International Journal of Analysis and Applications
ISSN 2291-8639
Volume 8, Number 1 (2015), 63-68
http://www.etamaths.com

SECOND HANKEL DETERMINANT FOR ANALYTIC

FUNCTIONS DEFINED BY RUSCHEWEYH DERIVATIVE

T. YAVUZ

Abstract. Let S denote the class of analytic and univalent functions in the

open unit disk D = {z : |z| < 1} with the normalization conditions. In the
present article an upper bound for the second Hankel determinant

∣∣a2a4 −a23∣∣
is obtained for the analytic functions defined by Ruscheweyh derivative.

1. INTRODUCTION

Let D be the unit disk {z : |z| < 1} , A be the class of functions analytic in D,
satisfying the conditions

(1.1) f(0) = 0 and f′(0) = 1.

Then each function f in A has the Taylor expansion

(1.2) f(z) = z +

∞∑
n=2

anz
n

because of the conditions (1.1) . Let S denote class of analytic and univalent func-
tions in D with the normalization conditions (1.1) .

The qth determinant for q ≥ 1 and n ≥ 0 is stated by Noonan and Thomas [13]
as

(1.3) Hq (n) =

∣∣∣∣∣∣∣∣∣
an an+1 · · · an+q+1
an+1 · · · . . .

...
...

an+q−1 · · · an+2q−2

∣∣∣∣∣∣∣∣∣
.

This determinant has also been considered by several authors. For example, Noor
in [14] determined the rate of growth of Hq (n) as n →∞ for functions f given by
(1.1) with bounded boundary. Ehrenborg in [2] stadied the Hankel determinant of
exponential polynomials. The Hankel transform of an integer sequence and some of
its properties were discussed by Layman’s article [9]. It is well known that [1] that for
f ∈ S and given by (1.2) the sharp inequality

∣∣a3 −a22∣∣ ≤ 1 holds. This corresponds
to the Hankel determinant with q = 2 and k = 1. After that, Fekete-Szegö further
generalized the estimate

∣∣a3 −µa22∣∣ with real µ and f ∈ S. For a given class of
functions in A, the sharp bound for the nonlinear functional

∣∣a2a4 −a23∣∣ is known
as the second Hankel determinant. This corresponds to the Hankel determinant

2010 Mathematics Subject Classification. Primary 30C45, Secondary 33C45.
Key words and phrases. univalent functions, starlike functions, convex functions, Hankel de-

terminant, Ruscheweyh derivative.

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.

63



64 YAVUZ

with q = 2 and k = 2. In particular, sharp bounds on H2 (2) were obtained by
several authors of articles [7], [17], [5], [6], [18] and [12] for different subclasses of
univalent functions.

Let f(z) = z +

∞∑
n=2

anz
n and g(z) = z +

∞∑
n=2

bnz
n be analytic functions in D. The

Hadamard product (convolution) of f and g, denoted by f ∗g is defined by

(1.4) (f ∗g) (z) = z +
∞∑
n=2

anbnz
n, z ∈ D.

Let n ∈ N0 = {0, 1, 2, . . .} . The Ruscheweyh derivative [15] of the nth order of
f, denoted by Dnf (z) , is defined by

(1.5) Dnf (z) =
z

(1 −z)n+1
∗f (z) = z +

∞∑
k=2

Γ (n + k)

Γ (n + 1) (k − 1)!
akz

k.

The Ruscheweyh derivative gave an impulse for various generalization of well known
classes of functions. By using the Ruscheweyh Derivative, we can generalize the
class of the starlike and convex functions functions, denoted by S∗ and C,which are
defined as

(1.6) S∗ =

{
f(z) ∈ S : Re

(
zf′ (z)

f (z)

)
> 0, z ∈ D

}
and

(1.7) C =

{
f(z) ∈ S : Re

(
1 +

zf′′ (z)

f′ (z)

)
> 0, z ∈ D

}
.

The class Rn was studied by Singh and Singh [16], which is given by the following
definition

(1.8) Re
z (Dnf (z))

′

Dnf (z)
> 0, z ∈ D.

We denote that R0 = S
∗ and R1 = C. In the present paper, we obtain an upper

bound for functional
∣∣a2a4 −a23∣∣ in the class Rn.

