EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 2, 2023, 1167-1179
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Coefficient Problems for Star-like Functions with
Respect to Symmetric Conjugate Points Connected to

the Sine Function

Daud Mohamad1, Nur Hazwani Aqilah Abdul Wahid1,∗, Nurul Natasya Hasni1

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

Abstract. In this paper, we introduce the subclass of star-like functions with respect to symmetric
conjugate points associated with the sine function. Some coefficient functionals for this class are
considered. Bounds of Taylor coefficients, logarithmic coefficients, and the Hankel and Toeplitz
determinants whose entries are logarithmic coefficients are provided.

2020 Mathematics Subject Classifications: 30C45, 30C50

Key Words and Phrases: Star-like functions with respect to symmetric conjugate points, sine
function, coefficient estimates, logarithmic coefficients, Hankel and Toeplitz determinants of loga-
rithmic coefficients, subordination

1. Introduction

Let A and S denote the classes of analytic and univalent functions, respectively. These
classes are defined in the form of

A =
{
f ∈ K (E) : f (0) = f ′ (0) − 1 = 0, z ∈ E

}
and

S = {f ∈ A: f is univalent in E} ,

where K (E) is the set of analytic functions in the open unit disk E = {z ∈ C : |z| < 1} .
If f ∈ A, then it can be expressed in the series representation of the form

f (z) = z +

∞∑
n=2

anz
n, z ∈ E. (1)

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v16i2.4779

Email addresses: daud201@uitm.edu.my (D. Mohamad),
hazwaniaqilah@uitm.edu.my (N. H. A. A. Wahid), nurulnatasya530@gmail.com (N. N. Hasni)

https://www.ejpam.com 1167 © 2023 EJPAM All rights reserved.



D. Mohamad, N. H. A. A. Wahid, N. N. Hasni / Eur. J. Pure Appl. Math, 16 (2) (2023), 1167-1179 1168

Let H denotes the class of Schwarz functions υ which are analytic in E given by

υ (z) =

∞∑
k=1

bkz
k, z ∈ E

and satisfying υ (0) = 0 and |υ (z)| < 1. Given two functions f, g ∈ A. We let ≺ to denote
the subordination. The analytic function f is subordinate to another analytic function
g if there exists a Schwarz function υ ∈ H such that f (z) = g (υ (z)) for all z ∈ E.
Furthermore, if g is univalent in E, then we have the following equivalence

f ≺ g ⇔ f (0) = g (0)

and
f (E) = g (E) .

Let P (A,B) denotes the class of analytic functions defined in the form of

P (A,B) =

{
p ∈ A: p (z) ≺ 1 + Az

1 + Bz
, − 1 ⩽ B < A ⩽ 1, z ∈ E

}
,

where p has a series form given by

p (z) = 1 +
∞∑
n=1

pnz
n, z ∈ E.

The class P (A,B) is known as the class of Janowski and was introduced by Janowski [13].
If P (1,−1) , then it reduces to the class P, the well-known class of functions with positive
real part consists of functions p that satisfy Re p (z) > 0 and p (0) = 1.
If p ∈ P, then a Schwarz function υ ∈ H exists with υ (0) = 0 and |υ (z)| < 1 such that

p (z) =
1 + υ (z)

1 − υ (z)
, z ∈ E.

We now introduce the subclass of star-like functions with respect to symmetric conju-
gate points connected to the sine function as follows:

Definition 1. Let S∗
SC (sin z) be the class of functions defined by

zf ′ (z)

h (z)
≺ φ (z) , z ∈ E, (2)

where h (z) = f(z)−f(−z)
2 and φ (z) = 1 + sin z.

It is observed that the classes S∗
SC and S∗

SC (A,B) consisting of star-like functions with
respect to symmetric conjugate points defined by El-Ashwah and Thomas [11] and Ping
and Janteng [26], respectively, are obtained if the right-hand side of (2) is changed to
φ (z) = 1+z

1−z and φ (z) = 1+Az
1+Bz , i.e.,

S∗
SC =

{
f ∈ A : Re

(
zf ′ (z)

h (z)

)
> 0, z ∈ E

}



D. Mohamad, N. H. A. A. Wahid, N. N. Hasni / Eur. J. Pure Appl. Math, 16 (2) (2023), 1167-1179 1169

and

S∗
SC (A,B) =

{
f ∈ A :

zf ′ (z)

h (z)
≺ 1 + Az

1 + Bz
, − 1 ⩽ B < A ⩽ 1, z ∈ E

}
,

where h (z) = f(z)−f(−z)
2 . Besides that, some subclasses of star-like functions with respect

to symmetric conjugate points are also studied from a different perspective by Halim [1]
and Mohamad et al. [24]. These subclasses are defined as follows:

