GENERALIZED HANKEL DETERMINANT FOR A GENERAL
SUBCLASS OF UNIVALENT FUNCTIONS
S. YALC¸ IN1, S¸. ALTINKAYA2, S. OWA3, §
Abstract. Making use of the generalized Hankel determinant, in this work, we consider
a general subclass of univalent functions. Moreover, upper bounds are obtained for
a3−µa22
,whereµ∈R.
Keywords: Analytic and univalent functions, Fekete-Szeg¨o inequality, Hankel determi-nant.
AMS Subject Classification: 30C45, 30C50.
1.
Introduction
Let
A
represent the class of functions
f
which are analytic in the open unit disk
U
=
{
z
:
z
∈
C
and
|
z
|
<
1
}
with in the form
f
(
z
) =
z
+
∞
X
n=2
a
nz
n.
(1)
Let
S
be the subclass of
A
consisting of the form (1) which are also univalent in
U.
Let
P
βdenote the class of functions consisting of
p
, such that
p
(
z
) = 1 +
p1z
+
p2z
2+
· · ·
= 1 +
∞
X
n=1
p
nz
n,
which are regular in the open unit disc
U
and satisfy
<
(
p
(
z
))
> β
for some
β
(0
≤
β <
1)
and for any
z
∈
U
.
The Fekete-Szeg¨
o functional
a3
−
µa
22for normalized univalent functions of the form
given by (1) is well known for its rich history in Geometric Function Theory. Its origin
was in the disproof by Fekete and Szeg¨
o of the 1933 conjecture of Littlewood and Paley
that the coefficients of odd univalent functions are bounded by unity (see [3]). The
func-tional has since received great attention, particularly in many subclasses of the family of
univalent functions. Nowadays, it seems that this topic had become an interest among the
researchers (see, for example, [1], [4], [5], [6]).
1
Department of Mathematics, Faculty of Arts and Science, Uludag University, 16059 Bursa, Turkey. e-mail: [email protected]; ORCID: https://orcid.org/0000-0002-0243-8263.
2 e-mail: [email protected]; ORCID: http://orcid.org/0000-0002-7950-8450. 3
Department of Mathematics, Faculty of Education, Yamato University, Katayama 2-5-1, Suita-Osaka 564-0082, Japan.
e-mail: [email protected]; ORCID: http://orcid.org/0000-0002-8842-2464.
§ Manuscript received: October 4, 2016; accepted: October 24, 2016.
TWMS Journal of Applied and Engineering Mathematics, Vol.8, No.1a cI¸sık University, Department of Mathematics, 2018; all rights reserved.
The
q
thHankel determinant for
n
≥
0 and
q
≥
1 is stated by Noonan and Thomas ([7])
as
H
q(
n
) =
a
na
n+1· · ·
a
n+q−1a
n+1a
n+2· · ·
a
n+q..
.
..
.
..
.
..
.
a
n+q−1a
n+q· · ·
a
n+2q−2(
a1
= 1)
.
This determinant has also been investigated by several authors. For example, Noor [8]
determined the rate of growth of
H
q(
n
) as
n
→ ∞
for functions
f
given by (1) with
bounded boundary. In particular, sharp upper bounds on
H
2(2) were obtained by the
authors of articles ([8], [10]) for different classes of functions.
It is interesting to note that
H2
(1) =
a1
a2
a
2a
3=
a3
−
a
22and
H2
(2) =
a
2a
3a
3a
4=
a2a4
−
a
23.
The Hankel determinant
H
2(1) =
a
3−
a
22is well-known as Fekete-Szeg¨
o functional.
Definition 1.1.
[2]
A function
f
∈
A
is said to be in the class
Q
λ(
β
)
,
if the following
condition is satisfied:
<
(1
−
λ
)
f
(
z
)
z
+
λf
0(
z
)
> β
;
0
≤
β <
1
, λ
≥
0
, z
∈
U.
In order to derive our main results, we require the following lemmas.
Lemma 1.1.
[9]
If the function
p
∈
P
β, then
2(1
−
β
)
p2
=
p
21+
x
(4(1
−
β
)
2−
p
21)
4(1
−
β
)
2p
3=
p
31+ 2(4(1
−
β
)
2−
p
12)
p
1x
−
p
1(4(1
−
β
)
2−
p
21)
x
2+2(1
−
β
)(4(1
−
β
)
2−
p
21)(1
− |
x
|
2)
z
for some
x, z
with
|
x
| ≤
1
and
|
z
| ≤
1
.
