EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 4, 2022, 1937-1947
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Toeplitz Determinants for the Class of Functions with
Bounded Turning

Nur Hazwani Aqilah Abdul Wahid1,∗, Daud Mohamad1, Nur Maziah Kamarozzaman1,
Ainurraziqin Ahmad Shahminan1

1 Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA, 40450 Shah
Alam, Selangor, Malaysia

Abstract. In this paper, we obtain the upper bounds of the Toeplitz determinants for the class
of functions with bounded turning. We also present some consequences of our main results. Some
estimates obtained on Toeplitz determinants are sharp.

2020 Mathematics Subject Classifications: 30C45, 30C50
Key Words and Phrases: Bounded turning functions, Toeplitz determinant, univalent functions

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.

Let P denote the class of positive real part functions p (z) of the form

p (z) = 1 +
∞∑
n=1

pnz
n, (2)

which satisfy Re p (z) > 0 for z ∈ E. This class is also known as the class of Carathéodory
functions.

Let G (α, δ) be the class of normalized functions f (z) ∈ A satisfying the condition
Re

(
eiαf ′ (z)

)
> δ, z ∈ E, where |α| < π, 0 ⩽ δ < 1, and cosα > δ. This class was

introduced by Mohamad [10].

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i4.4559

Email addresses: hazwaniaqilah@fskm.uitm.edu.my (N. H. A. A. Wahid),
daud@fskm.uitm.edu.my (D. Mohamad), nurmaziah16@gmail.com (N. M. Kamarozzaman),
ainurraziqin.ahmad@gmail.com (A. A. Shahminan)

https://www.ejpam.com 1937 © 2022 EJPAM All rights reserved.



N. H. A. A. Wahid et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1937-1947 1938

Remark 1. For the specific values of the parameters α and δ, we obtain the special cases
of G (α, δ) as follows:

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

(ii) If we let α = 0, then we have the class G (0, δ) ≡ 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 let δ = 0, then we have the class G (α, 0) ≡ R (α) which satisfies Re
(
eiαf ′ (z)

)
>

0.

Goel and Mehrok [8], Macgregor [9], and Silverman and Silvia [17] were among the first
researchers to study the classes R, R (δ), and R (α). Recently, the investigation into the
class of bounded turning functions and coefficient problems such as the Hankel determinant
for the higher order has been extensively studied by other researchers, see for example
[3, 4]. We may point interested readers to recent advances in the class of bounded turning
functions connected to a three-leaf-shaped domain and Bernoulli’s lemniscate as well as
their coefficient problems like the Hankel determinant, logarithmic coefficients, and the
Hankel determinant with logarithmic coefficients, which point in a different direction than
the current study, see [16, 23].

Finding estimates on the functional involving coefficients of f (z) ∈ A has been a
major research area in geometric function theory since the development of the Bieberbach
conjecture. Toeplitz determinant, for example, whose elements are the coefficients of
f (z) ∈ A has been appealing to many researchers because it is related to the coefficient
problems. Toeplitz determinant appeared in all branches of pure and applied mathematics,
statistics and probability, image processing, quantum mechanics, queuing networks, signal
processing, and time series analysis (see Ye and Lim [24] and references therein). Here we
consider the symmetric Toeplitz determinant and it is defined by [21]

Tq (n) =

∣∣∣∣∣∣∣∣∣
an
an+1

...
an+q−1

an+1

an
...

an+q−2

· · ·
· · ·
...
· · ·

an+q−1

an+q−2
...
an

∣∣∣∣∣∣∣∣∣ , a1 = 1.

The estimates of the Toeplitz determinant were obtained for different classes of univalent
functions. For instance, Ali et al. [2] studied Toeplitz matrices whose elements are the
coefficients of bounded turning, starlike, close-to-convex, and univalent functions, Radhika
et al. [12] obtained sharp bounds for Toeplitz determinants for the class of bounded
turning functions, Zhang et al. [25] considered Toeplitz determinants of starlike functions
connected with the sine function, and Zulfiqar et al. [26] investigated the fourth-order
Toeplitz determinant for convex functions connected with the sine function. Much of the
recent history of the development of this problem can also be found in [1, 5, 11, 13–15, 18–
20, 22]. Thus, inspired by these works, in this paper, we aim to investigate the upper



N. H. A. A. Wahid et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1937-1947 1939

bounds of the second, third, and fourth-order Toeplitz determinants for functions of the
class G (α, δ). In particular, we find the bounds for the following determinants:

T2 (n) =

∣∣∣∣ an an+1

an+1 an

∣∣∣∣ , n ⩾ 2, (3)

