EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 14, No. 1, 2021, 53-64
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Coefficient problems in a class of functions with
bounded turning associated with Sine function

Muhammad Ghaffar Khan1, Bakhtiar Ahmad2, Janusz Sokó l3, Zubair Muhammad1,
Wali Khan Mashwani1, Ronnason Chinram 4, Pattarawan Petchkaew5,∗

1 Institute of Numerical Sciences, Kohat University of Science & Technology, Kohat,
Pakistan
2 Government Degree College Mardan, 23200 Mardan, Pakistan
3 University of Rzeszów, College Natural Sciences, ul. Prof. Pigonia 1, 35-310 Rzeszów,
Poland
4 Algebra and Applications Research unit, Division of Computational Science, Faculty of
Sciences, Prince of Songkla University, Hat Yai, Songkhla 90110 Thailand
5 Program in Mathematics, Faculty of Science and Technology, Songkha Rajabhat University,
Songkhla, 90000, Thailand

Abstract. The Hankel determinant for a function having power series was first defined by Pom-
merenke. The growth of Hankel determinant has been evaluated for different subcollections of
univalent functions. Many subclasses with bounded turning have several interesting geometric
properties. In this paper, some classes of functions with bounded turning which connect to the
sine functions, are studied in the region of the unit disc in order. Our purpose is to obtain some
upper bounds for the third and fourth Hankel determinants related to such classes.

2020 Mathematics Subject Classifications: 30C45, 30C50

Key Words and Phrases: Holomorphic functions, Subordinations, Trigonometric function,
Hankel determinant

1. Introduction and Preliminaries

LetA be the class of all functions f(z) which are holomorphic in the region D = {z ∈ C : |z| < 1}
with the normalization f (0) = f ′(0)− 1 = 0. Therefore, for f(z) ∈ A, one has

f(z) = z +
∞∑
k=2

akz
k (z ∈ D) . (1)

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v14i1.3902

Email addresses: ghaffarkhan020@gmail.com (M. G. Khan), pirbakhtiarbacha@gmail.com (B. Ahmad),
jsokol@ur.edu.pl (J. Sokó l), zubair.math1984@gmail.com (Z. Muhammad), walikhan@kust.edu.pk
(W. K. Mashwani), ronnason.c@psu.ac.th (R. Chinram), pattarawan.pe@gmail.com (P. Petchkaew)

http://www.ejpam.com 53 c© 2021 EJPAM All rights reserved.



P. Petchkaew et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 53-64 54

Let S ⊂ A represent all functions that are univalent in D. For a function f ∈ S of the form
(1) , Bieberbach conjectured in 1916 that |an| ≤ n, n = 2, 3, . . .. De Branges proved this
in 1985, see [7]. During this period, a lot of coefficients results were established for some
subfamilies of S. For example, the class S∗ of starlike functions, K of convex functions
and R of bounded turning functions:

S∗ =

{
f ∈ S :

zf ′(z)

f(z)
≺ 1 + z

1− z
, z ∈ D

}
, (2)

K =

{
f ∈ S :

(zf ′(z))′

f ′(z)
≺ 1 + z

1− z
, z ∈ D

}
,

R =

{
f ∈ S : f ′(z) ≺ 1 + z

1− z
, z ∈ D

}
,

where ” ≺ ” represents the subordination.
We write g1 ≺ g2, if there is an analytic function v in D, with limitations v (0) = 0

and |v(z)| < 1, such that g1(z) = g2 (v(z)) , z ∈ D. In case of univalency of g2 in D, the
following relation holds;

g1(z) ≺ g2(z), z ∈ D ⇐⇒ g1(0) = g2(0) and g1(D) ⊂ g2(D).

By varying the function right hand side of subordinations in (2), we can define some
subclasses of the set S which have several interesting geometric properties, see [9–12, 15–
17, 23, 28, 29, 35]. From among these subfamilies we recall here the families that are
associated with trigonometric function as follows;

Ksin =

{
f ∈ A : 1 +

zf ′′(z)

f ′(z)
≺ 1 + sin(z), z ∈ D

}
, (3)

S∗sin =

{
f ∈ A :

zf ′(z)

f(z)
≺ 1 + sin(z), z ∈ D

}
, (4)

Rsin =
{
f ∈ A : f ′(z) ≺ 1 + sin(z), z ∈ D

}
. (5)

The set defined in (4) was established by Cho et.al [10] and studied the radii problems.
Here we investigated only the class (5) .