Lemma 1.2.
[9]
If the function
p
∈
P
β, then
|
p
n| ≤
2(1
−
β
)
(
n
∈
N
=
{
1
,
2
, . . .
}
)
.
Lemma 1.3.
[9]
If the function
p
∈
P
β, then, for all
n
and
s
(1
≤
s < n
)
,
|
(1
−
α
)
µp
n−
p
n−sp
s| ≤
2(2
−
µ
)(1
−
β
)
2;
µ
≤
1
2(1
−
β
)
2µ
;
µ
≥
1
.
The purpose of this paper is to find the upper bounds of generalized Hankel determinant
a
na
n+2−
µa
2n+12.
Main results
Theorem 2.1.
Let
f
given by (1) be in the class
Q
λ(
β
)
,
0
≤
β <
1
and
n
= 2
,
3
, . . . .
Then
a
na
n+2−
µa
2n+1≤
4(1−β)2
1+2nλ+(n2−1)λ2
h
1
−
1+2nλ+(n2−1)λ2(1+nλ)2
µ
i
;
µ
≤
0
4(1−β)2
1+2nλ+(n2−1)λ2
;
0
≤
µ
≤
(1+nλ)2 1+2nλ+(n2−1)λ2
4(1−β)2
1+2nλ+(n2−1)λ2
h
1+2nλ+(n2−1)λ2(1+nλ)2
µ
−
1
i
;
µ
≥
1+2nλ(1++(nλn2)−1)2 λ2.
(2)
The equality is satisfied for the function
f
(
z
) = (1
−
λ
)
z
+ (1
−
2
β
)
z
2
1
−
z
−
λ
[(1
−
2
β
)
z
+ 2(1
−
β
) log(1
−
z
)] (
µ
≤
0)
.
Proof.
Let
f
∈
Q
λ(
β
)
.
Then
(1
−
λ
)
f
(
z
)
z
+
λf
0(
z
) =
p
(
z
)
or equivalently,
1 + (1 +
λ
)
a
2z
+
· · ·
+ [1 + (
n
−
1)
λ
]
a
nz
n−1+ [1 +
nλ
]
a
n+1z
n+ [1 + (
n
+ 1)
λ
]
a
n+2z
n+1+
· · ·
= 1 +
p
1z
+
· · ·
+
p
n−1z
n−1+
p
nz
n+
p
n+1z
n+1+
· · ·
.
(3)
It follows that
a
n=
p
n−11 + (
n
−
1)
λ
, a
n+1=
p
n1 +
nλ
and
a
n+2=
p
n+11 + (
n
+ 1)
λ
.
This gives us that
a
na
n+2−
µa
2n+1=
1
1 + 2
nλ
+ (
n
2−
1)
λ
2p
n−1pn+1−
1 + 2
nλ
+ (
n
2−
1)
λ
2(1 +
nλ
)
2µp
2 n
.
Applying Lemma 1.3, we get
a
na
n+2−
µa
2n+1≤
4(1−β)2
1+2nλ+(n2−1)λ2
h
1
−
1+2nλ+(n2−1)λ2(1+nλ)2
µ
i
;
µ
≤
0
4(1−β)2
1+2nλ+(n2−1)λ2
;
0
≤
1+2nλ+(n2−1)λ2
(1+nλ)2
µ
≤
1
4(1−β)2 1+2nλ+(n2−1)λ2
h
1+2nλ+(n2−1)λ2
(1+nλ)2
µ
−
1
i
;
1+2nλ(1++(nλn2)−1)2 λ2µ
≥
1
This gives the bound on
a
na
n+2−
µa
2n+1as asserted in (2).
Theorem 2.2.
Let
f
given by (1) be in the class
Q
λ(
β
)
and
µ
∈
R
.
Then
a
3−
µa
22≤
2(1−β) 1+2λ
h
1
−
2(1−β)(1+2λ)(1+λ)2
µ
i
;
µ
≤
0
2(1−β)
1+2λ
;
0
≤
µ
≤
(1+λ)2 (1−β)(1+2λ)
2(1−β) 1+2λ
h
2(1−β)(1+2λ) (1+λ)2
µ
−
1
i
;
µ
≥
(1−(1+β)(1+2λ)2 λ)The equality is satisfied for the function
f
(
z
) = (1
−
λ
)
z
+ (1
−
2
β
)
z
2
1
−
z
−
λ
[(1
−
2
β
)
z
+ 2(1
−
β
) log(1
−
z
)] (
µ
≤
0)
.