T3 (1) =

∣∣∣∣∣∣
1 a2 a3
a2 1 a2
a3 a2 1

∣∣∣∣∣∣ , (4)

T3 (2) =

∣∣∣∣∣∣
a2 a3 a4
a3 a2 a3
a4 a2 a2

∣∣∣∣∣∣ , (5)

T3 (3) =

∣∣∣∣∣∣
a3 a4 a5
a4 a3 a4
a5 a4 a3

∣∣∣∣∣∣ , (6)

and

T4 (1) =

∣∣∣∣∣∣∣∣
1 a2 a3 a4
a2 1 a4 a3
a3 a4 1 a2
a4 a3 a2 1

∣∣∣∣∣∣∣∣ , (7)

where the elements are the coefficients of the functions f (z) of the form (1) in G (α, δ).
Besides, we point out several special cases and the consequences of our results.

We shall need the following lemmas in order to prove our main results.

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

pnz
n. Then

|pn| ⩽ 2, n ⩾ 1.

The inequality is sharp for the function p (z) = 1+z
1−z .

Lemma 2. ([7]) Let p (z) ∈ 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.

2. Main Results

In this section, we state and prove the main results of our present investigation.



N. H. A. A. Wahid et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1937-1947 1940

Theorem 1. Let f (z) ∈ G (α, δ) be of the form (1). Then

|T2 (n)| ⩽ 4tαδ
2

[
1

n2
+

1

(n + 1)2

]
,

where tαδ = cosα− δ. The inequality is sharp.

Proof. Let a function f (z) ∈ G (α, δ) given by (1). Then there exists a function
p (z) ∈ P of the form (2) such that [10]

eiαf ′ (z) − i sinα− δ

tαδ
= p (z) , (8)

where tαδ = cosα− δ.
Rearranging (8) and hence using the series representations for f ′ (z) and p (z), we get

1+2a2z+3a3z
2+4a4z

3+ · · · = e−iα
[
tαδ

(
1 + p1z + p2z

2 + p3z
3 + · · ·

)
+ i sinα + δ

]
. (9)

Equating the coefficients of like powers of zn, n ⩾ 1 yields

an =
tαδe

−iαpn−1

n
, n ⩾ 2. (10)

Then, applying Lemma 1, we get

|an| =
tαδ |pn−1|

n
⩽

2tαδ
n

(11)

and so

|an+1| =
tαδ |pn|
n + 1

⩽
2tαδ
n + 1

. (12)

Clearly from (3) leads to

|T2 (n)| =
∣∣an2 − an+1

2
∣∣ ⩽ ∣∣an2∣∣ +

∣∣an+1
2
∣∣ . (13)

Thus, making use of (11) and (12) gives the desired inequality. The inequality is sharp

for the function eiαf ′(z)−i sinα−δ
tαδ

= 1+iz
1−iz .

Theorem 2. Let f (z) ∈ G (α, δ) be of the form (1). Then

|T3 (1)| ⩽ 1

9

(
9 + 18tαδ

2 + 4tαδ
2
√

9tαδ2 − 6tαδ cosα + 1
)
,

where tαδ = cosα− δ. The inequality is sharp.



N. H. A. A. Wahid et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1937-1947 1941

Proof. By making use of (10) for n = 2, 3, from (4), we obtain

T3 (1) = 1 − 2a2
2 + 2a2

2a3 − a3
2

= 1 − 2

(
tαδe

−iαp1
2

)2

+ 2

(
tαδe

−iαp1
2

)2(
tαδe

−iαp2
3

)
−
(
tαδe

−iαp2
3

)2

=
1

18

(
18 − 9tαδ

2e−2iαp1
2 − 2tαδ

2e−2iαp2
2 + 3tαδ

3e−3iαp1
2p2

)
.

(14)
Further, we can rearrange (14) as

|T3 (1)| =
1

18

∣∣18 − 9tαδ
2e−2iαp1

2 − 2tαδ
2e−2iαp2

(
p2 − µp1

2
)∣∣ , (15)

where µ = 3tαδe
−iα

2 .
Thus, by the triangle inequality along with Lemma 1 and Lemma 2, we get

|T3 (1)| ⩽ 1

9

(
9 + 18tαδ

2 + 4tαδ
2
√

9tαδ2 − 6tαδ cosα + 1
)
. (16)

This inequality is sharp for the function eiαf ′(z)−i sinα−δ
tαδ

= 1+iz
1−iz .

Theorem 3. Let f (z) ∈ G (α, δ) be of the form (1). Then

|T3 (2)| ⩽ 7tαδ
3

3
,

where tαδ = cosα− δ.

