International Journal of Analysis and Applications
ISSN 2291-8639
Volume 6, Number 2 (2014), 170-177
http://www.etamaths.com

HANKEL DETERMINANT FOR A CLASS OF ANALYTIC

FUNCTIONS RELATED WITH LEMNISCATE OF BERNOULLI

ASHOK KUMAR SAHOO1 AND JAGANNATH PATEL2,∗

Abstract. The object of the present investigation is to solve Fekete-Szegö

problem and determine the sharp upper bound to the second Hankel determi-

nant for a new class R̃ of analytic functions in the unit disk.

1. Introduction and preliminaries

Let A be the class of functions f of the form

(1.1) f(z) = z +

∞∑
n=2

anz
n

which are analytic in the open unit disk U = {z ∈ C : |z| < 1}.
A function f ∈ A is said to be starlike of order ρ and convex of order ρ, if and

only if Re{zf′(z)/f(z)} > ρ and Re{(1 + zf′′(z))/f′(z)} > ρ for 0 ≤ ρ < 1 and
z ∈U. By usual notations, we write these classes of functions by S ?(ρ) and K (ρ),
respectively. We denote S ?(0) = S ? and K (0) = K , the familiar subclasses of
starlike and convex functions in U.

Further, we say that a function f ∈ A is in the class R(ρ), if it satisfies the
inequality:

(1.2) Re{f′(z)} > ρ (z ∈U)

We note that R(ρ) is a subclass of close-to-convex functions order
ρ(0 ≤ ρ < 1) in U. We write R(0) = R, the familiar class functions in A whose
derivatives have a positive real part in U.

A function f is said to be subordinate to a function g, written as f ≺ g, if
there exists a Schwarz function w with w(0) = 0 and |w(z)| < 1 such that f(z) =
g(w(z)),z ∈ U. In particular, if g is univalent in U, then f(0) = g(0) and f(U) ⊂
g(U).

Let P denote the class of analytic functions φ normalized by

(1.3) φ(z) = 1 + p1z + p2z
2 + · · · (z ∈U)

such that Re{φ(z)} > 0 in U.

2010 Mathematics Subject Classification. 30C45.
Key words and phrases. Analytic function; Subordination; Fekete-Szegö problem; Hankel

determinant.

c©2014 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

170



ANALYTIC FUNCTIONS RELATED WITH LEMNISCATE OF BERNOULLI 171

Definition. A function f ∈ A is said to be in the class R̃, if it satisfies the
condition

(1.4)
∣∣∣(f′(z))2 − 1∣∣∣ < 1 (z ∈U).

It follows from (1.4) and the definition of subordination that a function f ∈ R̃
satisfies the following subordination relation

(1.5) f′(z) ≺
√

1 + z (z ∈U).

To bring out the geometrical significance of the class R̃, we set

h(z) =
√

1 + z, z ∈U

and note that

ω = h(eiθ) =
√

1 + eiθ (0 ≤ θ ≤ 2π).
which yields ω2 − 1 = eiθ or |ω2 − 1| = 1. Letting ω = u + iv, we deduce that

(u2 + v2)2 = 2(u2 −v2).

Thus, h(U) is the region bounded by the right half of the lemniscate of Bernoulli
given by

{
u + iv ∈ C : (u2 + v2)2 = 2(u2 −v2)

}
, which implies that the derivative

of functions in R̃ have a positive real part and hence univalent in U [1].
Noonan and Thomas [12] defined the q-th Hankel determinant of the function f,

given by (1.1) by

(1.6) Hq(n) =

∣∣∣∣∣∣∣∣∣
an an+1 · · · an+q−1
an+1 an+2 · · · an+q

...
...

...
...

an+q−1 an+q · · · an+2q−2

∣∣∣∣∣∣∣∣∣
(a1 = 1,n,q ∈ N).

The determinant given in (1.6) has been studied by several authors with the
subject of inquiry ranging from the rate of growth of Hq(n) (as n →∞) [13] to the
determination of precise bounds with specific values of n and q for certain subclasses
of analytic functions in the unit disc U.

