International Journal of Analysis and Applications
ISSN 2291-8639
Volume 8, Number 1 (2015), 22-29
http://www.etamaths.com

SECOND HANKEL DETERMINANT FOR BI-UNIVALENT

ANALYTIC FUNCTIONS ASSOCIATED WITH HOHLOV

OPERATOR

G. MURUGUSUNDARAMOORTHY∗ AND K. VIJAYA

Abstract. In the present paper, we consider a subclass of the function class

Σ of bi-univalent analytic functions in the open unit disk ∆ associated with
Hohlov operator and we obtain the functional |a2a4−a23| for the function class
Σ . Our result gives corresponding |a2a4 −a23| for the subclasses of Σ defined
in the literature.

1. Introduction

Let A be the class of functions given by the power series

(1.1) f(z) = z +

∞∑
n=2

anz
n (z ∈ ∆).

and analytic in the open unit disk

∆ := {z : z ∈ C and |z| < 1}.
Also let Ω be the family of functions f ∈A which are univalent in ∆ and satisfying
the normalization conditions (see[4]):

f(0) = f′(0) − 1 = 0.
The well-known Koebe one-quarter theorem (see[4]) asserts that the image of ∆

under every univalent function f ∈ Ω contains a disk of radius 1
4
. Thus, the inverse

of f ∈ Ω is a univalent analytic function on the disk ∆ρ := {z : z ∈ C and |z| <
ρ; ρ ≥ 1

4
}. Therefore, for each function f(z) = w ∈ Ω, there is an inverse function

f−1(w) of f(z) defined by

f−1(f(z)) = z (z ∈ ∆)
and

f(f−1(w)) = w (w ∈ ∆ρ)
where

(1.2) g(w) = f−1(w) = w −a2w2 + (2a22 −a3)w
3 − (5a32 − 5a2a3 + a4)w

4 + ....

A function f ∈ Ω is said to be bi-univalent in ∆ if both f and f−1 are univalent in ∆.
Let Σ denote the class of bi-univalent function in ∆ given by (1.1). The concept of
bi-univalent analytic functions was introduced by Lewin [14] in 1967 and he showed

2010 Mathematics Subject Classification. 30C45.
Key words and phrases. Univalent functions; Analytic functions; Bi-univalent functions;

Hohlov operator; Coefficient bounds.

c©2015 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

22



BI-UNIVALENT ANALYTIC FUNCTIONS 23

that |a2| < 1.51. Subsequently, Brannan and Clunie [1] conjectured that |a2| ≤
√

2.
Netanyahu [18], on the other hand, showed that maxf∈Σ |a2| = 43. The coefficient
estimate problem for each of the following Taylor-Maclaurin coefficients |an| (n ∈
N \ {1, 2})is presumably still an open problem. In [3](see also [2, 7, 20, 22, 23]),
certain subclasses of the bi-univalent analytic functions class Σ were introduced
and non-sharp estimates on the first two coefficients |a2| and |a3| were found.

In 1976, Noonan and Thomas [19] defined the qth Hankel determinant of f for
q ≥ 1 by

Hq(n) =

∣∣∣∣∣∣∣∣∣
an an+1 . . . an+q−1
an+1 an+2 . . . an+q

...
...

...
...

an+q−1 an+q . . . an+2q−2

∣∣∣∣∣∣∣∣∣
.

Further, Fekete and Szegö [6] considered the Hankel determinant of f ∈ A for

q = 2 and n = 1, H2(1) =

∣∣∣∣ a1 a2a2 a3
∣∣∣∣ . They made an early study for the estimates

of |a3 −µa22| when a1 = 1 with µ real. The well known result due to them states
that if f ∈A, then

|a3 −µa22| ≤




4µ− 3 if µ ≥ 1,
1 + 2 exp(−2µ

1−µ) if 0 ≤ µ ≤ 1,
3 − 4µ if µ ≤ 0.