Proof. Using (10) for n = 2, 3, 4, from (5), it follows that

T3 (2) = a2
3 − 2a2a3

2 + 2a3
2a4 − a2a4

2

=

(
tαδe

−iαp1
2

)3

− 2

(
tαδe

−iαp1
2

)(
tαδe

−iαp2
3

)2

+ 2

(
tαδe

−iαp2
3

)2(
tαδe

−iαp3
4

)
−
(
tαδe

−iαp1
2

)(
tαδe

−iαp3
4

)2

=
tαδ

3e−3iα

288

(
36p1

3 − 32p1p2
2 + 16p2

2p3 − 9p1p3
2
)
.

(17)
Rearranging the terms in (17) and hence applying the triangle inequality, then we can
rewrite it as

|T3 (2)| ⩽ tαδ
3

288

[
36|p1|3 + 32 |p1| |p2|2 + 16 |p3|

∣∣p4 − η1p2
2
∣∣ + 16 |p3| |p4 − η2p1p3|

]
, (18)

where η1 = 1 and η2 = 9
16 .

Further, by implementing Lemma 1 and Lemma 2, thus we obtain

|T3 (2)| ⩽ 7tαδ
3

3
.

This concludes the proof.



N. H. A. A. Wahid et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1937-1947 1942

Theorem 4. Let f (z) ∈ G (α, δ) be of the form (1). Then

|T3 (3)| ⩽ 112tαδ
3

135
,

where tαδ = cosα− δ.

Proof. Using the values of a3, a4, and a5 from (10) and in view of (6), it can be seen
that

T3 (3) = a3
3 − 2a3a4

2 + 2a4
2a5 − a3a5

2

=

(
tαδe

−iαp2
3

)3

− 2

(
tαδe

−iαp2
3

)(
tαδe

−iαp3
4

)2

+ 2

(
tαδe

−iαp3
4

)2(
tαδe

−iαp4
5

)
−
(
tαδe

−iαp2
3

)(
tαδe

−iαp4
5

)2

=
tαδ

3e−3iα

5400

(
200p2

3 − 225p2p3
2 + 135p3

2p4 − 72p2p4
2
)
.

(19)
After rearranging the terms and using triangular inequalities, (19) yields

|T3 (3)| ⩽ tαδ
3

5400

[
200|p2|3 + 225 |p2| |p3|2 + 135 |p4|

∣∣p6 − η1p3
2
∣∣ + 135 |p4| |p6 − νp2p4|

]
,

(20)
where η1 = 1 and ν = 72

135 .
Finally, by applying Lemma 1 and Lemma 2, we get

|T3 (3)| ⩽ 112tαδ
3

135
.

This completes the proof.

Theorem 5. Let f (z) ∈ G (α, δ) be of the form (1). Then

|T4 (1)| ⩽ 1

1296

(
1296 + 4392tαδ

2 + 3456tαδ
3 + 1921tαδ

4
)
,

where tαδ = cosα− δ.



N. H. A. A. Wahid et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1937-1947 1943

Proof. From the expansion of (7) and using the values of a2, a3, and a4 from (10), we
get

T4 (1) = 1 − 2a2
2 + a2

4 − 2a3
2 + a3

4 − 2a4
2 + a4

4 − 2a2
2a3

2 − 2a2
2a4

2

−2a3
2a4

2 + 8a2a3a4

= 1 − 2

(
tαδe

−iαp1
2

)2

+

(
tαδe

−iαp1
2

)4

− 2

(
tαδe

−iαp2
3

)2

+

(
tαδe

−iαp2
3

)4

−2

(
tαδe

−iαp3
4

)2

+

(
tαδe

−iαp3
4

)4

− 2

(
tαδe

−iαp1
2

)2(
tαδe

−iαp2
3

)2

−2

(
tαδe

−iαp1
2

)2(
tαδe

−iαp3
4

)2

− 2

(
tαδe

−iαp2
3

)2(
tαδe

−iαp3
4

)2

+8

(
tαδe

−iαp1
2

)(
tαδe

−iαp2
3

)(
tαδe

−iαp3
4

)
=

1

20736

(
20736 − 10368tαδ

2e−2iαp1
2 + 1296tαδ

4e−4iαp1
4 − 4608tαδ

2e−2iαp2
2

+256tαδ
4e−4iαp2

4 − 2592tαδ
2e−2iαp3

2 + 81tαδ
4e−4iαp3

4 − 1152tαδ
4e−4iαp1

2p2
2

−648tαδ
4e−4iαp1

2p3
2 − 288tαδ

4e−4iαp2
2p3

2 + 6912tαδ
3e−3iαp1p2p3

)
.