For given parameters q, n ∈ N = {1, 2, . . .}, the Hankel determinant Hq,n (f) was
defined by Pommerenke [32, 33] for a function f ∈ S having power series expansion (1) 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

∣∣∣∣∣∣∣∣∣ . (6)

The growth of Hq,n (f) has been evaluated for different subcollections of univalent func-
tions. Exceptionally, for each of the sets K, S∗ and R the sharp bound of the determinant
H2,2 (f) =

∣∣a2a4 − a23∣∣ were found by Janteng et al. [13, 14] while for the family of close-
to-convex functions the sharp estimation is still unknown (see, [38]). On the other hand,



P. Petchkaew et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 53-64 55

for the set of Bazilevič functions, the best estimate of |H2,2 (f)| was proved by Krishna
and RamReddy [21]. For more work on H2,2 (f) , see [5, 26, 27, 30, 31].
The determinant

H3,1 (f) =

∣∣∣∣∣∣
1 a2 a3
a2 a3 a4
a3 a4 a5

∣∣∣∣∣∣ (7)

is known as third order Hankel determinant and the estimation of this determinant
|H3,1 (f)| is a challenging task. In 2010, the first article on H3,1 (f) by Babalola [4], in
which he obtained the upper bound of |H3,1 (f)| for the groups of S∗, K and R. Later on, a
few creators distributed their work regarding |H3,1 (f)| for various subcollections of holo-
morphic and univalent functions, see [1, 2, 6, 8, 19, 22, 37, 39]. In 2017, the consequences
of Babalola [4] improved by Zaprawa [40], by proving

|H3,1 (f)| ≤


1, for f ∈ S∗,
49
540 , for f ∈ K,
41
60 , for f ∈ R.

and asserted that these inequalities are as yet not sharp. Additionally for the sharpness,
he thought about the subfamilies of S∗, C and R comprising of functions with m-fold
symmetry and acquired the sharp bounds. Recently in 2018, Kowalczyk et.al [20] and
Lecko et.al [25] evaluated the sharp inequalities

|H3,1 (f)| ≤ 4/135, and |H3,1 (f)| ≤ 1/9,

for the recognizable sets K and S∗ (1/2) respectively, where the symbol S∗ (1/2) indicates
the family of starlike functions of order 1/2. Additionally in 2018, the authors [24] got
an improved bound |H3,1 (f)| ≤ 8/9 for f ∈ S∗, yet not best possible. Now in this paper,
our main purpose is to study third and fourth order Hankel determinants family defined
in (5) .

2. A Set of Lemmas

Let P be the family of functions p that are holomorphic in D with Rep(z) > 0 and the
power series form as follow;

p(z) = 1 +
∞∑
n=1

cn z
n (z ∈ D) . (8)

Lemma 1. If p ∈ P be expressed in series expansion (8), then

|cn| ≤ 2 for n ≥ 1, (9)∣∣∣∣c2 − c21
2

∣∣∣∣ ≤ 2− |c1|
2

2
, (10)

|ci+j − µcicj | ≤ 2, for 0 ≤ µ ≤ 1. (11)



P. Petchkaew et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 53-64 56

and for complex number ρ, we have∣∣c2 − ρc21∣∣ ≤ 2 max {1, |2ρ− 1|} . (12)

where the inequalities (9) , (10) , (11) are taken from [34] and (12) is obtained in [18].

Lemma 2. If p(z) ∈ P be expressed in series expansion (8), then

2c2 = c21 + x
(
4− c21

)
for some x, |x| ≤ 1 and

4c3 = c31 + 2
(
4− c21

)
c1x−

(
4− c21

)
c1x

2 + 2
(
4− c21

) (
1− |x|2

)
z

for some z, |z| ≤ 1.

Lemma 3. ([3]) Let p ∈ P has power series (8), then∣∣Jc31 −Kc1c2 + Lc3
∣∣ ≤ 2 |J |+ 2 |K − 2J |+ 2 |J −K + L| (13)

Corollary 1. ([34]) Let p ∈ P has power series (8), then∣∣c31 − 2c1c2 + c3
∣∣ ≤ 2,

Lemma 4. ([36]) Let m,n, l and a satisfy the inequalities 0 < m < 1, 0 < r < 1, and

8r (1− r)
[
(mn− 2l)2 + (m (r +m)− n)2

]
+m (1−m) (n− 2rm)2 ≤ 4m2 (1−m)2 r (1− r) .