S∗
SC (δ) =

{
Re

(
zf ′ (z)

h (z)

)
> δ, 0 ⩽ δ< 1, z ∈ E

}
(3)

and

S∗
SC (α, δ,A,B) =

{
f ∈ A :

(
eiα

zf ′ (z)

h (z)
− δ − i sinα

)
1

ταδ
≺ 1 + Az

1 + Bz

}
, (4)

where h (z) = f(z)−f(−z)
2 , ταδ = cosα− δ, 0 ⩽ δ < 1, and |α| < π

2 .
The problem of computing the bounds of the Hankel determinant has been studied in

almost every subclass of A and has consistently piqued the interest of geometric functions
theory researchers. The Hankel determinant of a function f ∈ A whose elements are
Taylor coefficients of f ∈ A is defined as [27, 28]

Hq,n (f) =

∣∣∣∣∣∣∣∣∣
an an+1 · · · an+q−1

an+1 an+2 · · · an+q
...

...
...

...
an+q−1 an+q · · · an+2q−2

∣∣∣∣∣∣∣∣∣ ,
where q, n ∈ N and a1 = 1. It plays an important role in the study of singularities and
power series with integral coefficients [7, 8]. On the other hand, it is known that the
Toeplitz matrices are closely related to the Hankel matrices and one of the well-studied
classes of structured matrices. Unlike the Hankel matrices, which have constant entries
along the reverse diagonals, Toeplitz matrices have constant entries along the diagonals.
Toeplitz matrices have a wide range of uses in both pure and applied mathematics, which
has sparked some of the most important advances in research on the Toeplitz determinants,
kernel, operators, and q-deformed Toeplitz matrices (see Ye and Lim [35]). Thomas and
Halim [32] introduced the symmetric Toeplitz determinant of a function f ∈ A whose
elements are Taylor coefficients of f ∈ A and it is defined as

Tq,n (f) =

∣∣∣∣∣∣∣∣∣
an
an+1

...
an+q−1

an+1

an
...

an+q−2

· · ·
· · ·
...
· · ·

an+q−1

an+q−2
...
an

∣∣∣∣∣∣∣∣∣ ,
where q, n ∈ N and a1 = 1. It is worth noting that the exact bounds of the Hankel and
Toeplitz determinants for some subclasses of S are still not sharp and are yet undiscovered.
For recent work especially related to the class consisting of star-like functions with respect



D. Mohamad, N. H. A. A. Wahid, N. N. Hasni / Eur. J. Pure Appl. Math, 16 (2) (2023), 1167-1179 1170

to other points, i.e., symmetric points, conjugate points, and symmetric conjugate points,
see ([2, 16, 22, 24, 30, 31, 34] and reference therein).

Recently, the Hankel and Toeplitz determinants of a function f ∈ A whose elements
are logarithmic coefficients of f ∈ A have been introduced by Kowalczyk and Lecko [17, 18]
and Giri and Kumar [12], respectively, as follows:

Hq,n (Ff/2) =

∣∣∣∣∣∣∣∣∣
γn
γn+1

...
γn+q−1

γn+1

γn+2
...

γn+q

· · ·
· · ·
...
· · ·

γn+q−1

γn+q
...

γn+2q−2

∣∣∣∣∣∣∣∣∣ ,
and

Tq,n (γf ) =

∣∣∣∣∣∣∣∣∣
γn
γn+1

...
γn+q−1

γn+1

γn
...

γn+q−2

· · ·
· · ·
...
· · ·

γn+q−1

γn+q−2
...
γn

∣∣∣∣∣∣∣∣∣ ,
where γn, n ⩾ 1, the logarithmic coefficients, are defined in the series form

log
f (z)

z
= 2

∞∑
n=1

γnz
n.