(21)
Rearranging the terms in (21) and applying the triangle inequality, as well as some calcu-
lations, we can rewrite it in the following expression:

|T4 (1)| ⩽ 1

20736

[
20736 + 10368tαδ

2|p1|2 + 256tαδ
4|p2|4 + 81tαδ

4|p3|4 + 2592tαδ
2|p3|2

+648tαδ
4|p1|2|p3|2 + 288tαδ

4|p2|2|p3|2 + 1296tαδ
4|p1|2

∣∣p2 − η1p1
2
∣∣

+4608tαδ
2 |p2|

∣∣p2 − υ1p1
2
∣∣ + 6912tαδ

3 |p1| |p2| |p3 − υ2p1p2|
]
,

(22)

where η1 = 1, υ1 = 9tαδ
2e−2iα

32 , and υ2 = tαδe
−iα

6 .
Now, with the help of Lemma 1 and Lemma 2, we obtain

|T4 (1)| ⩽ 1

1296

(
1296 + 4392tαδ

2 + 3456tαδ
3 + 1921tαδ

4
)
.

Thus, this completes the proof.

3. Corollaries and Consequences

In this section, we shall give the consequences of our main results.
For α = 0 and δ = 0 in Theorem 1, Theorem 2, Theorem 3, Theorem 4, and Theorem 5,
we get the estimates for the class R.

Corollary 1. Let f (z) ∈ R. Then



N. H. A. A. Wahid et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1937-1947 1944

(i) |T2 (n)| ⩽ 4
[

1
n2 + 1

(n+1)2

]
.

(ii) |T3 (1)| ⩽ 35
9 .

(iii) |T3 (2)| ⩽ 7
3 .

(iv) |T3 (3)| ⩽ 112
135 .

(v) |T4 (1)| ⩽ 11065
1296 .

For α = 0 in Theorem 1, Theorem 2, Theorem 3, Theorem 4, and Theorem 5, we
obtain the estimates for the class R (δ).

Corollary 2. Let f (z) ∈ R (δ). Then

(i) |T2 (n)| ⩽ 4(1 − δ)2
[

1
n2 + 1

(n+1)2

]
.

(ii) |T3 (1)| ⩽ 1
9

(
9 + 18(1 − δ)2 + 4(1 − δ)2 (3δ − 2)

)
.

(iii) |T3 (2)| ⩽ 7(1−δ)3

3 .

(iv) |T3 (3)| ⩽ 112(1−δ)3

135 .

(v) |T4 (1)| ⩽ 1
1296

(
1296 + 4392(1 − δ)2 + 3456(1 − δ)3 + 1921(1 − δ)4

)
.

By choosing δ = 0 in Theorem 1, Theorem 2, Theorem 3, Theorem 4, and Theorem 5,
we obtain the estimates for the class R (α).

Corollary 3. Let f (z) ∈ R (α). Then

(i) |T2 (n)| ⩽ 4cos2α
[

1
n2 + 1

(n+1)2

]
.

(ii) |T3 (1)| ⩽ 1
9

(
9 + 18cos2α + 4cos2α

√
3cos2α + 1

)
.

(iii) |T3 (2)| ⩽ 7 cosα3

3 .

(iv) |T3 (3)| ⩽ 112cos3α
135 .

(v) |T4 (1)| ⩽ 1
1296

(
1296 + 4392cos2α + 3456cos3α + 1921cos4α

)
.

We remark that the inequalities in Corollary 1(i), Corollary 1(ii), and Corollary 1(iii)
coincide with the results of Ali et al. [2]. It is also shown in [2] that the results in Corollary
1(i) and Corollary 1(ii) were sharp. In the existing literature, no bounds for |T2 (n)|, n ⩾ 2,
|T3 (n)| , n = 1, 2, 3, and |T4 (1)| for the classes G (α, δ), R (δ), and R (α) were obtained.
Additionally, the results on |T3 (3)| and |T4 (1)| for functions in the class R had never been
studied before.



REFERENCES 1945

4. Conclusions

In the present paper, we have considered the Toeplitz determinants whose elements are
coefficients of univalent functions. We have obtained the upper bounds of |T2 (n)|, n ⩾ 2,
|T3 (n)| , n = 1, 2, 3, and |T4 (1)| not only for functions of the class G (α, δ), but also for
some classes of functions with bounded turning namely R, R (δ), and R (α). Some results
obtained are reduced to the estimates proven in [2] for specific choices of parameters
α and δ. For the class G (α, δ), we have determined the sharp estimates for |T2 (n)|,
n ⩾ 2 and |T3 (1)|. The results obtained perhaps give an opportunity for researchers to
further investigate inequalities problems for functions of the class G (α, δ) as well as other
subclasses of S.

Acknowledgements

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

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