If p(z) ∈ P and has power series (8) then∣∣∣∣lc41 + rc22 + 2mc1c3 −
3

2
nc21c2 − c4

∣∣∣∣ ≤ 2.

3. Improved bound of |H3,1 (f)| for the Set Rsin

Theorem 1. If f(z) of the form (1) belongs to Rsin, then

|ak| ≤
1

k
, k = 2, 3, 4, 5. (14)

The results are sharp.

Proof. Since f(z) ∈ Rsin, form subordination definition there exists a Schwarz function
v(z) with v (0) = 0 and |v(z)| < 1, in such a way that

f ′(z) = 1 + sin (v(z)) , (z ∈ D) .

Since,
f ′(z) = 1 + 2a2z + 3a3z

2 + 4a4z
3 + 5a5z

4 + · · · . (15)



P. Petchkaew et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 53-64 57

Define a function

h(z) =
1 + v(z)

1− v(z)
= 1 + c1z + c2z

2 + · · · . (16)

Clearly, we have h(z) ∈ P and

v(z) =
h(z)− 1

h(z) + 1
=

c1z + c2z
2 + c3z

3 + · · ·
2 + c1z + c2z2 + c3z3 + · · ·

.

This gives

1 + sin (v(z)) = 1 +
1

2
c1z +

(
c2
2
− c21

4

)
z2 +

(
5c31
48
− c1c2

2
+
c3
2

)
z3

+

(
− 1

32
c41 +

5

16
c21c2 −

1

2
c3c1 −

1

4
c22 +

1

2
c4

)
z4 + · · · . (17)

By comparing (15) and (17), we may get

a2 =
c1
4
, (18)

a3 =
1

3

(
c2
2
− c21

4

)
, (19)

a4 =
1

4

(
5

48
c31 +

c3
2
− c1c2

2

)
, (20)

a5 =
1

5

(
c4
2

+
5

16
c21c2 −

c41
32
− c1c3

2
− c22

4

)
. (21)

Now implementing (9), in (18), we obtain

|a2| ≤
1

2
.

Now using (10), in (19), we get

|a3| ≤
1

6

(
2− |c1|

2

2

)
The maximum value of above function at c1 = 0.

|a3| ≤
1

3
.

Implementation of triangle inequality and Lemma 3, in (20), leads us to

|a4| ≤
1

4
.

By applying Lemma 4 in (21), it provides

|a5| ≤
1

5
.



P. Petchkaew et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 53-64 58

If for k = 2, 3, 4, 5, we take the functions fk(z) = z + · · · such that

f ′k(z) = 1 + sin(zk−1), (z ∈ D),

then f ′k(z) ≺ 1 + sin z and so fk ∈ Rsin and

fk(z) = z +
1

k
zk − 1

3!(3k − 2)
z3k−2 + · · · , (z ∈ D) (22)

which shows that the bounds are sharp.

Conjecture If f(z) of the form (1) belongs to Rsin, then

|an| ≤
1

n
, n ≥ 6. (23)

Theorem 2. If f(z) of the form (1) belongs to Rsin, then for any complex number ρ∣∣a3 − ρa22∣∣ ≤ 1

3
max

{
1,

3|ρ|
4

}
. (24)

The result is sharp.

Proof. Utilizing (18) and (19), we may get∣∣a3 − ρa22∣∣ =

∣∣∣∣c26 − c21
12
− ρ

16
c21

∣∣∣∣ .
This gives ∣∣a3 − ρa22∣∣ =

1

6

∣∣∣∣{c2 − (4 + 3ρ

8

)
c21

}∣∣∣∣ .
Application of (12), leads us to∣∣a3 − ρa22∣∣ ≤ 1

3
max

{
1,
|3ρ|
4

}
.

For the sharpness of (24) consider (22), with k = 2:

f2(z) = z +
1

2
z2 − 1

4!
z4 + · · · , (z ∈ D),

which gives equality in (24) when |ρ| ≥ 4/3, namely∣∣a3 − ρa22∣∣ = |ρa22| =
|ρ|
4
.

For the case |ρ| ≤ 4/3 consider

f3(z) = z +
1

3
z3 − 1

42
z7 + · · · , (z ∈ D),

which gives ∣∣a3 − ρa22∣∣ = |a3| =
1

3
=

1

3
max

{
1,

3|ρ|
4

}
.



