where the entries are logarithmic coeffi-
cients of functions and inverse functions inGG (α, δ), that is, |V2,1 (γf )| , |V2,2 (γf )| ,

∣∣V2,1

(
Γf−1

)∣∣ ,
and

∣∣V2,1

(
Γf−1

)∣∣ .
Theorem 7. Let f (z) ∈ GG (α, δ) . Then

|V2,1(γf )| ≤
5ταδ
24

and
|V2,2(γf )| ≤

ταδ
48

[
7 +

∣∣7 + ταδe
−iα

∣∣] ,
where ταδ = cosα− δ.



N. H. A. A. Wahid, S. C. Soh / Eur. J. Pure Appl. Math, 18 (2) (2025), 6081 12 of 16

Proof. From (16), we have

|V2,1(γf )| = |γ2 − γ1| ≤ |γ2|+ |γ1| (44)

and
|V2,2(γf )| = |γ3 − γ2| ≤ |γ3|+ |γ2| . (45)

Substituting |γ1| ≤ ταδ
8 and |γ2| ≤ ταδ

12 into (44), as well as |γ2| ≤ ταδ
12 and |γ3| ≤

ταδ
48

[
3 +

∣∣ταδe−iα + 7
∣∣] into (45), respectively, yields

|V2,1(γf )| ≤
ταδ
12

+
ταδ
8

=
5ταδ
24

and
|V2,2(γf )| ≤

ταδ
48

[
3 +

∣∣ταδe−iα + 7
∣∣]+ ταδ

12
=

ταδ
48

[
7 +

∣∣ταδe−iα + 7
∣∣] .

This concludes the proof of Theorem 7.

Theorem 8. Let f (z) ∈ GG (α, δ) . Then∣∣V2,1

(
Γf−1

)∣∣ ≤ 5ταδ
24

and ∣∣V2,2

(
Γf−1

)∣∣ ≤ ταδ
48

[
7 +

∣∣7 + 4ταδe
−iα

∣∣]
where tαδ = cosα− δ.

Proof. Using (18), we can establish∣∣V2,1

(
Γf−1

)∣∣ = |Γ2 − Γ1| ≤ |Γ2|+ |Γ1| (46)

and ∣∣V2,2

(
Γf−1

)∣∣ = |Γ3 − Γ2| ≤ |Γ3|+ |Γ2| . (47)

Using the result of Theorem 4, we obtain∣∣V2,1

(
Γf−1

)∣∣ ≤ ταδ
12

+
ταδ
8

=
5ταδ
24

and ∣∣V2,2

(
Γf−1

)∣∣ ≤ ταδ
48

[
3 +

∣∣4ταδe−iα + 7
∣∣]+ ταδ

12
=

ταδ
48

[
7 +

∣∣7 + 4ταδe
−iα

∣∣]
This concludes the proof of Theorem 8.



N. H. A. A. Wahid, S. C. Soh / Eur. J. Pure Appl. Math, 18 (2) (2025), 6081 13 of 16

4. Consequences and corollaries

This section explores several new implications of Theorems 1-8, as GG (α, δ) general-
izes the classes GG (α), GG (δ), and GG.

Selecting δ = 0 from Theorems 1-8, we get the following estimates bounds for the class
GG (α).

Corollary 1. Let f (z) = z+
∞∑
n=2

anz
n and f−1 (w) = w+

∞∑
n=2

Anw
n be in the class GG (α).

Then

(i) |an| ≤ cosα
2n , n = 2, 3, 4, 5

(ii) |A2| ≤ cosα
4 , |A3| ≤ cosα

6 , |A4| ≤ cosα
24

[∣∣5e−iα cosα+ 7
∣∣+ 3

]
(iii) |γ1| ≤ cosα

8 , |γ2| ≤ cosα
12 , |γ3| ≤ cosα

48

[
3 +

∣∣e−iα cosα+ 7
∣∣]

(iv) |Γ1| ≤ cosα
8 , |Γ2| ≤ cosα

12 , |Γ3| ≤ cosα
48

[
3 +

∣∣4e−iα cosα+ 7
∣∣]

(v) |V2,2 (f)| ≤ 5 cosα
12

(vi)
∣∣V2,2

(
f−1

)∣∣ ≤ 5 cosα
12

(vii) |V2,1(γf )| ≤ 5 cosα
24 , |V2,2(γf )| ≤ cosα

48

[
7 +

∣∣7 + e−iα cosα
∣∣]

(viii)
∣∣V2,1

(
Γf−1

)∣∣ ≤ 5 cosα
24 ,

∣∣V2,2

(
Γf−1

)∣∣ ≤ cosα
48

[
7 +

∣∣7 + 4e−iα cosα
∣∣]

Taking into account α = 0 in Theorems 1-8, we obtain the following estimates bounds for
the class GG (δ).

