Paper-03-22-2010. 118KB Jun 04 2011 12:03:08 AM

Acta Universitatis Apulensis
ISSN: 1582-5329

No. 22/2010
pp. 23-32

SUBORDINATION PROPERTIES FOR CERTAIN CLASS OF
ANALYTIC FUNCTIONS DEFINED BY AN EXTENDED
MULTIPLIER TRANSFORMATION

M. K. Aouf and M. M. Hidan
Abstract. In this paper we derive several subordination results for certain
class of analytic functions defined by an extended multiplier transformation.
2000 Mathematics Subject Classification: 30C45.
1. Introduction
Let A denote the class of functions of the form:
∞
X
f (z) = z +
ak z k
k=2

which are analytic in the open unit disc U = {z : |z| < 1}. We also denote by K the
class of functions f (z) ∈ A that are convex in U .
Many essentially equivalent definitions of multiplier transformation have been
given in literature(see [5], [6], and [10]). In [4] Catas defined the operator I m (λ, ℓ)
as follows:
Definition 1 . [4] Let the function f (z) ∈ A. For m ∈ No = N ∪ {0}, where
N = {1, 2, 3, ...}, λ ≥ 0, ℓ ≥ 0. The extended multiplier transformation I m (λ, ℓ) on
A is defined by the following infinite series:

∞ 
X
1 + λ(k − 1) + ℓ m
m
ak z k .
(1)
I (λ, ℓ)f (z) = z +
1+ℓ
k=2

It follows form(1.1) that

I o (λ, ℓ)f (z)

= f (z),

λzI m ((λ, ℓ)f (z))´= (1 + ℓ)I m+1 (λ, ℓ)f (z) − (1 − λ + ℓ)I m (λ, ℓ)f (z)

(λ > 0) (2)

and
I m1 (λ, ℓ)(I m2 (λ, ℓ))f (z)) = I m1 +m2 (λ, ℓ)f (z) = I m2 (λ, ℓ)(I m1 (λ, ℓ)f (z)).
for all integers m1 and m2 .
23

(3)

M. K. Aouf, M. M. Hidan - Subordination properties for certain class of...

We note that:

(i) Im (1, 0)f (z) = Dm f (z) (see[8]);
(ii) I m (λ, 0)f (z) = Dλm f (z) (see [1]);
(iii) I m (1, ℓ)f (z) = I m (ℓ)f (z) (see [5] and [6]);
(iv) I m (1, 1)f (z) = I m f (z) (see[10])..
Also if f (z) ∈ A, then we can write

where



I m (λ, ℓ)f (z) = f ∗ ϕm
λ,ℓ (z),
ϕm
λ,ℓ (z)

=z+


∞ 
X
1 + λ(k − 1) + ℓ m

1+ℓ

k=2

zk .

(4)

Let Gm (λ, ℓ, µ, b) denote the subclass of A consisting of functions f (z) which satisfy:



I m (λ, ℓ)f (z)
1
′
m
+ µ (I (λ, ℓ)f (z)) − 1
>0
(5)
(1 − µ)

Re 1 +
b
z
or which satisfy the following inequality:


 (1 − µ) I m (λ,ℓ)f (z) + µ (I m (λ, ℓ)f (z))′ − 1 


z
 0; ℓ ≥ 0; m ∈ No ; b ∈ C∗ = C\{0}.
We note that:
(1)Gm (1, 0, µ, b) = Gm (µ, b) (see Aouf [2]);





1
Dm f (z)

′
m
= f ∈ A : Re 1 +
+ µ (D f (z)) − 1
> 0; z ∈ U
(1 − µ)
b
z
(2) Gm (λ, 0, µ, b) = Gm (λ, µ, b)





Dλm f (z)
1
′
m
= f ∈ A : Re 1 +
+ µ (Dλ f (z)) − 1

> 0; z ∈ U ;
(1 − µ)
b
z

(7)

(8)

(3) Gm (1, ℓ, µ, b) = Gm (ℓ, µ, b)





I m (ℓ)f (z)
1
′
m
(1 − µ)

+ µ (I (ℓ)f (z)) − 1
> 0; z ∈ U ;
= f ∈ A : Re 1 +
b
z
(9)
(4) Gm (1, 1, µ, b) = Gm (µ, b)