P. Petchkaew et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 53-64 59

Corollary 2. If f ∈ Rsin and |ρ| ≤ 4/3, then∣∣a3 − ρa22∣∣ ≤ 1

3
. (25)

Theorem 3. If f of the form (1) belongs to Rsin, then

|a2a3 − a4| ≤
1

4
. (26)

The result is sharp.

Proof. From (18),(19) and (20), we have

|a2a3 − a4| =
∣∣∣∣− 3

64
c31 +

1

6
c2c1 −

1

8
c3

∣∣∣∣ =

∣∣∣∣ 3

64
c31 −

c1c2
6

+
c3
8

∣∣∣∣ .
Implementation of triangle inequality and Lemma 3, in (20), leads us to

|a2a3 − a4| ≤
1

4
.

The sharpness of (26) shows f4(z) = z + z4/4− z10/60 + · · · which was defined in (22).

Theorem 4. If f(z) of the form (1) belongs to Rsin, then∣∣a2a4 − a23∣∣ ≤ 1

9
. (27)

The result is sharp.

Proof. Now since from (18), (19), and (20), we have

∣∣a2a4 − a23∣∣ =

∣∣∣∣c1c332
− c21c2

288
− c41

2304
− c22

36

∣∣∣∣ .
Now in terms of Lemma 2, we obtain

∣∣a2a4 − a23∣∣ =

∣∣∣∣c1c332
− c21c2

288
− c41

2304
− c22

36

∣∣∣∣
=

∣∣∣∣∣∣− c41
768
−
c21x

2
(
4− c21

)
128

−
x2
(
4− c21

)2
144

+
c1
(
4− c21

) (
1− |x|2

)
z

64

∣∣∣∣∣∣ .
Let |z| = 1 , |x| = t, t ∈ [0, 1], |c1| = c ∈ [0, 2] . Then, using the triangle inequality, we get

∣∣a2a4 − a23∣∣ ≤ c4

768
+
t2c2

(
4− c2

)
128

+
t2
(
4− c2

)2
144

+

(
1− t2

)
c
(
4− c2

)
64

.



P. Petchkaew et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 53-64 60

Putting

H (c, t) =
c4

768
+
t2c2

(
4− c2

)
128

+
t2
(
4− c2

)2
144

+

(
1− t2

)
c
(
4− c2

)
64

,

then,
∂H (c, t)

∂t
=
t
(
c2 − 18c+ 32

) (
4− c2

)
576

> 0,

which shows that H (c, t) increases on [0, 1] with respect t. That is H (c, t) have maximum
value at t = 1, which is

maxH (c, t) = H (c, 1) =
c4

768
+
c2
(
4− c2

)
128

+

(
4− c2

)2
144

.

Setting

G (c) =
c4

768
+
c2
(
4− c2

)
128

+

(
4− c2

)2
144

,

then we have

G
′
(c) =

c3

192
+
c
(
4− c2

)
64

− c3

64
−
c
(
4− c2

)
36

.

If G
′
(c) = 0, then the root is c = 0. Further, since G

′′
(c) = − 7

144 < 0, so the function
G (c) can attain the maximum value at c = 0, which is∣∣a2a4 − a23∣∣ ≤ 1

9
.

The sharpness of (27) shows f3(z) = z + z3/3− z7/42 + · · · which was defined in (22).

Theorem 5. If f(z) = z + a2z
2 + a3z

3 + · · · belongs to Rsin, then

|H3,1 (f)| ≤ 359

2160
= 0.16620 . . . . (28)

Proof. Third order Hankel determinant form equation (7) one may written as;

H3,1 (f) = a3
(
a2a4 − a23

)
− a4 (a4 − a2a3) + a5

(
a3 − a22

)
.

where a1 = 1. This provides that

|H3,1 (f)| ≤ |a3|
∣∣a2a4 − a23∣∣+ |a4| |a4 − a2a3|+ |a5|

∣∣a3 − a22∣∣ .
By implementing (14), (25), (26) and (27 ), we obtain our desired result.