Corollary 2. Let f (z) = z+
∞∑
n=2

anz
n and f−1 (w) = w+

∞∑
n=2

Anw
n be in the class GG (δ).

Then

(i) |an| ≤ 1−δ
2n , n = 2, 3, 4, 5

(ii) |A2| ≤ 1−δ
4 , |A3| ≤ 1−δ

6 , |A4| ≤ 5(1−δ)(3−δ)
24

(iii) |γ1| ≤ 1−δ
8 , |γ2| ≤ 1−δ

12 , |γ3| ≤ (1−δ)(11−δ)
48

(iv) |Γ1| ≤ 1−δ
8 , |Γ2| ≤ 1−δ

12 , |Γ3| ≤ (1−δ)(7−2δ)
24

(v) |V2,2 (f)| ≤ 5(1−δ)
12

(vi)
∣∣V2,2

(
f−1

)∣∣ ≤ 5(1−δ)
12

(vii) |V2,1(γf )| ≤ 5(1−δ)
24 , |V2,2(γf )| ≤ (1−δ)(15−δ)

48



N. H. A. A. Wahid, S. C. Soh / Eur. J. Pure Appl. Math, 18 (2) (2025), 6081 14 of 16

(viii)
∣∣V2,1

(
Γf−1

)∣∣ ≤ 5(1−δ)
24 ,

∣∣V2,2

(
Γf−1

)∣∣ ≤ (1−δ)(9−2δ)
24

Putting α = 0 and δ = 0 in Theorems 1-8, we have the following results for the class GG.

Corollary 3. Let f (z) = z +
∞∑
n=2

anz
n and f−1 (w) = w +

∞∑
n=2

Anw
n be in the class GG.

Then

(i) |an| ≤ 1
2n , n = 2, 3, 4, 5

(ii) |A2| ≤ 1
4 , |A3| ≤ 1

6 , |A4| ≤ 5
8

(iii) |γ1| ≤ 1
8 , |γ2| ≤

1
12 , |γ3| ≤

11
48

(iv) |Γ1| ≤ 1
8 , |Γ2| ≤ 1

12 , |Γ3| ≤ 7
24

(v) |V2,2 (f)| ≤ 5
12

(vi)
∣∣V2,2

(
f−1

)∣∣ ≤ 5
12

(vii) |V2,1(γf )| ≤ 5
24 , |V2,2(γf )| ≤ 5

16

(viii)
∣∣V2,1

(
Γf−1

)∣∣ ≤ 5
24 ,

∣∣V2,2

(
Γf−1

)∣∣ ≤ 3
8

5. Conclusion

Recent research has sparked considerable interest in Vandermonde determinants. This
has inspired us to study the Vandermonde determinant of functions and inverse functions
belonging to the class GG (α, δ) of analytic functions, which is associated with general-
ized bounded turning and the generating functions of Gregory coefficients. Furthermore,
we have defined Vandermonde determinants whose entries are logarithmic coefficients of
functions and inverse functions in S. Thus, in this paper, we have obtained estimates
for Taylor coefficients, logarithmic coefficients, and the second-order Vandermonde deter-
minant whose entries are Taylor coefficients and logarithmic coefficients of functions and
inverse functions belonging to the class GG (α, δ). This extends not only the properties
of the class GG (α, δ) but also those of GG (α), GG (δ), and GG as shown in Corollaries
1-3. The lemmas in the preliminary section have proven invaluable in establishing upper
bounds for coefficient functionals in Theorems 1-8. The findings of this work could be
used to further investigate the upper bounds for the second-order Hankel, Toeplitz, and
higher-order Vandermonde determinants, specifically within bounded turning functions
connected to Gregory coefficients.



N. H. A. A. Wahid, S. C. Soh / Eur. J. Pure Appl. Math, 18 (2) (2025), 6081 15 of 16

Acknowledgements

The authors gratefully appreciate the referees’ insightful remarks and express their deepest
gratitude to Kementerian Pendidikan Tinggi (KPT) for giving the funding (Reference No.
FRGS-EC/1/2024/STG06/UITM/02/28) that supported this research. The authors also
thank the Universiti Teknologi MARA for their support in publishing this paper.

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