In particular, for a function given in (1), the logarithmic coefficients γn, n = 1, 2, 3, 4 are
given as follows:

γ1 =
1

2
a2, (5)

γ2 =
1

2

(
a3 −

1

2
a2

2

)
, (6)

γ3 =
1

2

(
a4 − a2a3 +

1

3
a2

3

)
, (7)

and

γ4 =
1

2

(
a5 − a2a4 + a2

2a3 −
1

2
a3

2 − 1

4
a2

4

)
. (8)

The logarithmic coefficients have great importance, for instance, these coefficients
helped Kayumov [14] to solve Brennan’s conjecture for conformal mapping and estimation
of the logarithmic coefficients can be transferred to the Taylor coefficients of univalent
functions via the Lebedev–Milin inequalities (see [9, 19–21] for details). Due to the great
importance of logarithmic coefficients and the Hankel and Toeplitz determinants, some
recent works on this problem that relate to the theory of univalent functions have been
studied in [3–5, 12, 15, 16, 18, 23, 25, 29, 33, 36] but only a few papers have been published
for the class of star-like functions with respect to other points. Motivated by these works,
in this paper, we obtain the upper bounds of the Taylor coefficients |an| , n = 2, 3, 4, 5,



D. Mohamad, N. H. A. A. Wahid, N. N. Hasni / Eur. J. Pure Appl. Math, 16 (2) (2023), 1167-1179 1171

logarithmic coefficients |γn| , n = 1, 2, 3, 4, and hence some cases of the Hankel deter-
minant as well as Toeplitz determinant, whose both entries are logarithmic coefficients,
i.e., |H2,1 (Ff/2)|, |H2,2 (Ff/2)|, |T2,1 (γn)|, and |T2,2 (γn)| for the functions in the class
S∗
SC (sin z) as defined in Definition 1.

2. Preliminary results

In this section, we give some lemmas to prove our main results.

Lemma 1. ([9]) For a function p ∈ P of the form p (z) = 1 +
∞∑
n=1

pnz
n, z ∈ E, the sharp

inequality |pn| ⩽ 2 holds for each n ⩾ 1. Equality holds for the function p (z) = 1+z
1−z .

Lemma 2. ([10]) Let p ∈ P of the form p (z) = 1 +
∞∑
n=1

pnz
n 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.

Lemma 3. ([6]) Let p ∈ P of the form p (z) = 1 +
∞∑
n=1

pnz
n, z ∈ E and α, β, γ ∈ ℜ. Then

∣∣αp13 − βp1p2 + γp3
∣∣ ⩽ 2 |α| + 2 |β − 2α| + 2 |α− β + γ| .

3. Main results

This section is devoted to the proof of our main results. We will now determine the
upper bounds of the Taylor coefficients, logarithmic coefficients, and Hankel and Toeplitz
determinants of logarithmic coefficients, respectively, as follows:

3.1. Taylor coefficients

Theorem 1. If f is of the form (1) belongs to S∗
SC (sin z) , then

|a2| ⩽
1

2
,

|a3| ⩽
1

2
,

|a4| ⩽
1

4
,

and

|a5| ⩽
1

2
.



D. Mohamad, N. H. A. A. Wahid, N. N. Hasni / Eur. J. Pure Appl. Math, 16 (2) (2023), 1167-1179 1172

Proof. Since f ∈ S∗
SC (sin z) , from definition of subordination, there exists a Schwarz

function υ with υ (0) = 0 and |υ (z)| < 1, and from (2) we have

zf ′ (z)

h (z)
= 1 + sin υ (z) , z ∈ E, (9)

where h (z) = f(z)−f(−z)
2 .

Assuming that

p (z) =
1 + υ (z)

1 − υ (z)
= 1 +

∞∑
n=1

knz
n ∈ P.

This leads to

υ (z) =
p (z) − 1

p(z) + 1
.

Hence, from the right-hand side of (9), we obtain

1 + sin υ (z) = 1 +
1

2
k1z +

(
k2
2

− k1
2

4

)
z2 +

(
5k1

3

48
− k1k2

2
+

k3
2

)
z3

+

(
k4
2

+
5k1

2k2
16

− k2
2

4
− k1k3

2
− k1

4

32

)
z4 + · · · .

On the other hand, since f of the form (1), this gives

zf ′ (z) = z + 2a2z
2 + 3a3z

3 + 4a4z
4 + 5a5z

5 + · · ·

and
h (z) = z + a3z

3 + a5z
5 + · · · .

Further, we have from (9) that

z + 2a2z
2 + 3a3z

3 + 4a4z
4 + 5a5z

5 + · · ·

=
(
z + a3z

3 + a5z
5 + · · ·

) [
1 +

1

2
k1z +

(
k2
2

− k1
2

4

)
z2 +

(
5k1

3

48
− k1k2

2
+

k3
2

)
z3

+

(
k4
2

+
5k1

2k2
16

− k2
2

4
− k1k3

2
− k1

4

32

)
z4 + · · ·

]
.

