7_744_darus.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 4, No. 1, 2011, 59-66

ISSN 1307-5543 – www.ejpam.com

On New Subclasses of Analytic Functions Involving

Generalized Differential and Integral Operators

Maslina Darus,∗, Rabha W. Ibrahim

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia,

Bangi 43600, Selangor Darul Ehsan, Malaysia

Abstract. We define a generalized differential and integral operators on the class A of analytic func-

tions f (z) = z +
∑∞

n=2
anzn in the unit disk U := {z ∈ C : |z| < 1} involving k−th Hadamard product

(convolution) as follows

Dk
α,λ

f (z) = z +

∞∑

n=2

[(n− 1)(λ−α) + n]kanzn, (z ∈ U).

These operators are generalized for some of well known operators for example Sǎlǎgean operator.

New classes containing these operators are investigated. Characterization and other properties of

these classes are studied.
2000 Mathematics Subject Classifications: 30C45
Key Words and Phrases: Hadamard product; Integral operator; Differential operator; Sǎlǎgean oper-

ator.

1. Introduction and Preliminaries

Let H be the class of functions analytic in U := {z ∈ C : |z| < 1} and H [a, n] be the

subclass ofH consisting of functions of the form f (z) = a+ anzn + an+1zn+1 + . . .. LetA be

the subclass of H consisting of functions of the form

f (z) = z +

∞∑

n=2

anzn, (z ∈ U). (1)

Given two functions f , g ∈A , f (z) = z+
∑∞

n=2 anzn and g(z) = z+
∑∞

n=2 bnzn their convolu-

tion or Hadamard product f (z) ∗ g(z) is defined by

f (z) ∗ g(z) = z +

∞∑

n=2

an bnzn, (z ∈ U).

∗Corresponding author.

Email addresses: maslina�ukm.my (M. Darus), rabhaibrahim�yahoo.
om (R. Ibrahim)

http://www.ejpam.com 59 c© 2010 EJPAM All rights reserved.



M. Darus, R. Ibrahim / Eur. J. Pure Appl. Math, 4 (2011), 59-66 60

And for several functions f1(z), . . . , fm(z) ∈A

f1(z) ∗ . . . ∗ fm(z) = z +

∞∑

n=2

(a1n...amn)z
n, (z ∈ U).

Our aim is to use the Hadamard product of k−th order to define generalized differential and

integral operators.

For a function f inA of the form (1) first, we define the following generalized differential

operator:

D0 f (z) = f (z)

= z +

∞∑

n=2

anzn,

D1
α,λ f (z) = (α−λ) f (z) + (λ−α+ 1)z f ′(z)

= z +

∞∑

n=2

[(n− 1)(λ−α) + n]anzn,

...

Dk
α,λ f (z) = D1

α,λ

�

Dk−1
α,λ,

f (z)
�

= z +

∞∑

n=2

[(n− 1)(λ−α) + n]kanzn
(2)

for α ≥ 0,λ ≥ 0 and k ∈ N0 = N ∪ {0} with Dk
α,λ

f (0) = 0. Note that when α = λ we get

Sǎlǎgean’s differential operator [see 11].

Let M (µ) be the subclass of the class A consisting of functions f (z) which satisfy the

inequality

ℜ{
z f ′(z)

f (z)
} < µ, (z ∈ U)

for some µ(µ > 1). And let N (µ) be the subclass of the class A consisting of functions f (z)

which satisfy the inequality

ℜ{
z f ′′(z)

f ′(z)
}< µ, (z ∈ U)

for some µ(µ > 1). Then f ∈ N (µ) if and only if z f ′ ∈ M (µ). In this paper we define and

study the following subclasses involving the generalized differential operator (2). LetM k
α,λ
(µ)

be the subclass of the classA consisting of functions f (z) which satisfy the inequality

ℜ{
z[Dk

α,λ
f (z)]′

Dk
α,λ

f (z)
}< µ, (z ∈ U)



M. Darus, R. Ibrahim / Eur. J. Pure Appl. Math, 4 (2011), 59-66 61

for some µ(µ > 1). And letN k
α,λ
(µ) be the subclass of the classA consisting of functions f (z)

which satisfy the inequality

ℜ{
z[Dk

α,λ
f (z)]′′

[Dk
α,λ

f (z)]′
} < µ, (z ∈ U)

for some µ(µ > 1). Then f ∈ N k
α,λ
(µ) if and only if z f ′ ∈M k

α,λ
(µ)(µ).