Furthermore, Hummel [9, 10] obtained sharp estimates for |a3 − µa22| when f
is convex functions and also Keogh and Merkes [13] obtained sharp estimates for
|a3 − µa22| when f is close-to-convex, starlike and convex in ∆. Here we consider
the Hankel determinant of f ∈A for q = 2 and n = 2,

H2(2) =

∣∣∣∣ a2 a3a3 a4
∣∣∣∣ .

For the functions f, g ∈A and given by the series

f(z) =

∞∑
n=0

anz
n and g(z) =

∞∑
n=0

bnz
n (z ∈ ∆),

the Hadamard product (or convolution) of f and g denoted by f ∗g is defined as

(f ∗g)(z) =
∞∑
n=0

anbnz
n = (g ∗f)(z) (z ∈ ∆).

By using the Hadamard product (or convolution ), Hohlov (cf.[11]) introduced and
studied the linear operator Ia,bc : Ω → Ω defined by

Ia,bc f(z) = z2F1(a,b; c; z) ∗f(z) (f ∈ Ω,z ∈ ∆),

where 2F1(z) known as Gaussian hypergeometric function is defined by
(1.3)

2F1(z) = 2F1(a,b; c; z) =

∞∑
n=0

(a)n(b)n
(c)n(1)n

zn (a,b ∈ C,c ∈ C\Z−0 =: {0,−1,−2, . . .})



24 MURUGUSUNDARAMOORTHY AND VIJAYA

and (λ)n is the Pochhamer symbol or shifted factorial, written in terms of the
gamma function Γ, by

(λ)n =
Γ(λ + n)

Γ(λ)
=

{
1, n = 0

λ(λ + 1)....(λ + n− 1), n ∈ N := {1, 2, 3, .....}.

Note that 2F1(z) is symmetric in a and b and that the series (1.3) terminates if at
least one of the numerator parameter a and b is zero or a negative integer.Observe
that for the function f of the form (1.1), we have

Ia,bc f(z) = z +
∞∑
n=2

(a)n−1(b)n−1
(c)n−1(1)n−1

anz
n

= z +

∞∑
n=2

Φnanz
n (z ∈ ∆),(1.4)

where

Φn =
(a)n−1(b)n−1
(c)n−1(1)n−1

.

Making use of Hohlov operator we consider a new subclass of Σ due to Panigarhi
and Murugusundaramoorthy[20] as given below

Definition 1.1. [20] A function f ∈ Σ and of the form (1.1)is said to be in the
class Ma,b;cΣ (β,λ) if the following conditions are satisfied:

(1.5) <
[
(1 −λ)

Ia,bc f(z)
z

+ λ
(
Ia,bc f(z)

)′]
> β (0 ≤ β < 1,λ ≥ 1,z ∈ ∆)

and

(1.6) <
[
(1 −λ)

Ia,bc g(w)
w

+ λ
(
Ia,bc g(w)

)′]
> β (0 ≤ β < 1,λ ≥ 1,w ∈ ∆)

where the function g is the inverse of f given by (1.2).

It is of interest to note that by taking a = b and c = 1 we state the following
subclass FΣ(β,λ) due to Frasin et al.[7].

Example 1.2. [7] A function f ∈ Σ and of the form (1.1) is said to be in the class
FΣ(β,λ) if the following conditions are satisfied:

(1.7) <
[
(1 −λ)

f(z)

z
+ λf′(z)

]
> β (0 ≤ β < 1,λ ≥ 1,z ∈ ∆)

and

(1.8) <
[
(1 −λ)

g(w)

w
+ λg′(w)

]
> β (0 ≤ β < 1,λ ≥ 1,w ∈ ∆)

where the function g is the inverse of f given by (1.2).