Proof.
From (3)
(1 + 2
λ
)
a
3−
µ
(1 +
λ
)
2a
22=
p
2−
µp
21,
(4)
and from this equation (4), we obtain
(1 + 2
λ
)
a
3−
µ
(1 +
λ
)
21 + 2
λ
a
2 2=
p
2−
µp
21.
Our result now follows by an application of Lemma 4:
a
3−
µa
22≤
1
1 + 2
λ
2(1
−
β
)
h
1
−
2(1−β)(1+2λ)(1+λ)2
µ
i
;
µ
≤
0
2(1
−
β
);
0
≤
(1+2λ)(1+λ)2
µ
≤
1 1−β2(1
−
β
)
h
2(1−β)(1+2λ)(1+λ)2
µ
−
1
i
;
(1+2λ)(1+λ)2
µ
≥
1 1−β.
Remark 2.1.
Putting
λ
= 0
in Theorem 2.2 we have the generalized Hankel determinant
for the well-known class
Q
λ(
β
) =
Q
(
β
)
as in
[10]
.
Corollary 2.1.
Let
f
given by (1) be in the class
Q
(
β
)
and
0
≤
β <
1
.
Then
a3
−
µa
22≤
2(1
−
β
) [1
−
2(1
−
β
)
µ
] ;
µ
≤
0
2(1
−
β
);
0
≤
µ
≤
11−β
2(1
−
β
) [2(1
−
β
)
µ
−
1] ;
µ
≥
1−1β.
Remark 2.2.
Putting
λ
= 1
in Theorem 2.2 we have the generalized Hankel determinant
for the well-known class
Q
λ(
β
) =
R
(
β
)
as in
[10]
.
Corollary 2.2.
Let
f
given by (1) be in the class
R
(
β
)
and
0
≤
β <
1
.
Then
a
3−
µa
22≤
2(1−β) 3
1
−
32
(1
−
β
)
µ
;
µ
≤
0
2
3
(1
−
β
);
0
≤
µ
≤
4 3(1−β)
2(1−β) 3
32
(1
−
β
)
µ
−
1
;
µ
≥
3(1−4β).
Theorem 2.3.
Let
f
given by (1) be in the class
Q
λ(
β
)
and
0
≤
β <
1
.
Then
a
2a
4−
µa
23≤
3B2−4B+9
8(1+λ)(1+3λ)(1−B)
;
f or
(1+2λ)2
2(1+λ)(1+3λ)
≤
µ
≤
(1+2λ)2 (1+λ)(1+3λ)
,
where
B
=
(1+λ)(1+3λ)(1+2λ)2
µ.
Proof.
Using (3), one can see easily that
a
2a
4−
µa
23=
(1+λ)(1+31 λ)p
1p
3−
(1+λ)(1+3λ) (1+2λ)2
µp
Then, Lemma 2.1 gives us that
p
1p
3−
(1+λ)(1+3λ) (1+2λ)2
µp
2 2
=
1 4(1−β)2
p
4 1
+ 2
n
4 (1
−
β
)
2−
p
21o
p
21x
−
n
4 (1
−
β
)
2−
p
21o
p
21x
2+2 (1
−
β
)
n
4 (1
−
β
)
2−
p
21o
p
11
− |
x
|
2z
−
(1+λ)(1+3λ)(1+2λ)2
µ
p
41+
h
4 (1
−
β
)
2−
p
21i2
x
2+ 2
n
4 (1
−
β
)
2−
p
21o
p
21x
.
Letting
|
p
1|
=
p
, for
η
=
|
x
| ≤
1
,
|
z
| ≤
1 we get
p
1p
3−
(1+λ)(1+3λ) (1+2λ)2
µp
2 2
≤
1 4(1−β)2
nh
1
−
(1+λ)(1+3λ)(1+2λ)2
µ
i
p
4+2
h
1
−
(1+λ)(1+3λ)(1+2λ)2
µ
i h
4 (1
−
β
)
2−
p
2i
p
2η
+
h
4 (1
−
β
)
2−
p
2i h
p
2+
(1+λ)(1+3λ) (1+2λ)2µ
h
4 (1
−
β
)
2−
p
2i
−
2 (1
−
β
)
p
i
η
2+2 (1
−
β
)
h
4 (1
−
β
)
2−
p
2i
p
o
=
4(1−F(η)β)2
.
