EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 4, 2024, 2738-2752
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Bounds for Certain Determinants of Logarithmic
Coefficients for the Class of Functions with Bounded

Turning

Nur Hazwani Aqilah Abdul Wahid1,∗, Ilya Qursiah Amirnuddin1, Nurul Izzah
Mohammad Azmi1

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

Abstract. This paper aims to estimate the logarithmic coefficients for the class of functions with
bounded turning. Hence, the upper bounds of the second-order for three types of determinants
(Hankel, Toeplitz, and Vandermonde) whose entries are logarithmic coefficients for this class of
functions are obtained. Some interesting consequences of these results are also highlighted, offering
new findings within the class of functions with bounded turning.

2020 Mathematics Subject Classifications: 30C45, 30C50

Key Words and Phrases: Univalent functions, bounded turning functions, logarithmic coeffi-
cients, Hankel determinant, Toeplitz determinant, Vandermonde determinant

1. Introduction

Let A denote the class of all functions f (z) of the form

f (z) = z +

∞∑
n=2

anz
n, (1)

which are analytic in the open unit disk E = {z ∈ C : |z| < 1}. We denote by S the
subclass of A consisting of univalent functions in E.

A typical problem in geometric function theory is to study a functional consisting of
combinations of the Taylor coefficients an, n ≥ 2 for the subclass of univalent functions
such as Hankel and Toeplitz determinants, but this is not limited to this. The unknown
upper bounds of these determinants for the class of univalent functions have attracted
researchers, making this an open and intriguing topic for further study. The Hankel

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5374

Email addresses: hazwaniaqilah@uitm.edu.my (N. H. A. A. Wahid),
ilyaqursiah17@gmail.com (I. Q. Amirnuddin), izzah.nurul63.ni@gmail.com (N. I. M. Azmi)

https://www.ejpam.com 2738 Copyright: © 2024 The Author(s). (CC BY-NC 4.0)



N.H.A.A. Wahid, I.Q. Amirnuddin, N.I.M. Azmi / Eur. J. Pure Appl. Math, 17 (4) (2024), 2738-2752 2739

determinant is an extremely useful tool in the study of singularities. This is particularly
important when analyzing power series with integral coefficients [6, 9]. Meanwhile, the
Toeplitz determinant has a variety of applications in both pure and applied mathematics,
statistics, and probability; for example, it is used in algebra, quantum mechanics, queuing
networks, signal processing, partial differential equations, and time series analysis [49].

Pommerenke [38, 39] and Ali et al. [7] defined the Hankel determinant Hq,n (f) and
Toeplitz determinant Tq,n (f) , n, q ≥ 1, whose elements are Taylor coefficients an, n ≥ 2
for functions f (z) ∈ A, respectively, as follows:

Hq,n (f) =

∣∣∣∣∣∣∣∣∣
an an+1 · · · an+q−1

an+1 an+2 · · · an+q
...

...
. . .

...
an+q−1 an+q · · · an+2q−2

∣∣∣∣∣∣∣∣∣ , a1 = 1 (2)

and

Tq,n (f) =

∣∣∣∣∣∣∣∣∣
an
an+1
...

an+q−1

an+1

an
...

an+q−2

· · ·
· · ·
...
· · ·

an+q−1

an+q−2
...
an

∣∣∣∣∣∣∣∣∣ . (3)

A recent work delves into the interesting world of Hankel and Toeplitz determinants in
the context of considering logarithmic coefficients as the entries. This idea generalizes
the traditional concept of both determinants (2) and (3) by replacing their entries with
the logarithmic coefficients of f (z) ∈ A. Kowalczyk and Lecko [22, 23], as well as Giri,
Kumar, and Mohamad et al. [15, 35], introduced the Hankel and Toeplitz determinants
of logarithmic coefficients γn, n ⩾ 1 for functions f (z) ∈ A, respectively, as follows:

Hq,n (γf ) =

∣∣∣∣∣∣∣∣∣
γn
γn+1
...

γn+q−1

γn+1

γn+2
...

γn+q

· · ·
· · ·
...
· · ·

γn+q−1

γn+q
...