It is of interest to note that by taking a = b; c = 1 and λ = 1 we state the
following subclass HΣ(β) due to Srivastava et al.[22]. By taking a = b; c = 1 and
we state the following :



BI-UNIVALENT ANALYTIC FUNCTIONS 25

Example 1.3. [22] A function f ∈ Σ and of the form (1.1) is said to be in the
class HΣ(β) if the following conditions are satisfied:

< [f′(z)] > β (0 ≤ β < 1,z ∈ ∆)

and

< [g′(w)] > β (0 ≤ β < 1,w ∈ ∆)
where the function g is the inverse of f given by (1.2).

The object of the present paper is to determine the functional |a2a4 − a23| for
the function f ∈ Ma,b;cΣ (β,λ). Our result gives corresponding |a2a4 − a

2
3| for the

subclasses of Σ defined in the Examples 1.2 and 1.3.

2. coefficient bounds for the function class Ma,b;cΣ (β,λ)

We need the following lemma for our investigation.

Lemma 2.1. (see [4], p. 41) Let P be the class of all analytic functions p(z) of
the form

(2.1) p(z) = 1 +

∞∑
n=1

pnz
n

satisfying <(p(z)) > 0 (z ∈ ∆) and p(0) = 1. Then

|pn| ≤ 2 (n = 1, 2, 3, ...).

This inequality is sharp for each n. In particular, equality holds for all n for the
function

p(z) =
1 + z

1 −z
= 1 +

∞∑
n=1

2zn.

Lemma 2.2. If the function p ∈P is given by the series

(2.2) 2p2 = p
2
1 + x(4 −p

2
1),

(2.3) 4p3 = p
3
1 + 2(4 −p

2
1)p1x−p1(4 −p

2
1)x

2 + 2(4 −p21)(1 −|x|
2z),

for some x,z with |x| ≤ 1 and |z| ≤ 1.

Lemma 2.3. [8] The power series for p given in (2.1) converges in ∆ to a function
in P if and only if the Toeplitz determinants

(2.4) Dn =

∣∣∣∣∣∣∣∣∣
2 c1 c2 · · · cn
c−1 2 c1 · · · cn−1

...
...

...
...

...
c−n c−n+1 c−n+2 · · · 2

∣∣∣∣∣∣∣∣∣
, n = 1, 2, 3, . . .

and c−k = ck, are all nonnegative. They are strictly positive except for

p(z) =

m∑
k=1

ρkp0(e
itkz), ρk > 0, tk real

and tk 6= tj for k 6= j in this case Dn > 0 for n < m− 1 and Dn = 0 for n ≥ m.

In the following theorem we determine the second hankel coefficient results for



26 MURUGUSUNDARAMOORTHY AND VIJAYA

Theorem 2.4. Let f ∈Ma,b;cΣ (β,λ) be given by (1.1). Then

(2.5)

|a2a4−a23| ≤




4(1 −β2)
[

(1+λ)3Φ32+4(1−β)
2(1+3λ)Φ4

(1+λ)4(1+3λ)Φ42Φ4

]
, β ∈

[
0, 1 −

√
(1+λ)3 Φ32
8(1+3λ)Φ4

]
9(1+λ)2(1−β)2Φ22

2(1+3α)Φ4[(1+λ)3Φ
3
2−2(1−β)2(1+3λ)Φ4]

, β ∈
(

1 −
√

(1+λ)3 Φ32
8(1+3λ)Φ4

, 1

)
.

Proof. Since f ∈ Ma,b;cΣ (β,λ), there exists two functions φ(z) and ψ(z) ∈ P satis-
fying the conditions of Lemma 2.1 such that

(2.6) (1 −λ)
Ia,bc f(z)

z
+ λ

(
Ia,bc f(z)

)′
= β + (1 −β)φ(z)

and

(2.7) (1 −λ)
Ia,bc g(w)

w
+ λ

(
Ia,bc g(w)

)′
= β + (1 −β)ψ(z)

where

(2.8) φ(z) = 1 + c1z + c2z
2 + c3z

3 + ...

and

(2.9) ψ(w) = 1 + d1w + d2w
2 + d3w

3 + ....