2. Preliminary Results

The following lemmas are required to prove our main results. Let P be the family
of all functions p analytic in D for which Re (p(z)) > 0 and

(2.1) p(z) = 1 + c1z + c2z + · · · .

Lemma 1. (Duren, [1]) If p ∈ P, then |ck| ≤ 2 for each k ∈ N.

Lemma 2. (Grenander&Szegö, [4]) The power series for p(z) given by (2.1) con-
verges in D to a function in P if and only if the Toeplitz determinants

(2.2) Dn =

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

...
...

...
...

c−n c−n+1 c−n+2 · · · 2

∣∣∣∣∣∣∣∣∣
, n = 1, 2, · · · .



UNIVALENT FUNCTIONS 65

and c−k = ck, are all nonnegative. They are strictly positive except for p(z) =
m∑
k=1

ρkp0
(
eitkz

)
, 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.

We may assume that without restriction that c1 > 0. On using Lemma 2.2, for
n = 2 and n = 3 respectively, we get

(2.3) D2 =

∣∣∣∣∣∣
2 c1 c2
c1 2 c1
c2 c1 2

∣∣∣∣∣∣ = 8 + 2 Re
{
c21c2

}
− 2 |c2|

2 − 4c21 ≥ 0,

which is equivalent to

(2.4) 2c2 = c
2
1 + x

(
4 − c21

)
for some x, |x| ≤ 1. If we consider the determinant

(2.5) Dn =

∣∣∣∣∣∣∣∣
2 c1 c2 c3
c1 2 c1 c2
c2 c1 2 c1
c3 c2 c1 2

∣∣∣∣∣∣∣∣ ≥ 0,
we get the following inequality

(2.6)
∣∣∣(4c3 − 4c1c2 + c31)(4 − c21) + c1 (2c2 − c21)2∣∣∣ ≤ 2 (4 − c21)2−2 ∣∣(2c2 − c21)∣∣2 .

From (2.4) and (2.6), it is obtained that

(2.7) 4c3 = c
3
1 + 2c1

(
4 − c21

)
x− c1

(
4 − c21

)
x2 + 2c1

(
4 − c21

)(
1 −|x|2

)
z

for some z, |z| ≤ 1.

3. Main Results

We prove the following theorem by using thecniques of Libera and Zlotkiewicz
[10], [11].

Theorem 1. Let the function f given by (1.2) be in the class in Rn. Then

(3.1)
∣∣a2a4 −a23∣∣ ≤




1, n = 0
1
8
, n = 1

12(n−1)
(n+1)2(n+2)2(n+3)

, n > 1

Proof. Since f ∈ Rn, there exists an analytic function p ∈ P in the unit disk D
with p(0) = 1 and Re (p(z)) > 0 such that

(3.2)
z (Dnf (z))

′

Dnf (z)
= p(z)

Let

(3.3) F(z) = Dnf(z) = z +

∞∑
k=2

Akz
k,

where

(3.4) Ak =
Γ (n + k)

Γ (n + 1) (k − 1)!
ak,



66 YAVUZ

then we have

(3.5)
zF ′(z)

F(z)
= p(z).

By using the series expansion of F(z) and p(z) as in (3.3) and (2.1) , equating
coefficients in (3.5) yields

a2 =
1

n + 1
c1

a3 =
1

(n + 1) (n + 2)

{
c2 + c

2
1

}
(3.6)

a4 =
1

(n + 1) (n + 2) (n + 3)

{
2c3 + 3c1c2 + c

3
1

}
.

Hence, we get from (3.6)

(3.7) a2a4 −a23 = A(n)
{

2c1c3 + 3c
2
1c2 + c

4
1 −B(n)

(
c2 + c

2
1

)2}
,

where

(3.8) A(n) =
1

(n + 1) (n + 2) (n + 3)
,

and

(3.9) B(n) =

(
n + 3

n + 2

)
, n = 0, 1, 2, · · · .