γn+2q−2

∣∣∣∣∣∣∣∣∣ , (4)

and

Tq,n (γf ) =

∣∣∣∣∣∣∣∣∣
γn
γn+1
...

γn+q−1

γn+1

γn
...

γn+q−2

· · ·
· · ·
...
· · ·

γn+q−1

γn+q−2
...
γn

∣∣∣∣∣∣∣∣∣ . (5)

The logarithmic coefficients γn, n ⩾ 1 of f (z) ∈ A are defined by

log
f (z)

z
= 2

∞∑
n=1

γnz
n. (6)

Differentiating (6) and equating coefficients of zn provides the logarithmic coefficients in
terms of Taylor coefficients for f (z) ∈ A, which specifically, for n = 1, 2, 3, 4:

γ1 =
1

2
a2, (7)



N.H.A.A. Wahid, I.Q. Amirnuddin, N.I.M. Azmi / Eur. J. Pure Appl. Math, 17 (4) (2024), 2738-2752 2740

γ2 =
1

2

(
a3 −

1

2
a2

2

)
, (8)

γ3 =
1

2

(
a4 − a2a3 +

1

3
a2

3

)
, (9)

and

γ4 =
1

2

(
a5 − a2a4 + a2

2a3 −
1

2
a3

2 − 1

4
a2

4

)
. (10)

Milin [31–33] highlighted the importance of logarithmic coefficients for estimating the Tay-
lor coefficients of univalent functions. Subsequently, this led to de Branges [4] establishing
the Bieberbach conjecture. Logarithmic coefficients also play a significant role in conformal
mapping, which helped Kayumov [20] solve Brennan’s conjecture. Since then, numerous
studies on logarithmic coefficients have continued, with examples found in [3, 12, 14, 42].

On the other hand, Vijayalakshmi et al. [44] introduced the Vandermonde determinant
Vq,n (f), where n, q ≥ 1 and an, n ≥ 2 are the coefficients of the Taylor series 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. (11)

This determinant has many applications in a variety of domains. For example, it is used in
digital signal processing to compute the discrete Fourier transform (DFT) and the inverse
discrete Fourier transform (IDFT), and it also plays an important part in approximation
problems [44]. The Vandermonde determinant, often known as a discriminant, is also an
important tool in linear algebra; refer to [26] and the references therein for details.

Therefore, following the generalization of the Hankel and Toeplitz determinants in (2)
and (3), where their entries are replaced by logarithmic coefficients, and acknowledging the
significance of both the Vandermonde determinant and logarithmic coefficients, we now
define the Vandermonde determinant of logarithmic coefficients for functions f (z) ∈ A as
follows:

Vq,n (γf ) =

∣∣∣∣∣∣∣∣∣
1
1
...
1

γn
γn+1
...

γn+q−1

· · ·
· · ·
...
· · ·

γn
q−1

γn+1
q−1

...
γn+q−1

q−1

∣∣∣∣∣∣∣∣∣ . (12)

Consequently, if q = 2 and n = 2, then from (4), (5), and (12), respectively, yield the
second-order of three types of determinants, namely Hankel, Toeplitz, and Vandermonde,
as follows:

H2,2 (γf ) = γ2γ4 − γ3
2, (13)

T2,2 (γf ) = γ2
2 − γ3

2, (14)



N.H.A.A. Wahid, I.Q. Amirnuddin, N.I.M. Azmi / Eur. J. Pure Appl. Math, 17 (4) (2024), 2738-2752 2741

and
V2,2 (γf ) = γ3 − γ2. (15)

In [2, 5, 8, 24, 28, 43, 46, 48], sharp bounds for the Hankel determinant of logarithmic
coefficients were recently established for several subclasses of univalent functions. Works
by [1, 35] also investigated both Hankel and Toeplitz determinants with logarithmic coef-
ficients as entries, specifically for the subclass of starlike functions with respect to other
points. While there has been limited study on Toeplitz determinants in this context, it is
important to note that, in general, the upper bounds for both Hankel and Toeplitz deter-
minants remain unknown for classes of functions. In fact, to the best of our knowledge,
no one has yet studied the Vandermonde determinant of logarithmic coefficients.