(10)

Expanding the series and comparing the coefficients of zn, n = 1, 2, 3, 4, 5 on both sides
of (10) yields

a2 =
k1
4
, (11)

a3 =
1

8

(
2k2 − k1

2
)
, (12)

a4 =
1

96

(
k1

3 − 9k1k2 + 12k3
)
, (13)



D. Mohamad, N. H. A. A. Wahid, N. N. Hasni / Eur. J. Pure Appl. Math, 16 (2) (2023), 1167-1179 1173

and

a5 =
1

64

(
8k4 + 3k1

2k2 − 2k2
2 − 8k1k3

)
. (14)

Using triangle inequality and Lemma 1 in (11), we get

|a2| ⩽
1

2
.

Now, applying Lemma 2 in (12) and Lemma 3 in (13), respectively, implies that

|a3| =
1

8

∣∣2k2 − k1
2
∣∣ ⩽ 1

4

[
2max

{
1,

∣∣∣∣2(1

2

)
− 1

∣∣∣∣}] =
1

2

and

|a4| =
1

96

∣∣k13 − 9k1k2 + 12k3
∣∣ ⩽ 1

96
[2 |1| + 2 |9 − 2 (1)| + 2 |1 − 9 + 12|] =

1

4
.

Rearranging the terms in (14), we can rewrite it as

|a5| =
1

64

∣∣8 (k4 − ν1k1k3) − 2k2
(
k2 − ν2k1

2
)∣∣ ,

where ν1 = 1 and ν2 = 3
2 .

Consequently, by applying Lemma 1 and Lemma 2 as well as the triangle inequality, we
obtain

|a5| ⩽
1

2
.

This completes the proof of Theorem 1.

3.2. Logarithmic coefficients

Theorem 2. If f is of the form (1) belongs to S∗
SC (sin z) , then

|γ1| ⩽
1

4
,

|γ2| ⩽
1

4
,

|γ3| ⩽
1

8
,

and

|γ4| ⩽
7

16
.

Proof. Putting (11)-(14) in (5)-(8), we obtain

γ1 =
k1
8
, (15)



D. Mohamad, N. H. A. A. Wahid, N. N. Hasni / Eur. J. Pure Appl. Math, 16 (2) (2023), 1167-1179 1174

γ2 =
1

2

[
1

8

(
2k2 − k1

2
)
− 1

2

(
k1
4

)2
]

=
1

8

(
k2 −

5

8
k1

2

)
,

(16)

γ3 =
1

2

[
1

96

(
k1

3 − 9k1k2 + 12k3
)
−
(
k1
4

)(
1

8

(
2k2 − k1

2
))

+
1

3

(
k1
4

)3
]

=
1

128

(
3k1

3 − 10k1k2 + 8k3
)
,

(17)

and

γ4 =
1

2

[
1

64

(
8k4 + 3k1

2k2 − 2k2
2 − 8k1k3

)
− k1

4

(
1

96

(
k1

3 − 9k1k2 + 12k3
))

+

(
k1
4

)2(1

8

(
2k2 − k1

2
))

− 1

2

(
1

8

(
2k2 − k1

2
))2

− 1

4

(
k1
4

)4
]

=
1

6144

(
384k4 + 360k1

2k2 − 192k2
2 − 480k1k3 − 59k1

4
)
.

(18)

The bounds of |γ1| , |γ2| , and |γ3| follow from Lemma 1, Lemma 2, and Lemma 3, respec-
tively.
On the other hand, rearranging the terms in (18), we get

γ4 =
1

6144

(
384

(
k4 − µk2

2
)
− k1

(
αk1

3 − βk1k2 + γk3
))

,

where µ = 1
2 , α = 59, β = 360, and γ = 480.

Hence, implementing Lemma 2 and Lemma 3, we get the desired bound of |γ4| . This
completes the proof of Theorem 2.

3.3. Hankel determinant of logarithmic coefficients

Theorem 3. If f ∈ S∗
SC (sin z) and has the series representation (1), then

|H2,1 (Ff/2)| ⩽ 87

1024
.

Proof. In view of (15)-(17), we have

H2,1 (Ff/2) = γ1γ3 − γ2
2

=
k1
8

(
1

128

(
3k1

3 − 10k1k2 + 8k3
))

−
(

1

8

(
k2 −

5

8
k1

2

))2

=
1

64

(
3

16
k1

4 − 10

16
k1

2k2 +
1

2
k1k3 − k2

2 +
5

4
k1

2k2 −
25

64
k1

4

)
=

1

4096

(
−13k1

4 + 40k1
2k2 + 32k1k3 − k2

2
)
.