For
F
(
η
)
,
we see that
F
0(
η
)
=
2
h
4 (1
−
β
)
2−
p
2i
×
1
−
(1 +
λ
) (1 + 3
λ
)
(1 + 2
λ
)
2µ
p
2+
p
2+
(1+λ)(1+3λ)(1+2λ)2
µ
h
4 (1
−
β
)
2−
p
2i
−
2 (1
−
β
)
p
η
.
Therefore, if
(1 + 2
λ
)
22(1 +
λ
) (1 + 3
λ
)
≤
µ
≤
(1 + 2
λ
)
2(1 +
λ
) (1 + 3
λ
)
,
then
F
0(
η
) satisfies
F
0(
η
)
=
2
h
4 (1
−
β
)
2−
p
2i h
p
2+
124 (1
−
β
)
2−
p
2−
2 (1
−
β
)
p
i
η
=
h
4 (1
−
β
)
2−
p
2i
[
p
−
2 (1
−
β
)]
2η
≥
0
,
because 0
≤
η
≤
1 and 0
≤
p
≤
2 (1
−
β
)
.
Writing that
F
(1)
≡
G
(
p
)
=
−
1
A
2(1
−
B
)
p
4−
A
(3
−
4
B
)
p
2+
A
2B
,
where
A
= 4 (1
−
β
)
2and
B
=
(1+λ)(1+3λ)we obtain that
G
0(
p
)
=
−
1
A
8(1
−
B
)
p
3−
2
A
(3
−
4
B
)
p
=
−
8(1
−
B
)
p
A
p
2−
A
(3
−
4
B
)
4(1
−
B
)
=
0
for
p
= 0 and
p
=
(1−β)√ 3−4B
√
1−B
.
(i)
If 0
≤
3−4B1−B
≤
4
,
then
(1+λ)(1+3λ) (1+2λ)2
µ
≤
3 4
.
In this case,
G
(
p
) has the maximum value
G
(
p
)
=
−
1
4(1
−
β
)
22(1
−
β
)
4(3
−
4
B
)
21
−
B
−
4(1
−
β
)
4(3
−
4
B
)
21
−
B
+ 16(1
−
β
)
4
B
=
(1
−
β
)
24(1
−
3
B
) +
1
2(1
−
B
)
.
(ii)
If
3−41−BB>
4
,
then we have the contradiction for
G
(
p
)
.
(iii)
If
p
= 0
,
then
G
(
p
) takes its minimal value.
Consequently, we say that
p1p3
−
(1 +
λ
) (1 + 3
λ
)
(1 + 2
λ
)
2µp
2 2
≤
G
(
p
)
4(1
−
β
)
2=
3
B
2−
4
B
+ 9
8(1
−
B
)
.
This completes the proof of the theorem.
References
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[3] Fekete, M. and Szeg¨o, G., (1933), Eine Bemerkung ¨Uber Ungerade Schlichte Funktionen, J. Lond. Math. Soc., 2, pp. 85-89.
[4] El-Ashwah, R. and Kanas, S., (2015), Fekete-Szeg¨o inequalities for quasi-subordination functions classes of complex order, Kyungpook Math. J., 55, pp. 679-688.
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Sibel Yal¸cın received her Ph.D. degree in Mathematics in 2001 from Uludag Uni-versity, Bursa, Turkey. She became a full Professor in 2011. She is currently with the Department of Mathematics, Uludag University. Her research interests include harmonic mappings, geometric function theory, meromorphic functions, analytic func-tions, bi-univalent funcfunc-tions, convolution operators.
S¸ahsene Altınkayawas born in 1990. She received her B.S. degree in mathemat-ics (2012) from Erciyes University and her M.S. degree in mathematmathemat-ics (2014) from Uludag University, Turkey. She is a Research Assistant of the Department of Mathe-matics Faculty of Arts and Science, Uludag University since 2013. Her research areas include geometric function theory, analytic functions and bi-univalent functions.