Thus, motivated by the previous studies, in this paper, we aim to estimate the upper
bounds of the logarithmic coefficients |γn| , specifically for n = 1, 2, 3, 4. Hence, we focus
on estimating the upper bounds of the second-order Hankel, Toeplitz, and Vandermonde
determinants whose entries are logarithmic coefficients, as given in (7)-(10), for functions
belonging to the following class of bounded turning functions:

Definition 1. A function f (z) given by (1) is said to be in the class G (α, δ) if the
following condition is satisfied:

Re
(
eiαf ′ (z)

)
> δ, z ∈ E,

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

This class was introduced by Mohamad [34].

Remark 1. Selecting specific values for the parameters α and δ in the class G (α, δ) yields
the following classes:

(i) If we choose α = δ = 0, then G (α, δ) reduces to R which satisfies Re f ′ (z) > 0. The
functions from R are said to be of bounded turning.

(ii) If we choose α = 0, then G (α, δ) reduces to R (δ) which satisfies Re (f ′ (z)) > δ. The
class R (δ) is called the class of bounded turning functions of order δ.

(iii) If we choose δ = 0, then G (α, δ) reduces to R (α) which satisfies Re
(
eiαf ′ (z)

)
> 0.

Pioneering researchers like Goel and Mehrok [16], Macgregor [30], Noshiro [37], Silverman
and Silvia [45], and Warschawski [47] were among those who explored the classes R, R (δ),
and R (α), and further investigation into the class of bounded turning functions has also
been extensively studied by other researchers, see, for example, [13, 18, 19, 21, 25, 27, 36,
40], suggesting different directions than the current study.



N.H.A.A. Wahid, I.Q. Amirnuddin, N.I.M. Azmi / Eur. J. Pure Appl. Math, 17 (4) (2024), 2738-2752 2742

2. Preliminary results

Let P denote the class of positive real part functions p (z), also known as Carathéodory
functions, of the form

p (z) = 1 +

∞∑
n=1

pnz
n, (16)

which satisfy Re p (z) > 0 for z ∈ E.
To verify our main findings, we require a few sharp estimates in the form of lemmas valid
for functions with a positive real part, as follows:

Lemma 1. ([10]) For a function p (z) ∈ P of the form (16), the sharp inequality |pn| ⩽ 2
holds for each n ⩾ 1. Equality holds for the function p (z) = 1+z

1−z .

Lemma 2. ([11]) Let p (z) ∈ P be a function of the form (16) and µ ∈ C. Then

|pn − µpkpn−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.

3. Main results

This section presents the proof of our main findings, primarily focusing on the upper
bounds of logarithmic coefficients and three types of determinants (Hankel, Toeplitz, and
Vandermonde) for the class G (α, δ).

3.1. Logarithmic coefficients for G(α, δ)

We now estimate the upper bounds of the logarithmic coefficients for functions belonging
to G (α, δ).

Theorem 1. If f (z) = z +
∞∑
n=2

anz
n ∈ G (α, δ) , then

|γ1| ≤
tαδ
2
,

|γ2| ≤
tαδ
3
,

|γ3| ≤
tαδ
4

+
tαδ

3

6
,

and

|γ4| ≤
tαδ
5

+
tαδ

2

4
+

tαδ
4

8
,

where tαδ = cosα− δ.



N.H.A.A. Wahid, I.Q. Amirnuddin, N.I.M. Azmi / Eur. J. Pure Appl. Math, 17 (4) (2024), 2738-2752 2743

Proof. Let a function f (z) ∈ G (α, δ) given by (1). Then there exists a function
p (z) ∈ P such that [34]

eiαf ′ (z)− i sinα− δ

tαδ
= p(z),

where tαδ = cosα− δ, p (z) = 1 +
∞∑
n=1

pnz
n, and f ′ (z) = 1 + n

∞∑
n=2

anz
n−1.

Moreover, it can be observed that

an =
tαδe

−iαpn−1

n
, n ⩾ 2, (17)

and specifically, for n = 2, 3, 4, 5, we get

a2 =
tαδe

−iαp1
2 ,

a3 =
tαδe

−iαp2
3 ,

a4 =
tαδe

−iαp3
4 ,

a5 =
tαδe

−iαp4
5 .