REFERENCES 61

4. Bound of |H4,1 (f)| for the Set Rsin

First we can write H4,1 (f) in the form

H4,1 (f) = a7H3,1 (f)− 2a4a6
(
a2a4 − a23

)
− 2a5a6 (a2a3 − a4)− a26

(
a3 − a22

)
+a25

(
a2a4 − a23

)
+ a25

(
a2a4 + 2a23

)
− a35 + a44 − 3a3a

2
4a5. (29)

Also ∣∣a2a4 + 2a23
∣∣ ≤ ∣∣a2a4 − a23∣∣+ 3 |a3|2 ,

using (14) and (27) , we get ∣∣a2a4 + 2a23
∣∣ ≤ 1

9
+

1

3

=
4

9
. (30)

Theorem 6. If f(z) = z + a2z
2 + a3z

3 + · · · belongs to Rsin, then

|H4,1 (f)| ≤ 0.10556.

Proof. Using triangle inequality in (29) , we obtain

|H4,1 (f)| ≤ |a7| |H3,1 (f)|+ 2 |a4| |a6|
∣∣a2a4 − a23∣∣+ 2 |a5| |a6| |a2a3 − a4|+ |a6|2

∣∣a3 − a22∣∣
+ |a5|2

∣∣a2a4 − a23∣∣+ |a5|2
∣∣a2a4 + 2a23

∣∣+ |a5|3 + |a4|4 + 3 |a3| |a4|2 |a5| .

By using(14), (23) , (25), (26) , (27 ) , (28) and (30) , we get the required result.

Acknowledgements

The authors would like to express our appreciation to the anonymous referees for the
comprehensive reading of this paper and their valuable comments and suggestions.

References

[1] S Altinkaya and S Yalçin. Third hankel determinant for bazilevič functions. Advances
in Mathematics, 5:91–96, 2016.

[2] M Arif, K M Noor, and M Raza. Hankel determinant problem of a subclass of analytic
functions. Journal of Inequalities and Applications, Art. 22:7 pages, 2012.

[3] M Arif, M Raza, H Tang, S Hussain, and H Khan. Hankel determinant of order three
for familiar subsets of analytic functions related with sine function. Open Mathemat-
ics, 17:1615–1630, 2019.

[4] K O Babalola. On h3 (1) hankel determinant for some classes of univalent functions.
Inequality Theo ry and Applications, 6:1–7, 2010.



REFERENCES 62

[5] D Bansal. Upper bound of second hankel determinant for a new class of analytic
functions. Applied Mathematics Letters, 23:103–107, 2013.

[6] D Bansal, S Maharana, and J K Prajapat. Third order hankel determinant for certain
univalent functions. Journal of the Korean Mathematical Society, 52:1139–1148, 2015.

[7] L De Branges. A proof of the bieberbach conjecture. Acta Mathematica, 154:137–152,
1985.

[8] N E Cho, B Kowalczyk, O S Kwon, A Lecko, and J Sim. Some coefficient inequali-
ties related to the hankel determinant for strongly starlike functions of order alpha.
Journal of Mathematical Inequalities, 11:429–439, 2017.

[9] N E Cho, S Kumar, V Kumar, V Ravichandran, and H M Srivastava. Starlike
functions related to the bell numbers. Symmetry, 11:219, 2019.

[10] N E Cho, V Kumar, S S Kumar, and V Ravichandran. Radius problems for starlike
functions associated with the sine function. Bulletin of the Iranian Mathematical
Society, 45:213–232, 2019.

[11] J Dziok, R K Raina, and J Sokó l. On a class of starlike functions related to a shell-
like curve connected with fibonacci numbers. Mathematical and Computer Modelling,
57:1203–1211, 2013.

[12] J̃ Sokó l and J Stankiewicz. Radius of convexity of some subclasses of strongly starlike
functions. Folia Scient. Univ. Tech. Resoviensis, 19:101–105, 1996.

[13] A Jangteng, S A Halim, and M Darus. Coefficient inequality for a function whose
derivative has a positive real part. Journal of Inequalities in Pure and Applied Math-
ematics, 7:1–5, 2006.

[14] A Jangteng, S A Halim, and M. Darus. Coefficient inequality for starlike and convex
functions. International Journal of Inequalities in Mathematical Analysis, 1:619–625,
2007.

[15] W Janowski. Extremal problems for a family of functions with positive real part and
for some related families. Annales Polonici Mathematici, 23:159–177, 1970.

[16] R Kargar, A Ebadian, and J Sokó l. Radius problems for some subclasses of analytic
functions. Complex Analysis and Operator Theory, 11:1639–1649, 2017.