I m f (z)
1
′
m
= f ∈ A : Re 1 +
(1 − µ)
+ µ (I f (z)) − 1
> 0; z ∈ U ; (10)

b
z
24

M. K. Aouf, M. M. Hidan - Subordination properties for certain class of...

(5) Gm (λ, ℓ, 0, b) = Gm (λ, ℓ, b)





1 I m (λ, ℓ)f (z)
= f ∈ A : Re 1 +
−1
> 0; z ∈ U ;
b
z
(6) Gm (λ, ℓ, 1, b) = Rm (λ, ℓ, b)






1 m
= f ∈ A : Re 1 +
(I (λ, ℓ)f (z))′ − 1 > 0; z ∈ U .
b

(11)

(12)

Definition 2 (Hadamard Product or Convolution). Given two functions f and g
in the class A, where f (z) is given by (1.1) and g(z) is given by
g(z) = z +

∞
X

bk z k

(13)

k=2

the Hadamard product (or Convolution) f ∗ g is defined (as usual)by
(f ∗ g) (z) = z +

∞
X

ak bk z k = (g ∗ f ) (z) (z ∈ U ) .

k=2

Definition 3 (Subordination Principal). For two functions f and g, analytic in
U, we say that the function f (z) is subordinate to g(z) in U , and write f (z) ≺
g(z) (z ∈ U ), if there exists a Schwarz function ω(z), which (by definition) is analytic in U with ω(0) = 0 and |ω(z)| < 1, such that f (z) = g (ω(z)) (z ∈ U ). Indeed
it is known that
f (z) ≺ g(z) (z ∈ U ) ⇒ f (0) = g(0)and f (U ) ⊂ g(U ).
Furthermore, if the function g is univalent in U , then we have the following equivalence [7,p.4]
f (z) ≺ g(z) (z ∈ U ) ⇔ f (0) = g(0)and f (U ) ⊂ g(U ).
Definition 4 (Subordinatiry Factor Sequence). A sequence {bk }∞
k=1 of complex numbers is said to be a subordinating factor sequence if, wherever f (z) of the form (1.1)
is analytic, univalent and convex in U , we have the subordination given by
∞
X

ak bk z k ≺ f (z) (z ∈ U ; a1 = 1) .

k=1

25

(14)

M. K. Aouf, M. M. Hidan - Subordination properties for certain class of...

2.main result
To prove our main result we need the following lemmas.
Lemma 1 [11]. The sequence {bk }∞
k=1 is a subordinating factor sequence if
(
)
∞
X
Re 1 + 2
bk z k > 0 (z ∈ U ) .
k=1

Now, we prove the following lemma which gives a sufficient condition for functions
belonging to the class Gm (λ, ℓ, µ, b).
Lemma 2 Let the function f (z) which is defined by (1.1) satisfies the following
condition:


∞
X
1 + λ(k − 1) + ℓ m
|ak | ≤ |b| (µ ≥ 0; λ > 0; ℓ ≥ 0; m ∈ No ; b ∈ C∗ ) ,
[1 + µ(k − 1)]
1+ℓ
k=2
(15)
then f (z) ∈ Gm (λ, ℓ, µ, b).
Proof. Suppose that the inequality (2.1) holds. Then we have for z ∈ U ,


m


(1 − µ) I (λ, ℓ)f (z) + µ (I m (λ, ℓ)f (z))′ − 1


z




I m (λ, ℓ)f (z)
′
m

+ µ (I (λ, ℓ)f (z)) + 2b − 1
− (1 − µ)
z



m
∞
X

1
+
λ(k
−
1)
+
ℓ


[1 + µ(k − 1)]
ak z k−1 
=


1+ℓ
k=2



m
∞


X
1
+
λ(k
−
1)
+
ℓ


− 2b +
[1 + µ(k − 1)]
ak z k−1 


1+ℓ
k=2

≤

∞
X
k=2

(

− 2|b| −

≤2

1 + λ(k − 1) + ℓ
[1 + µ(k − 1)]
1+ℓ

∞
X
k=2

(∞
X
k=2



m



1 + λ(k − 1) + ℓ
[1 + µ(k − 1)]
1+ℓ


1 + λ(k − 1) + ℓ
[1 + µ(k − 1)]
1+ℓ
26

m

|ak | |z|k−1

m

|ak | |z|k−1
)

|ak | − |b|

)