(18)

Substituting (18) into (7)-(10) yields

γ1 =
tαδe

−iαp1
4

, (19)

γ2 =
tαδe

−iα

48

(
8p2 − 3tαδe

−iαp21
)
, (20)

γ3 =
tαδe

−iα

48

(
6p3 − 4tαδe

−iαp1p2 + tαδ
2e−2iαp31

)
, (21)

and

γ4 =
tαδe

−iαp4
10

− tαδ
2e−2iαp22
36

− tαδ
2e−2iαp1p3

16
+

tαδ
3e−3iαp21p2

24
− tαδ

4e−4iαp41
128

. (22)

Hence, we can express (19)-(22) as follows:

|γ1 | =
∣∣∣∣ tαδe−iαp1

4

∣∣∣∣ , (23)

|γ2| =
∣∣∣∣ tαδe−iα

48

(
8

(
p2 −

3tαδe
−iα

8
p21

))∣∣∣∣ , (24)

|γ3| =
∣∣∣∣ tαδe−iα

48

(
6

(
p3 −

2tαδe
−iα

3
p1p2

)
+ tαδ

2e−2iαp31

)∣∣∣∣ , (25)



N.H.A.A. Wahid, I.Q. Amirnuddin, N.I.M. Azmi / Eur. J. Pure Appl. Math, 17 (4) (2024), 2738-2752 2744

|γ4| =
∣∣∣∣tαδe−iα

(
− 1

10

(
p4 −

10tαδe
−iα

36
p22

)
+

tαδe
−iαp1
16

(
p3 −

2tαδe
−iα

3
p1p2

)
+

tαδ
3e−3iαp41
128

)∣∣∣∣ .
(26)

Applying Lemma 2, it can be observed that∣∣∣p2 − 3tαδe
−iα

8 p21

∣∣∣ ≤ 2max
{
1,
∣∣∣3tαδe

−iα−4
4

∣∣∣} = 2,

∣∣∣p3 − 2tαδe
−iα

3 p1p2

∣∣∣ ≤ 2max
{
1,
∣∣∣4tαδe

−iα−3
3

∣∣∣} = 2,

∣∣∣p4 − 10tαδe
−iα

36 p22

∣∣∣ ≤ 2max
{
1,
∣∣∣5tαδe

−iα−9
9

∣∣∣} = 2,

∣∣∣p3 − 2tαδe
−iα

3 p1p2

∣∣∣ ≤ 2max
{
1,
∣∣∣4tαδe

−iα−3
3

∣∣∣} = 2.


(27)

Thus, the upper bounds of |γ1| and |γ2| result from applying Lemma 1 and Lemma 2,
respectively. Meanwhile, the upper bounds of |γ3| and |γ4| result from using both Lemma
1 and Lemma 2, as well as triangle inequality. This completes the proof of Theorem 1.

3.2. Second-Order Hankel Determinant of Logarithmic Coefficients for
G(α, δ)

Now, in this subsection, using the results from Theorem 1, we estimate the upper bound
of the second-order Hankel determinant of logarithmic coefficients, specifically for n = 2
and q = 2, for functions belonging to G (α, δ).

Theorem 2. If f (z) = z +
∞∑
n=2

anz
n ∈ G (α, δ) , then

|H2,2 (γf )| ≤
tαδ

2

2160

(
36
∣∣5tαδe−iα + 4

∣∣+ 9tαδ
∣∣5tαδe−iα + 12

∣∣+ 30tαδ
3 + 80tαδ + 135

)
,

where tαδ = cosα− δ.