Remark 1. When k = 0, then the classes

M 0
α,λ(µ)≡M (µ) and N 0

α,λ(µ)≡N (µ)

were introduced by

(i) For 1< µ ≤ 4

3
, k = 0, Uralegaddi et al. [13, 12].

(ii) For µ > 1, k = 0, Owa and Srivastava [9] and Owa and Nishiwaki [10].

(iii) For µ > 1, α= 1 Bulut [2].

2. Coefficient estimates.

In this section we derive sufficient conditions for f (z) to belongs to the classesM k
α,λ
(µ)

and N k
α,λ
(µ), which are obtained by using coefficient inequalities.

Theorem 1. If f (z) ∈A satisfies the inequality

∞∑

n=2

|[(n− 1)(λ−α) + n]k|
n

(n− κ) + |n+κ− 2µ|
o

|an| ≤ 2(µ− 1) (3)

for some 0≤ κ≤ 1 and µ > 1, then f ∈M k
α,λ
(µ).

Proof. Assume that the inequality (3) holds. It suffices to show that

�
�
�

z[Dk
α,λ

f (z)]′

Dk
α,λ

f (z)
− κ

z[Dk
α,λ

f (z)]′

Dk
α,λ

f (z)
− (2µ− κ)

�
�
� < 1, (z ∈ U).

We observe

�
�
�

z[Dk
α,λ

f (z)]′

Dk
α,λ

f (z)
− κ

z[Dk
α,λ

f (z)]′

Dk
α,λ

f (z)
− (2µ− κ)

�
�
�=

�
�
�

1− κ+
∑∞

n=2(n− κ)[(n− 1)(λ−α) + n]kanzn−1

1+ κ− 2µ+
∑∞

n=2(n+ κ− 2µ)[(n− 1)(λ−α) + n]kanzn−1

�
�
�

≤
1− κ+
∑∞

n=2(n− κ)|[(n− 1)(λ−α) + n]k||an||z|
n−1

2µ− 1− κ−
∑∞

n=2 |(n+ κ− 2µ)||[(n− 1)(λ−α) + n]k||an||z|n−1

<
1− κ+
∑∞

n=2(n− κ)|[(n− 1)(λ−α) + n]k||an|

2µ− 1− κ−
∑∞

n=2 |(n+ κ− 2µ)||[(n− 1)(λ−α) + n]k||an|
.



M. Darus, R. Ibrahim / Eur. J. Pure Appl. Math, 4 (2011), 59-66 62

The last expression is bounded above by 1 if

1−κ+
∞∑

n=2

(n−κ)|[(n−1)(λ−α)+n]k||an| < 2µ−1−κ−
∞∑

n=2

|(n+κ−2µ)||[(n−1)(λ−α)+n]k||an|

which is equivalent to assertion (3), hence the proof.

When k = 0, the next result can found in [10].

Corollary 1. If f (z) ∈A satisfies the inequality

∞∑

n=2

n

(n− κ) + |n+ κ− 2µ|
o

|an| ≤ 2(µ− 1) (4)

for some 0≤ κ≤ 1 and µ > 1, then f ∈M 0
α,λ
(µ)≡M (µ).

When κ = 1, we obtain the next result

Corollary 2. If f (z) ∈A satisfies the inequality

∞∑

n=2

(n−µ)|[(n− 1)(λ−α) + n]k||an| ≤ µ− 1 (5)

for 1< µ ≤ 3

2
, then f ∈M k

α,λ
(µ).

When k = 0,κ= 1 the next result can found in [10].

Corollary 3. If f (z) ∈A satisfies the inequality

∞∑

n=2

(n−µ)|an| ≤ µ− 1 (6)

for 1< µ ≤ 3

2
, then f ∈M (µ).

Theorem 2. If f (z) ∈A satisfies the inequality

∞∑

n=2

n|[(n− 1)(λ−α) + n]k|
n

n− κ+ 1+ |n+ κ− 2µ|
o

|an| ≤ 2(µ− 1) (7)

for some 0≤ κ≤ 1 and µ > 1, then f ∈ N k
α,λ
(µ).

When k = 0, the next result can be found in [10].

Corollary 4. If f (z) ∈A satisfies the inequality

∞∑

n=2

n
n

n− κ+ 1+ |n+ κ− 2µ|
o

|an| ≤ 2(µ− 1) (8)

for some 0≤ κ≤ 1 and µ > 1, then f ∈ N 0
α,λ
(µ)≡N (µ).