For n = 1,q = 2,a1 = 1 and n = q = 2, the Hankel determinant simplifies
to H2(1) = |a3 −a22| and H2(2) = |a2a4 −a23|. We refer to H2(2) as the second
Hankel determinant. It is known [1] that if the function f, given by (1.1) is analytic
and univalent in U, then the sharp inequality H2(1) = |a3 −a22| ≤ 1 holds. For a
family F of functions in A of the form (1.1), the more general problem of finding
the sharp upper bounds for the functionals |a3−µa22| (µ ∈ R or µ ∈ C) is popularly
known as Fekete-Szegö problem for the class F. The Fekete-Szegö problem for the
known classes of univalent functions, starlike functions, convex functions and close-
to-convex functions has been completely settled ([2], [5], [6], [7]). Recently, Janteng
et al. [3, 4] have obtained the sharp upper bounds to the second Hankel determinant
H2(2) for the family R. For initial work on the class R one may refer to the paper
by MacGregor [11].

In our present investigation, by following the techniques devised by Libera and
Zlotkiewicz [8, 9], we solve the Fekete-Szegö problem and also determine the sharp

upper bound to the second Hankel determinant H2(1) for the class R̃.
To establish our main results, we shall need the followings lemmas.



172 SAHOO AND PATEL

Lemma 1.1. Let the function φ, given by (1.3) be a member of the class P. Then

(1.7) |pk| ≤ 2 (k ≥ 1)

and

(1.8)
∣∣p2 −ν p21∣∣ ≤ 2 max{1, |2ν − 1|}.

The estimate (1.7) is sharp for the function ϕ(z) = (1+z)/(1−z),z ∈U, whereas the
estimate (1.8) is sharp for the functions given by ϕ and ψ(z) = (1+z2)/(1−z2), z ∈
U.

We note that the estimate (1.7) is contained in [1] and the estimate (1.8) is
obtained in [10].

Lemma 1.2 ([9],see also [8]). If the function φ, given by (1.3) belongs to the class
P, then

(1.9) p2 =
1

2

{
p21 + (4 −p

2
1)x
}

and

(1.10) p3 =
1

4

{
p31 + 2(4 −p

2
1)p1x− (4 −p

2
1)p1x

2 + 2(4 −p21)(1 −|x|
2)z
}

for some complex numbers x,z satisfying |x| ≤ 1 and |z| ≤ 1.

2. Main results

Now, we determine an upper bound for the Fekete-Szegö problem of the class

R̃.

Theorem 2.1. If the function f, given by (1.1) belongs to the class R̃, then for
any µ ∈ C

(2.1) |a3 −µa22| ≤
1

6
max

{
1,
|2 + 3 µ|

8

}
.

The estimate in (2.1) is sharp.

Proof. From (1.5), it follows that

(2.2) f′(z) =
√

1 + w(z) (z ∈U),

where w is analytic and satisfies the condition w(0) = 0 and |w(z)| < 1 in U.
Setting

(2.3) χ(z) =
1 + w(z)

1 −w(z)
= 1 + p1z + p2z

2 + · · · (z ∈U),

we see that χ ∈ P. From (2.3), we get

(2.4) w(z) =
χ(z) − 1
χ(z) + 1

(z ∈U)

so that by (2.2) and (2.4), we get

(2.5) f′(z) =

(
2χ(z)

1 + χ(z)

)1
2

(z ∈U).



ANALYTIC FUNCTIONS RELATED WITH LEMNISCATE OF BERNOULLI 173

Now, by substituting the series expansion of χ from (2.3) in (2.5), it is easily seen
that (

2χ(z)

1 + χ(z)

)1
2

= 1 +
1

4
p1z +

(
1

4
p2 −

5

32
p21

)
z2 +

(
1

4
p3 −

5

16
p1p2 +

13

128
p31

)
z3 + · · · .(2.6)

Differentiating the series expansion of f given by (1.1) with respect to z and com-
paring the coefficients of z,z2 and z3 in (2.6), we deduce that

a2 =
1

8
p1(2.7)

a3 =
1

12

(
p2 −

5

8
p21

)
(2.8)

a4 =
1

16

(
p3 −

5

4
p1p2 +

13

32
p31

)
.(2.9)

Thus, by using (2.7) and (2.8), we get

(2.10)
∣∣a3 −µa22∣∣ = 112

∣∣∣∣p2 − 116 (10 + 3µ)p21
∣∣∣∣

The expression in (2.10) with the aid of (1.8) yields the required estimate (2.1).
The estimate in (2.1) is sharp for the function f0 ∈ A defined by

(2.11) f′0(z) =

{√
1 + z2, |2 + 3 µ| ≤ 8
√

1 + z, |2 + 3 µ| > 8.