Proof. Using (8)–(10), we can establish

γ3
2 = tαδ

2e−2iα

2304

(
6p3 − 4tαδe

−iαp1p2 + tαδ
2e−2iαp31

)2
= tαδ

2e−2iα

2304

(
36p23 − 48tαδe

−iαp1p2p3 + 16tαδ
2e−2iαp21p

2
2

+12tαδ
2e−2iαp31p3 − 8tαδ

3e−3iαp41p2 + tαδ
4e−4iαp61

)
and

γ2γ4 =
tαδ

2e−2iα(8p2− 3tαδp
2
1e

−iα)
48

(
p4
10 − tαδp

2
2e

−iα

36 − tαδp1p3e
−iα

16 +
tαδ

2p21p2e
−2iα

24 − tαδ
3p41e

−3iα

128

)
= tαδ

2e−2iα

2304

(
192p2p4

5 − 32tαδe
−iαp32
3 − 24tαδe

−iαp1p2p3 + 20tαδ
2e−2iαp21p

2
2

− 9tαδ
3e−3iαp41p2 −

72tαδe
−iαp21p4
5 + 9tαδ

2e−2iαp31p3 +
9tαδ

4e−4iαp61
8

)
.



N.H.A.A. Wahid, I.Q. Amirnuddin, N.I.M. Azmi / Eur. J. Pure Appl. Math, 17 (4) (2024), 2738-2752 2745

Therefore, we have

H2,2 (γf ) =
tαδ

2e−2iα

2304

(
192p2p4

5 + 24tαδe
−iαp1p2p3 −

32tαδe
−iαp32
3 + 4tαδ

2e−2iαp21p2
2 − 36p23

−72tαδe
−iαp21p4
5 − 3tαδ

2e−2iαp31p3 − tαδ
3e−3iαp41p2 +

tαδ
4e−4iαp61

8

)
.

(28)
Taking the modulus of both sides of equation (28) and rearranging the terms according
to Lemma 2, we obtain

|H2,2 (γf )| =
tαδ

2

2304

∣∣∣∣∣ −192p2
5 (p4 − ν∗p1p3) +

32tαδe
−iαp22
3

(
p2 − ν∗∗p21

)
+ 36p23

+
72tαδe

−iαp21
5 (p4 − ν∗∗∗p1p3) + tαδ

3e−3iαp41
(
p2 − ν∗∗∗∗p21

) ∣∣∣∣∣ , (29)

where ν∗ = −5tαδe
−iα

8 , ν∗∗ = 3tαδe
−iα

8 , ν∗∗∗ = −5tαδe
−iα

24 , and ν∗∗∗∗ = tαδe
−iα

8 .
We see that

|p4 − ν∗p1p3| ≤
∣∣∣5tαδe

−iα+4
2

∣∣∣ ,∣∣p2 − ν∗∗p21
∣∣ ≤ 2,

|p4 − ν∗∗∗p1p3| ≤
∣∣∣5tαδe

−iα+12
6

∣∣∣ ,∣∣p2 − ν∗∗∗∗p21
∣∣ ≤ 2.


(30)

Thus, from (29), considering the triangle inequality, Lemma 1, and (30), we obtain the
desired inequality. This concludes the proof of Theorem 2.

3.3. Second-Order Toeplitz Determinant of Logarithmic Coefficients for
G(α, δ)

In this subsection, using the results from Theorem 1, we determine the upper bound of
the second-order Toeplitz determinant of logarithmic coefficients, specifically for n = 2
and q = 2, for functions belonging to G (α, δ).

Theorem 3. If f (z) = z +
∞∑
n=2

anz
n ∈ G (α, δ) , then

|T2,2 (γf ) | ≤
tαδ

2

144

(
16 + 37tαδ

2 + 4 tαδ
4 + 3

∣∣8tαδe−iα − 3
∣∣) ,

where tαδ = cosα− δ.

Proof. In light of (8) and (9) give

γ2
2 = 1

2304

(
8tαδe

−iαp2 − 3tαδ
2e−2iαp21

)2
= tαδ

2e−2iα

2304

(
64p22 − 48tαδe

−iαp21p2 + 9tαδ
2e−2iαp41

)



N.H.A.A. Wahid, I.Q. Amirnuddin, N.I.M. Azmi / Eur. J. Pure Appl. Math, 17 (4) (2024), 2738-2752 2746

and
γ3

2 = 1
2304

(
6tαδp3e

−iα − 4tαδ
2p1p2e

−2iα + tαδ
3p31e

−3iα
)2

= tαδ
2e−2iα

2304

(
36p23 − 48tαδp1p2p3e

−iα + 12tαδ
2p31p3e

−2iα

+16tαδ
2p21p

2
2e

−2iα − 8tαδ
3p41p2e

−3iα + tαδ
4p61e

−4iα

)
.