≤ 0,

M. K. Aouf, M. M. Hidan - Subordination properties for certain class of...

which shows that f (z) belongs to the class Gm (λ, ℓ, µ, b).
Let Gm
∗ (λ, ℓ, µ, b) denote the class of functions f (z) ∈ A whose coefficients satisfy
m
the condition(2.1). We note that Gm
∗ (λ, ℓ, µ, b) ⊆ G (λ, ℓ, µ, b).
Employing the technique used earlier by Attiya [3] and Srivastava and Attiya
[9], we prove
Theorem 3 Let f (z) ∈ Gm
∗ (λ, ℓ, µ, b). Then
m

(1 + µ) 1+λ+ℓ
1+ℓ
h

i (f ∗ g) (z) ≺ g(z) (z ∈ U )
m
2 (1 + µ) 1+λ+ℓ
+
|b|
1+ℓ

(16)

for every function g in K, and

Re (f (z)) > −

h

(1 + µ)

1+λ+ℓ
1+ℓ

(1 + µ)
m

The constant factor



(1+µ)( 1+λ+ℓ
1+ℓ )

m

2[(1+µ)( 1+λ+ℓ
+|b|]
1+ℓ )



m

1+λ+ℓ
1+ℓ

i
+ |b|
m
(z ∈ U ) .

(17)

in the subordination result (2.2) cannot be

replaced by a larger one.
Proof. Let f (z) ∈ Gm
∗ (λ, ℓ, µ, b) and let g(z) = z +

∞
X

ck z k ∈ K. Then we have

k=2

m
1+λ+ℓ
(1 + µ) 1+λ+ℓ
(1
+
µ)
1+ℓ
1+ℓ
i (f ∗ g) (z) = h
i
m
m
h


1+λ+ℓ
+
|b|
+
|b|
2 (1 + µ) 1+λ+ℓ
2
(1
+
µ)
1+ℓ
1+ℓ


m



z+

∞
X

ak ck z k

k=2

(18)
Thus, by Definition 3, the subordination result (2.2) will hold true if the sequence

m

∞


(1 + µ) 1+λ+ℓ
1+ℓ
i ak
m
h

(19)
 2 (1 + µ) 1+λ+ℓ

+
|b|
1+ℓ
k=1

is a subordinating factor sequence with a1 = 1.
In view of Lemma 1, this is equivalent to the following inequality:
m



1+λ+ℓ
∞


(1
+
µ)
X
1+ℓ
m

Re 1 +
ak z k > 0 (z ∈ U ) .


1+λ+ℓ
+ |b|
k=1 (1 + µ)
1+ℓ
27

(20)

!

.

M. K. Aouf, M. M. Hidan - Subordination properties for certain class of...

Now, since
Φ(k) = [1 + µ(k − 1)]



1 + λ(k − 1) + ℓ
1+ℓ

m

is an increasing function of k(k ≥ 2), we have


m
m



1+λ+ℓ
∞



(1 + µ) 1+λ+ℓ
(1
+
µ)
X
1+ℓ
1+ℓ


m
m
Re 1 +
ak z k = Re 1 +
z



1+λ+ℓ
1+λ+ℓ
(1
+
µ)
(1
+
µ)
+
|b|
+ |b|
k=1
1+ℓ
1+ℓ
∞
X

1

+
(1 + µ)



1+λ+ℓ
1+ℓ

−
(1 + µ)

1+λ+ℓ
1+ℓ

(1 + µ)

+ |b| k=2

1+λ+ℓ
1+ℓ

m

ak z k


m
(1 + µ) 1+λ+ℓ
1+ℓ
m

≥1−
r
1+λ+ℓ
+ |b|
(1 + µ) 1+ℓ

1


m



m

∞
X

+ |b| k=2



1 + λ(k − 1) + ℓ
[1 + µ(k − 1)]
1+ℓ





m

|ak |rk


m
(1 + µ) 1+λ+ℓ
1+ℓ
|b|
m
m


> 1−
r−
r = 1 − r > 0 (|z| = r < 1) ,
1+λ+ℓ
+
|b|
+
|b|
(1 + µ) 1+λ+ℓ
(1
+
µ)
1+ℓ
1+ℓ

where we have also made use of the assertion (2.1) of Lemma 2. Thus (2.6) holds
true in U . This proves the inequality (2.2). The inequality (2.3) follows from (2.2)
∞
P
z
by taking the convex function g(z) = 1−z
z k . To prove the sharpness
= z+
k=2

m

of the constant

(1+µ)( 1+λ+ℓ
1+ℓ )

m

2[(1+µ)( 1+λ+ℓ
+|b|]
1+ℓ )