M. Darus, R. Ibrahim / Eur. J. Pure Appl. Math, 4 (2011), 59-66 63

3. Integral operator.

Analogous to the generalized differential operator (2), we define and study a new integral

operator Ik
α,λ

:A →A as follows. Let

φ(z) :=
(λ−α)z

(1− z)2
−
(λ−α)z

1− z
+

z

(1− z)2

and

F(z) = φ(z) ∗ . . . ∗φ(z)
︸ ︷︷ ︸

k−t imes

= z +

∞∑

n=2

[(n− 1)(λ−α) + n]kzn

Now we define the integral operator Ik
α,λ

such that

Ik
α,λ := [F(z)]−1 ∗ f (z), (z ∈ U)

where f ∈A and

F(z) ∗ [F(z)]−1 =
z

1− z
= z +

∞∑

n=2

zn, (z ∈ U).

Implies

[F(z)]−1 = z +

∞∑

n=2

1

[(n− 1)(λ−α) + n]k
zn, (z ∈ U)

thus we have

Ik
α,λ f (z) = z +

∞∑

n=2

an

[(n− 1)(λ−α) + n]k
zn, (z ∈ U). (9)

Remark 2. Note that when α= λ, the integral operator (9) reduces to the integral operator

Ik
α,α f (z) = z +

∞∑

n=2

an

nk
zn, (z ∈ U),

which defined and studied by Sǎlǎgean [see 11].

Lemma 1. Let f ∈A . Then

(i) I0
α,λ

f (z) = f (z),

(ii) I1
α,α f (z) =
∫ z

0

f (t)

t
d t.

Proof.



M. Darus, R. Ibrahim / Eur. J. Pure Appl. Math, 4 (2011), 59-66 64

(i)

I0
α,λ f (z) = z +

∞∑

n=2

anzn = f (z),

(ii)
∫ z

0

f (t)

t
d t =

∫ z

0

[1+

∞∑

n=2

antn−1]d t

= z +

∞∑

n=2

an

n
zn

= I1
α,α f (z).

Define the subclasses involving the generalized integral operator (9). Let S k
α,β ,λ

(µ) be the

subclass of the classA consisting of functions f (z) which satisfy the inequality

ℜ{
z[Ik
α,λ

f (z)]′

Ik
α,λ

f (z)
} < µ, (z ∈ U)

for some µ(µ > 1). It is clear that

S 0
α,λ
(µ)≡M (µ).

And letK k
α,β ,λ
(µ) be the subclass of the classA consisting of functions f (z) which satisfy the

inequality

ℜ{
z[Ik
α,λ

f (z)]′′

[Ik
α,λ

f (z)]′
}< µ, (z ∈ U)

for some µ(µ > 1). Then f ∈K k
α,λ
(µ) if and only if z f ′ ∈ S k

α,λ
(µ)(µ). Also we have

K 0
α,λ
(µ)≡N (µ).

In the same manner of Theorem 1 and Theorem 2, we have the following results.

Theorem 3. If f (z) ∈A satisfies the inequality

∞∑

n=2

n

(n− κ) + |n+ κ− 2µ|
o

|[(n− 1)(λ−α) + n]k|
|an| ≤ 2(µ− 1) (10)

for some 0≤ κ≤ 1 and µ > 1, then f ∈ S k
α,λ
(µ).

Theorem 4. If f (z) ∈A satisfies the inequality

∞∑

n=2

n
n

n− κ+ 1+ |n+ κ− 2µ|
o

|[(n− 1)(λ−α) + n]k|
|an| ≤ 2(µ− 1) (11)

for some 0≤ κ≤ 1 and µ > 1, then f ∈K k
α,λ
(µ).



REFERENCES 65

4. Conclusion.

This work is a generalization for well known differential and integral operators of univa-

lent functions. Moreover, the classes which are studied here also generalized the ones studied

by different authors

M 0
α,λ(µ)≡ S

0
α,λ(µ)≡M (µ)

and

N 0
α,λ(µ)≡K

0
α,λ(µ)≡N (µ).

In fact, many other operators can be seen in [1,3-8,14] for different problems.

ACKNOWLEDGEMENTS The work presented here was supported by UKM-ST-06-FRGS0107-

2009.

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