Therefore, we obtain

T2,2 (γf ) =
tαδ

2e−2iα

2304

(
64p22 − 48tαδe

−iαp21p2 − 36p23 + 48tαδe
−iαp1p2p3 + 9tαδ

2e−2iαp41
− 12tαδ

2e−2iαp31p3 + 8tαδ
3e−3iαp41p2 − 16tαδ

2e−2iαp21p
2
2 − tαδ

4e−4iαp61

)
,

(31)
and we can express (31) as follows:

|T2,2 (γf ) | =
∣∣∣∣ tαδ2e−2iα

2304

(
−64p2

(
p2 − κ∗p21

)
+ 12tαδ

2e−2iαp31 ( p3 − κ∗∗p1p2)− 9tαδ
2e−2iαp41

+36p3 (p3 − κ∗∗∗p1p2) + 16tαδ
2e−2iαp21p

2
2 + tαδ

4e−4iαp61

)∣∣∣∣ ,
(32)

where κ∗ = 48tαδe
−iα

64 , κ∗∗ = 8tαδe
−iα

12 , and κ∗∗∗ = 48tαδe
−iα

36 .
According to Lemma 2, we can conclude that∣∣p2 − κ∗p21

∣∣ ≤ 2max
{
1,
∣∣∣3tαδe

−iα−2
2

∣∣∣} = 2,

|p3 − κ∗∗p1p2| ≤ 2max
{
1,
∣∣∣4tαδe

−iα−3
3

∣∣∣} = 2,

|p3 − κ∗∗∗p1p2| ≤ 2max
{
1,
∣∣∣8tαδe

−iα−3
3

∣∣∣} = 2
∣∣∣8tαδe

−iα−3
3

∣∣∣ .


(33)

Using Lemma 1, (33), and the triangle inequality, we obtain the desired bound from (32).
This concludes the proof of Theorem 3.

3.4. Second-Order Vandermonde Determinant of Logarithmic Coefficients
for G(α, δ)

In this subsection, we obtain the upper bound of the second-order Vandermonde determi-
nant of logarithmic coefficients, specifically for n = 2 and q = 2, for functions belonging
to G (α, δ).

Theorem 4. If f (z) = z +
∞∑
n=2

anz
n ∈ G (α, δ) , then

|V2,2 (γf )| ≤
tαδ
(
7 + 2tαδ

2
)

12
,

where tαδ = cosα− δ.

Proof. Through (8) and (9) yield

V2,2 (γf ) =
tαδe

−iα

48

(
6p3 − 4tαδe

−iαp1p2 + tαδ
2e−2iαp31 − 8p2 + 3tαδe

−iαp21
)
. (34)



N.H.A.A. Wahid, I.Q. Amirnuddin, N.I.M. Azmi / Eur. J. Pure Appl. Math, 17 (4) (2024), 2738-2752 2747

By rearranging the terms in (34) according to Lemma 2, we obtain

|V2,2 (γf )| =
∣∣∣∣ tαδe−iα

48

(
6 (p3 − η∗p1p2)− 8

(
p2 − η∗∗p21

)
+ tαδ

2e−2iαp31
)∣∣∣∣ , (35)

where η∗ = 4tαδe
−iα

6 and η∗∗ = 3tαδe
−iα

8 .
Furthermore, we discover that

|p3 − η∗p1p2| ≤ 2max
{
1,
∣∣∣4tαδe

−iα−3
3

∣∣∣} = 2,

∣∣p2 − η∗∗p21
∣∣ ≤ 2max

{
1,
∣∣∣3tαδe

−iα−4
4

∣∣∣} = 2.

 (36)

By implementing Lemma 1 and (36) into (35), as well as applying the triangle inequality,
we achieve the desired bound. This completes the proof of Theorem 4.

4. Consequences and corollaries

Since G (α, δ) generalizes R, R (δ), and R (α), several new consequences of Theorems
1-4 are highlighted out for specific choices of α and δ as follows:

Substituting α = 0 and δ = 0 in Theorems 1-4, we get the estimates bounds for the class
R.