[17] R Kargar, A Ebadian, and J Sokó l. On booth lemniscate and starlike functions.
Analysis and Mathematical Physics, 9:143–154, 2019.

[18] F Keogh and E Merkes. A coefficient inequality for certain subclasses of analytic
functions. Proceedings of the American Mathematical Society, 20:8–12, 1969.



REFERENCES 63

[19] M G Khan, B Ahmad, G Murugusundaramoorthy, R Chinram, and W K Mashwani.
Applications of modified sigmoid functions to a class of starlike functions. Journal of
Function Spaces, 2020:Article ID 8844814, 2020.

[20] B Kowalczyk, A Lecko, and Y J Sim. The sharp bound of the hankel determinant of
the third kind for convex functions. Bulletin of the Australian Mathematical Society,
97:435–445, 2018.

[21] D V Krishna and T RamReddy. Second hankel determinant for the class of bazilevic
functions. Studia Universitatis Babe-Bolyai Mathematica, 60:413–420, 2015.

[22] D V Krishna, B Venkateswarlua, and T RamReddy. Third hankel determinant for
bounded turning functions of order alpha. Journal of the Nigerian Mathematical
Society, 34:121–127, 2015.

[23] S Kumar and V Ravichandran. A subclass of starlike functions associated with a
rational function. Southeast Asian Bulletin of Mathematics, 40:199–212, 2016.

[24] O S Kwon, A Lecko, and Y J Sim. The bound of the hankel determinant of the third
kind for starlike functions. Bulletin of the Malaysian Mathematical Sciences Society,
42:1–14, 2018.

[25] A Lecko, Y J Sim, and B Śmiarowska. The sharp bound of the hankel determinant
of the third kind for starlike functions of order 1/2. Complex Analysis and Operator
Theory, 13:2231–2238, 2019.

[26] S K Lee, V Ravichandran, and S Supramaniam. Bounds for the second hankel deter-
minant of certain univalent functions. Journal of Inequalities and Applications, Art.
281, 2013.

[27] M S Liu, J F Xu, and M Yang. Upper bound of second hankel determinant for certain
subclasses of analytic functions. Abstract and Applied Analysis, Art. 603180, 2014.

[28] S Mahmood, H M Srivastava, and S N. Malik. Some subclasses of uniformly univalent
functions with respect to symmetric points. Complex Analysis and Operator Theory,
11:287, 2019.

[29] R Mendiratta, S Nagpal, and V Ravichandran. On a subclass of strongly starlike func-
tions associated with exponential function. Bulletin of the Malaysian Mathematical
Sciences Society, 38:365–386, 2015.

[30] J W Noonan and D K Thomas. On the second hankel determinant of areally mean
p-valent functions. Transactions of the American Mathematical Society, 223:337–346,
1976.

[31] H Orhan, N Magesh, and J Yamini. Bounds for the second hankel determinant of
certain bi-univalent functions. Turkish Journal of Mathematics, 40:679–687, 2016.



REFERENCES 64

[32] Ch Pommerenke. On the coefficients and hankel determinants of univalent functions.
Journal of the London Mathematical Society, 41:111–122, 1966.

[33] Ch Pommerenke. On the hankel determinants of univalent functions. Mathematika,
14:108–112, 1967.

[34] Ch Pommerenke. Univalent functions. Math, Lehrbucher, vandenhoeck and Ruprecht,
Gottingen,, 1975.

[35] R K Raina and J Sokó l. Some properties related to a certain class of starlike functions.
C. R. Math. Acad. Sci., 353:973–978, 2015.

[36] V Ravichandran and S Verma. Bound for the fifth coefficient of certain starlike
functions. Comptes Rendus Mathematique, 353:505–510, 2015.

[37] M Raza and S N Malik. Upper bound of third hankel determinant for a class of
analytic functions related with lemniscate of bernoulli. Journal of Inequalities and
Applications, Art. 412, 2013.

[38] D Răducanu and P Zaprawa. Second hankel determinant for close-to-convex functions.
Comptes Rendus Mathematique, 355:1063–1071, 2017.

[39] G Shanmugam, B A Stephen, and K O Babalola. Third hankel determinant for
α-starlike functions. Gulf Journal of Mathematics, 2:107–113, 2014.

[40] P Zaprawa. Third hankel determinants for subclasses of univalent functions. Mediter-
ranean Journal of Mathematics, 14:19, 2017.