. Equating the coefficients in (2.6) and (2.7)gives

(2.10) (1 + λ)Φ2a2 = (1 −β)c1

(2.11) (1 + 2λ)Φ3a3 = (1 −β)c2

(2.12) (1 + 3λ)Φ4a4 = (1 −β)c3
and

(2.13) −(1 + λ)Φ2a2 = (1 −β)d1

(2.14) (1 + 2λ)Φ3(2a
2
2 −a3) = (1 −β)d2

(2.15) −(1 + 3λ)Φ4(5a32 − 5a2a3 + a4) = (1 −β)d3
From (2.10) and (2.13) gives

(2.16) a2 =
1 −β

(1 + λ)Φ2
c1 = −

1 −β
(1 + λ)Φ2

d1

which implies

c1 = −d1
Now from(2.11) and (2.14), we obtain

(2.17) a3 =
(1 −β)2

(1 + λ)2 Φ22
c21 +

(1 −β)
4(1 + 2λ)Φ3

(c1 − c2).

On the other hand, subtracting (2.15) from (2.12) and using (2.16), we get
(2.18)

a4 =
1

2(1 + 3λ)Φ4

[
−5(1 + 3λ)(1 −β)3Φ4

(1 + λ)3Φ32
c31 +

5(1 + 3λ)(1 −β)Φ4
(1 + λ)Φ2

a3c1 + (1 −β)(c3 −d3)
]
.



BI-UNIVALENT ANALYTIC FUNCTIONS 27

Thus we establish that

(2.19) |a2a4 −a23| =
∣∣∣∣− (1 −β)4(1 + λ)4Φ42 c41 + (1 −β)

3c21(c2 −d2)
8(1 + λ)2(1 + 2λ)Φ22Φ3

+
(1 −β)2

2(1 + λ)(1 + 3λ)Φ4Φ2
c1(c3 −d3) − (1 −β)2(c2 −d2)2

∣∣∣∣ .
According to Lemma2.2 we have

2c2 = c
2
1 + x(4 − c

2
1), and 2d2 = d

2
1 + x(4 −d

2
1),

hence we have

(2.20) c2 = d2

and further

4c3 = c
3
1 + 2(4 − c

2
1)c1x− c1(4 − c

2
1)x

2 + 2(4 − c21)(1 −|x|
2z),

4d3 = d
3
1 + 2(4 −d

2
1)d1x−d1(4 −d

2
1)x

2 + 2(4 −d21)(1 −|x|
2z)

(2.21) c3 −d3 =
1

2
c31 + c1(4 − c

2
1)x−

1

2
c1(4 − c21)x

2

(2.22) |a2a4 −a23| =
∣∣∣∣ −(1 −β)4(1 + λ)4Φ42 c41 + (1 −β)

2

4(1 + λ)(1 + 3λ)Φ4Φ2
c41

+
(1 −β)2c21(4 − c21)x

2(1 + λ)(1 + 3λ)Φ4Φ2
−

(1 −β)2c21(4 − c21)x2

4(1 + λ)(1 + 3λ)Φ4Φ2

∣∣∣∣
Letting c1 = c, we may assume without restriction that c ∈ [0, 2] since φ ∈ P so
|c1| ≤ 2.Thus,applying triangle inequality on (2.19),with µ = |x| ≤ 1, we obtain

(2.23) |a2a4 −a23| ≤
(1 −β)4

(1 + λ)4Φ42
c4 +

(1 −β)2

4(1 + λ)(1 + 3λ)Φ4Φ2
c4

+
(1 −β)2c2(4 − c2)µ

2(1 + λ)(1 + 3λ)Φ4Φ2
+

(1 −β)2c2(4 − c2)µ2

4(1 + λ)(1 + 3λ)Φ4Φ2
= F(µ)

Differentiating F(µ), we get

F ′(µ) =
(1 −β)2c21(4 − c21)