This completes the proof of Theorem 2.1. �

Letting µ = 0(or µ = 1 respectively) in Theorem 2.1, we get

Corollary 2.1. If the function f, given by (1.1) belongs to the class R̃, then

|a3| ≤
1

6
and |a3 −a22| ≤

1

6
.(2.12)

The estimates in (2.12) are sharp for the function f0 ∈ A defined by

(2.13) f′0(z) =
√

1 + z2 (z ∈U).

If µ ∈ R, then Theorem 2.1 reduces to

Corollary 2.2. Let µ ∈ R. If the function f, given by (1.1) belongs to the class
R̃, then

(2.14)
∣∣a3 −µa22∣∣ ≤



−

2 + 3µ

48
, µ ≤−

10

3
1

6
, −

10

3
≤ µ ≤ 2

2 + 3µ

48
, µ > 2.

The estimates in (2.14) are sharp.



174 SAHOO AND PATEL

Proof. First, we assume that µ < −10/3. Then, (2+3µ)/8 < −1 so that |2+3µ|/8 >
1. Hence by using (2.1), we get

(2.15) |a3 −µa22| ≤
|2 + 3µ|

48
= −

2 + 3µ

48
.

Next, if −10/3 ≤ µ ≤ 2, then |2 + 3µ| ≤ 1 so that

(2.16) |a3 −µa22| ≤
1

6

again by the use of (2.1). Finally, if µ > 2, then (2 + 3µ)/8 > 1. Thus, by (2.1)

(2.17) |a3 −µa22| ≤
2 + 3µ

48
.

The estimates are sharp for the function f1 defined in U by f′1(z) =
√

1 + z,
for µ < −10/3 or µ > 2, and for the function f0 given by (2.13) in the case
−10/3 ≤ µ ≤ 2. �

In the following theorem, we find the sharp upper bound to the second Hankel

determinant for the class R̃.

Theorem 2.2. Let the function f, given by (1.1) be a member of the family R̃.
Then

(2.18)
∣∣a2a4 −a23∣∣ ≤ 136.

The estimate in (2.18) is sharp.

Proof. From (2.7), (2.8) and (2.9), we have∣∣a2a4 −a23∣∣ =
∣∣∣∣ 1128

(
p1p3 −

5

4
p21p2 +

13

32
p41

)
−

1

144

(
p22 −

5

4
p21p2 +

25

64
p41

)∣∣∣∣
=

1

16

∣∣∣∣18p1p3 − 5288p21p2 − 19p22 + 172304p41
∣∣∣∣ .(2.19)

Since the function χ, given by (2.3) and the function χ(eiθz) (θ ∈ R) are in the
class P simultaneously, we assume without loss of generality that p1 > 0. For
convenience of notation, we write p1 = p (0 ≤ p ≤ 2). Now, by using Lemma 2.2 in
(2.19), we get∣∣a2a4 −a23∣∣

=
1

16

∣∣∣∣
(

1

32
p4 +

1

16
(4 −p2)p2x−

1

32
(4 −p2)p2x2 +

1

16
(4 −p2)p(1 −|x|2)z

)
−
(

5

576
p4 +

5

576
(4 −p2)p2x

)
−
(

1

36
p4 +

1

18
(4 −p2)p2x +

1

36
(4 −p2)2x2

)
+

17

2304
p4
∣∣∣∣

=
1

16

∣∣∣∣ 52304p4 − 1576 (4 −p2)p2x− 1288{8(4 −p2) + 9p2}(4 −p2)x2
+

1

16
(4 −p2)p(1 −|x|2)z

∣∣∣∣
(2.20)