Using (2.4) and (2.7) in (3.7) ,we get∣∣a2a4 −a23∣∣ = A(n) ∣∣2c1c3 + 3c21c2 + c41 −B(n) (c22 + 2c1c2 + c41)∣∣
and

(3.10)
∣∣a2a4 −a23∣∣ = A(n)

∣∣∣∣3
(

1 −
3

4
B(n)

)
c41 +

3

2
(1 −B(n)) c21x

(
4 − c21

)
−
c21
2

(
4 − c21

)
x2 + c1

(
4 − c21

)(
1 −|x|2

)
z −B(n)

x2
(
4 − c21

)2
4

∣∣∣∣∣
Since the function p(eiθz), (θ ∈ R) is also in the class P , we assume that without
loss of generality that c1 > 0. For convenience of notation, we take c1 = c, c ∈ [0, 2] .
Applying the triangle inequality with the assumptions c1 = c ∈ [0, 2] , |x| = ρ and
|z| ≤ 1, it is obtained that∣∣a2a4 −a23∣∣ ≤ A(n) {3

∣∣∣∣1 − 34B(n)
∣∣∣∣c4 + 32 (B(n) − 1) c2ρ(4 − c2)(3.11)

+ρ2
(
4 − c2

)
c (c− 2)

2
+ c

(
4 − c2

)
+ B(n)ρ2

(
4 − c2

)2
4

}
= G(c,ρ).

We now maximize the function G(c,ρ) on the closed square [0, 2] × [0, 1] . Since

(3.12)
∂G(c,ρ)

∂ρ
=

3

2
(B(n) − 1) c2

(
4 − c2

)
−ρ
(
4 − c2

)
(2 − c)

{
c−

B(n)

2
(2 + c)

}



UNIVALENT FUNCTIONS 67

and B(n) ∈
[
1, 3

2

]
, we get the following inequality

(3.13)
∂G(c,ρ)

∂ρ
≥
ρ
(
4 − c2

)
(2 − c) (6 − c)

4
> 0.

Hence, G(c,ρ) can not have a maximum in the interior of the closed square [0, 2]×
[0, 1] . Hence, for fixed c ∈ [0, 2]

(3.14) max
0≤ρ≤1

G(c,ρ) = G(c, 1) = F(c).

One can obtain that

(3.15)
∣∣a2a4 −a23∣∣ ≤ A(n)F(c),

where
(3.16)

F(c) = 3

∣∣∣∣1 − 34B(n)
∣∣∣∣c4 + 32 (B(n) − 1) c2 (4 − c2) + c

(
4 − c2

)
2

+ B(n)

(
4 − c2

)2
4

.

Since
(3.17)

F ′(c) =




25
3
c3 + c

(
4 − c2

)
+ 3

2
c3, n = 0

8
3
c
(
1 − c2

)
, n = 1

(12 − 9B(n)) c3 + (B(n) − 1) c
(
4 − c2

)
− 3 (B(n) − 2) c3, n > 1

,

we have to consider following three cases:

Case 1. For n = 0, F ′(c) > 0. Hence F(c) ≤ F(2). We get the following result

(3.18)
∣∣a2a4 −a23∣∣ ≤ A(0) {48

∣∣∣∣1 − 34B(0)
∣∣∣∣
}

= 1.

This one coincides with the result in the article [8].

Case 2. After necessarly calculations, it is obtained that

(3.19) F ′(0) = 0 and F ′(1) = 0.

Since

F ′′(0) > 0 and F ′′(1) < 0,

F(c) has a maximum at c = 1. Hence, we obtain

(3.20)
∣∣a2a4 −a23∣∣ ≤ 18,

which is also stated in [8].

Case 3. Let n > 1. Then, F ′(c) can be rewrite as

(3.21) F ′(c) = c
{

(20 − 14B(n)) c2 + 8 (B(n) − 1)
}
.

Since 20−14B(n) > 0 and B(n)−1 > 0, we get F ′(0) = 0, F ′′(0) > 0 and F ′(c) > 0
in the interval (0, 2] . Therefore, it is obvious that

(3.22)
∣∣a2a4 −a23∣∣ ≤ A(n) {48

∣∣∣∣1 − 34B(n)
∣∣∣∣
}

=
12 (n− 1)

(n + 1)
2

(n + 2)
2

(n + 3)
.

This completes the proof of theorem.

�



68 YAVUZ

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International Journal of Mathematical Analysis Vol. 9(10), (2015), 493 - 498.

Gebze Technical University, Department of Mathematics, Kocaeli, TURKEY