4(1 + λ)(1 + 3λ)Φ4Φ2
+

(1 −β)2c2(4 − c2)µ
2(1 + λ)(1 + 3λ)Φ4Φ2

By using elementary calculus, one can show that F ′(µ) > 0 for µ > 0 hence F is
an increasing function and thus ,the upper bound for F(µ) corresponds to µ = 1,in
which case

(2.24) F(µ) = F(1) =

[
(1 −β)4

(1 + λ)4Φ42
+

(1 −β)2

4(1 + λ)(1 + 3λ)Φ4Φ2

]
c4

+
3(1 −β)2c2(4 − c2)

4(1 + λ)(1 + 3λ)Φ4Φ2
= G(c)

Assume that G(c) has a maximum value in an interior of c ∈ [0, 2], by elementary
calculations we find

(2.25) G′(c) =

[
4(1 −β)4

(1 + λ)4Φ42
−

2(1 −β)2

(1 + λ)(1 + 3λ)Φ4Φ2

]
c3+

6(1 −β)2c
(1 + λ)(1 + 3λ)Φ4Φ2

.



28 MURUGUSUNDARAMOORTHY AND VIJAYA

Then G′(c) = 0 implies the real critical point c01 = 0 or c02 =
√

3(1+λ)3 Φ32
(1+λ)3Φ32−2(1−β)2(1+3λ)Φ4

.

After some calculations we concluded following cases:

Case 1: When β ∈
[
0, 1 −

√
(1+λ)3 Φ32
8(1+3λ)Φ4

]
, we observe that c02 ≥ 2, that is,

c02 is out of the interval (0, 2). Therefore the maximum value of G(c) occurs at
c01 = 0 or c = c02 which contradicts our assumption of having the maximum value
at the interior point of c ∈ [0, 2]. Since G is an increasing function in the interval
[0, 2], maximum point of G must be on the boundary of c ∈ [0, 2], that is, c = 2.
Thus, we have

max
0≤c≤2

G1(p) = G(2) = 4(1 −β2)
[

(1 + λ)3Φ32 + 4(1 −β)2(1 + 3λ)Φ4
(1 + λ)4(1 + 3λ)Φ42Φ4

]
Case 2: When β ∈

(
1 −

√
(1+λ)3 Φ32
8(1+3λ)Φ4

, 1

)
, we observe that c02 ≤ 2, that is,

c02 is interior of the interval [0, 2]. Since G
′′(c02) < 0, the maximum value of G(c)

occurs at c = c02. Thus, we have

max
0≤c≤2

G(c) = G(c02) = G

(√
3(1 + λ)3 Φ32

(1 + λ)3Φ32 − 2(1 −β)2(1 + 3λ)Φ4

)

=
9(1 + λ)2(1 −β)2Φ22

2(1 + 3α)Φ4[(1 + λ)3Φ
3
2 − 2(1 −β)2(1 + 3λ)Φ4]

.

�

Concluding Remarks: Suitably specializing the parameter λ one can state the

Hankel coefficients for various subclasses ofMa,b;cΣ (β,λ). In fact, by choosing a = b
and c = 1 we have Φ2 = 1; Φ3 = 1; Φ4 = 1 hence we state the Hankel determinant
coefficients for the function f ∈FΣ(β,λ) studied in[7] as given below:
(2.26)

|a2a4 −a23| ≤




4(1 −β2)
[

(1+λ)3+4(1−β)2(1+3λ)
(1+λ)4(1+3λ)

]
, β ∈

[
0, 1 −

√
(1+λ)3

8(1+3λ)

]
9(1+λ)2(1−β)2

2(1+3α)[(1+λ)3−2(1−β)2(1+3λ)], β ∈
(

1 −
√

(1+λ)3

8(1+3λ)
, 1
)
.

Also by choosing λ = 1 one can easily derive Hankel determinant |a2a4 − a23| for
the functions f ∈HΣ studied by Srivastava et al.[22].

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School of Advanced Sciences, VIT University, Vellore - 632014, Tamilnadu, India

∗Corresponding author