Corollary 1. For any function f (z) given by (1) for the class G (0, 0) ≡ R, then

(i) |γ1| ≤ 1
2 , |γ2| ≤

1
3 , |γ3| ≤

5
12 , |γ4| ≤

23
40

(ii) |H2,2 (γf )| ≤ 301
432

(iii) |T2,2 (γf ) | ≤ 1
2

(iv) |V2,2 (γf )| ≤ 3
4

If we consider α = 0 in Theorems 1-4, we obtain the estimates bounds for the class R (δ).

Corollary 2. For any function f (z) given by (1) for the class G (0, δ) ≡ R (δ) , then

(i) |γ1| ≤ 1−δ
2 , |γ2| ≤ 1−δ

3 , |γ3| ≤ 1−δ
4 + (1−δ)3

6 , |γ4| ≤ 1−δ
5 + (1−δ)2

4 + (1−δ)4

8

(ii) |H2,2 (γf )| ≤ (1−δ)2

2160

(
36 |5 (1− δ) + 4|+ 9 (1− δ) |5 (1− δ) + 12|
+30(1− δ)3 + 80 (1− δ) + 135

)
(iii) |T2,2 (γf ) | ≤ (1−δ)2

144

(
16 + 37(1− δ)2 + 4 (1− δ)4 + 3 |8 (1− δ)− 3|

)
(iv) |V2,2 (γf )| ≤

(1−δ)(7+2(1−δ)2)
12



REFERENCES 2748

Putting δ = 0 in Theorems 1-4, we have the following results for the class R (α).

Corollary 3. For any function f (z) given by (1) for the class G (α, 0) ≡ R (α) , then

(i) |γ1| ≤ cosα
2 , |γ2| ≤ cosα

3 , |γ3| ≤ cosα
4 + cos3α

6 , |γ4| ≤ cosα
5 + cos2α

4 + cos4α
8

(ii) |H2,2 (γf )| ≤ cos2α
2160

(
36
∣∣5e−iα cosα+ 4

∣∣+ 9 cosα
∣∣5e−iα cosα+ 12

∣∣
+30cos3α+ 80 cosα+ 135

)
(iii) |T2,2 (γf ) | ≤ cos2α

144

(
16 + 37cos2α+ 4 cos4α+ 3

∣∣8e−iα cosα− 3
∣∣)

(iv) |V2,2 (γf )| ≤
cosα(7+2cos2α)

12

5. Conclusion

In this paper, we have obtained the estimates on logarithmic coefficients |γn| , n =
1, 2, 3, 4, thereby extending the properties of G(α, δ), R, R (δ), and R (α). Recent research
has sparked considerable interest in logarithmic coefficients and the Hankel, Toeplitz, and
Vandermonde determinants. This has inspired us to define the Vandermonde determinant
of logarithmic coefficients for functions f (z) ∈ A. As a result of determining the logarith-
mic coefficients, we have established the upper bounds for three types of determinants:
|H2,2 (γf )| , |T2,2 (γf ) | , and |V2,2 (γf )|, where the logarithmic coefficients are considered as
the entries, for functions from G(α, δ), as well as R, R (δ), and R (α). The lemmas from
the preliminary section have proven invaluable in establishing upper bounds for three
types of determinants of logarithmic coefficients. The findings in this paper could inspire
further research into determining upper bounds for Hankel, Toeplitz, and Vandermonde
determinants with logarithmic coefficients as entries, particularly within other subclasses
of univalent functions, while considering the inverse functions for G(α, δ). Additionally,
for new insights, one might refer to [41] for other coefficient-related problems in logarith-
mic functions such as Fekete Szegö inequality; however, consider subclasses of bi-univalent
functions, which could expand upon, for example, the works of [17, 29].

Acknowledgements

The authors deeply appreciate the referees’ thoughtful comments and extend their
heartfelt thanks to Universiti Teknologi MARA for supporting the publication of this
paper. This work is dedicated to Prof. Dr. Daud Mohamad upon his retirement after his
distinguished service at Universiti Teknologi MARA, in recognition of his pioneering role
in introducing the class of functions with bounded turning and tilted factors.

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