(19)



D. Mohamad, N. H. A. A. Wahid, N. N. Hasni / Eur. J. Pure Appl. Math, 16 (2) (2023), 1167-1179 1175

Rearranging the terms in (19), it becomes

H2,1 (Ff/2) =
1

4096

(
−k1

(
χk1

3 − λk1k2 + η32k3
)
− k2

2
)
,

where χ = 13, λ = 40, and η = −32.
By applying the triangle inequality as well as Lemma 1 and Lemma 3, we get the desired
inequality.

Theorem 4. If f ∈ S∗
SC (sin z) and has the series representation (1), then

|H2,2 (Ff/2)| ⩽ 33

256
.

Proof. In view of (16)-(18), we can establish

H2,2 (Ff/2) = γ2γ4 − γ3
2

=
1

8

(
k2 −

5

8
k1

2

)(
1

6144

(
384k4 + 360k1

2k2 − 192k2
2 − 480k1k3 − 59k1

4
))

−
(

1

128

(
3k1

3 − 10k1k2 + 8k3
))2

=
1

393216

(
3072k2k4 + 1440k1

2k2
2 − 1536k2

3 − 832k1
4k2 − 1920k1

2k4

+1248k1
3k3 + 79k1

6 − 1536k3
2
)
.

(20)
Further, rearranging the terms in (20), we can rewrite it in the following expression:

H2,2 (Ff/2) =
1

393216

[(
3072k4

(
k2 −

5

8
k1

2

)
− 1536k2

2

(
k2 −

15

16
k1

2

)
+k1

3
(
79k1

3 − 832k1k2 + 1248k3
)
− 1536k3

2
)]

.

Hence, making use of Lemma 1, Lemma 2, and Lemma 3 yields the desired bound.

3.4. Toeplitz determinant of logarithmic coefficients

Theorem 5. If f ∈ S∗
SC (sin z) , then∣∣γ12 − γ2

2
∣∣ ⩽ 65

256
.

Proof. Using (15) and (16), we obtain

γ1
2 − γ2

2 =
k1

2

64
− 1

64

(
k2 −

5

8
k1

2

)2

=
1

64

(
5

4
k1

2

(
k2 −

25

80
k1

2

)
+ k1

2 − k2
2

)
.

(21)

Applying the triangle inequality and Lemma 1 and Lemma 2, we get the desired inequality.



REFERENCES 1176

Theorem 6. If f ∈ S∗
SC (sin z) , then

|T2,2 (γn)| ⩽ 11

32
.

Proof. Making use of (16) and (17), upon simplification, we have

T2,2 (γn) = γ2
2 − γ3

2

=
1

64

(
k2 −

5

8
k1

2

)2

− 1

16384

(
3k1

3 − 10k1k2 + 8k3
)2

=
1

16384

(
256k2

2 − 320k1
2k2 + 100k1

4 − 9k1
6 + 60k1

4k2 − 100k1
2k2

2 − 48k1
3k3

+160k1k2k3 − 64k3
2
)

and equivalently,

T2,2 (γn) =
1

16384

[
256k2

(
k2 −

5

4
k1

2

)
− 100k1

2k2

(
k2 −

3

5
k1

2

)
+ 100k1

4 − 9k1
6

+k3
(
−48k1

3 + 160k1k2 − 64k3
)]

.

(22)

Applying Lemma 1, Lemma 2, and Lemma 3 on (22), we can obtain the desired bound.

4. Conclusion

This study was inspired by a number of previous studies. In this paper, we have ob-
tained the upper bounds of some coefficient problems for functions in the class S∗

SC (sin z)
including Taylor coefficients, logarithmic coefficients, and Hankel and Toeplitz determi-
nants of logarithmic coefficients. The results provided in this paper perhaps could be the
subject of further research related to the higher-order Hankel and Toeplitz determinants of
logarithmic coefficients and other coefficient problems, for instance, the Fekete-Szegö func-
tional, the Toeplitz and Hermitian-Toeplitz determinants, and the Krushkal inequality for
functions from the subclass of star-like functions with respect to other points. Addition-
ally, for another particular value of φ, several other classes of functions that are star-like
with respect to symmetric conjugate points can also be studied.

Acknowledgements

The authors truly appreciate the referees’ insightful comments. A special thanks to Uni-
versiti Teknologi MARA for supporting the publication of this paper.

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