ANALYTIC FUNCTIONS RELATED WITH LEMNISCATE OF BERNOULLI 175

for some x (|x| ≤ 1) and for some z (|z| ≤ 1). Applying the triangle inequality in
(2.20) and replacing |x| by y in the resulting equation, we get∣∣a2a4 −a23∣∣ ≤ 116

{
5

2304
p4 +

1

576
(4 −p2)p2y

+
1

288
(4 −p2)(2 −p)(16 −p)y2 +

1

16
(4 −p2)p

}
= G(p,y) (0 ≤ p ≤ 2, 0 ≤ y ≤ 1) (say).(2.21)

We next maximize the function G(p,y) on the closed rectangle [0, 2]× [0, 1]. Differ-
entiating the function G, given in (2.21) with respect to y, we deduce that

(2.22)
∂G
∂y

=
1

9216
(4 −p2)p2 +

1

2304
(4 −p2)(2 −p)(16 −p)y > 0

for 0 < p < 2 and 0 < y < 1. Thus, in view of (2.22), the function G(p,y) cannot
have a maximum in the interior on the closed rectangle [0, 2]× [0, 1]. Therefore, for
fixed p ∈ [0, 2]
(2.23) max

0≤y≤1
G(p,y) = G(p, 1) = F(p) (say),

where

F(p) =
1

16

{
5

2304
p4 +

1

576
(4 −p2)p2

+
1

288
(4 −p2)(2 −p)(16 −p) +

1

16
(4 −p2)p

}
(0 ≤ p ≤ 2).(2.24)

On differentiating the function F , given by (2.24) followed by a simple calculation
yields

F ′(p) = −
1

9216
(7p2 + 104)p < 0 which implies that the function F is a decreasing

function of p so that max0≤p≤2 F(p) occurs at p = 0. Thus, the upper bound in
(2.21) corresponds to p = 0 and y = 1 from which we get the required estimate
(2.18).

Equality holds in (2.18) for the function f0 ∈ A , given by (2.13) and the proof
of Theorem 2.2 is thus completed. �

Next, we determine the upper bound for the fourth coefficient of functions be-

longing to the class R̃.

Theorem 2.3. If the function f, given by (1.1) belongs to the class R̃, then

(2.25) |a4| ≤
1

8
and the estimate is sharp.

Proof. Using Lemma 1.1 in (2.9) and following the lines of proof of Theorem 1.2,
we deduce that

|a4| ≤
1

32

{
p3

16
+

(4 −p2)p
2

y +
(4 −p2)p

2
y2 + (4 −p2)(1 −y2)

}
=

1

32

{
p3

16
+

(4 −p2)p
2

t +
(4 −p2)(p− 2)

2
t2 + (4 −p2)

}
= G(p,t) (say),(2.26)



176 SAHOO AND PATEL

where p ∈ [0, 2] and y ∈ [0, 1]. We next maximize the function G(p,y) on the closed
rectangle [0, 2]×[0, 1]. Suppose that the maximum of G occurs at the interior point
of [0, 2] × [0, 1]. Differentiating the function G with respect to y, we get

∂G

∂y
=

1

128
(4 −p2){p + 4(p− 2)y}.

For y ∈ (0, 1) and fixed p ∈ (0, 2), it is easily seen that
∂G

∂y
> 0, which shows that

G is a decreasing function of y contradicting our assumption. Therefore,

(2.27) max{G(p,y)}0≤y≤1 = G(p, 0) =
1

32

{
p3

16
+ (4 −p2)

}
= F(p) (say).

From (2.27), we have

F ′(p) =
1

32

{
3

16
p2 − 2p)

}
and

F ′′(p) =
1

32

{
3

8
p− 2)

}
< 0

for p = 0. This implies that F attains its maximum at p = 0. Hence, we get the
required result.

The estimate in (2.25) is sharp for the function f ∈ A , defined by

f′(z) =
√

1 + z3 (z ∈U).
�

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ANALYTIC FUNCTIONS RELATED WITH LEMNISCATE OF BERNOULLI 177

1Department of Mathematics, Veer Surendra Sai University of Technology, Sidhi
Vihar, Burla-768 018, India

2Department of Mathematics, Utkal University, Vani Vihar, Bhubaneswar-751004,
India

∗Corresponding author