, we consider the function fo (z) ∈ Gm
∗ (λ, ℓ, µ, b)

given by
fo (z) = z −
(1 + µ)
Thus from (2.2), we have

|b|


1+λ+ℓ
1+ℓ

m z 2 .

m

(1 + µ) 1+λ+ℓ
1+ℓ
z
i fo (z) ≺
m
h

(z ∈ U ) .
1−z
+
|b|
2 (1 + µ) 1+λ+ℓ
1+ℓ

28

(21)

(22)

M. K. Aouf, M. M. Hidan - Subordination properties for certain class of...

Morover, it can easily be verified for the function fo (z) given by (2.7) that
m





(1 + µ) 1+λ+ℓ
1+ℓ
1
i fo (z) = − .
m

min Re h

2
|z|≤r 
+ |b|
2 (1 + µ) 1+λ+ℓ
1+ℓ

(23)

m

This shows that the constant

(1+µ)( 1+λ+ℓ
1+ℓ )

2[(1+µ)(

1+λ+ℓ m
+|b|
1+ℓ

)

]

is the best possible.

Putting ℓ = 0 in Theorem 1, we have
Corollary 4 Let the function f (z) defined by (1.1) be in the class Gm
∗ (λ, µ, b) and
suppose that g(z) ∈ K. Then
(1 + µ) (1 + λ)m
(f ∗ g) (z) ≺ g(z) (z ∈ U )
2 [(1 + µ)(1 + λ)m + |b|]
and
Re (f (z)) > −

[(1 + µ)(1 + λ)m + |b|]
(z ∈ U ) .
(1 + µ) (1 + λ)m

(24)

(25)

m

(1+µ)(1+λ)
The constant factor 2[(1+µ)(1+λ)
m +|b|] in the subordination result (2.10) cannot be
replaced by a larger one.

Putting λ = 1 in Theorem 1, we have
Corollary 5 Let the function f (z) defined by (1.1) be in the class Gm
∗ (ℓ, µ, b) and
suppose that g(z) ∈ K.Then

m
(1 + µ) 2+ℓ
1+ℓ
h

i (f ∗ g) (z) ≺ g(z) (z ∈ U )
m
(26)
2 (1 + µ) 2+ℓ
+
|b|
1+ℓ
for every function g in K, and

Re (f (z)) > −

h

(1 + µ)



(1 + µ)

m

The constsnt factor

(1+µ)( 2+ℓ
1+ℓ )
2[(1+µ)(

2+ℓ m
+|b|
1+ℓ

)

2+ℓ
1+ℓ

]



m

2+ℓ
1+ℓ

i
+ |b|
m
(z ∈ U ) .

(27)

in the subordination result (2.12) cannot be

replaced by a larger one.
Putting λ = ℓ = 1 in Theorem 1, we have
29

M. K. Aouf, M. M. Hidan - Subordination properties for certain class of...

Corollary 6 Let the function f (z) defined by (1.1) be in the class Gm
∗ (µ, b) and
suppose that g(z) ∈ K. Then

m
(1 + µ) 32

 (f ∗ g) (z) ≺ g(z) (z ∈ U )

m
(28)
2 (1 + µ) 32
+ |b|
for every function g in K and

Re (f (z)) > −

(1 + µ)
(1 +

m

The constant factor

(1+µ)( 23 )
2[(1+µ)(

placed by a larger one.

3 m
+|b|
2

)

]



3 m
+ |b|
2


3 m
µ) 2

(z ∈ U ) .

(29)

in the subordination result (2.14) cannot be re-

Putting µ = 0 in Theorem 1, we have
Corollary 7 Let the function f (z) defined by (1.1) be in the class Gm
∗ (λ, ℓ, b) and
suppose that g(z) ∈ K. Then
m

1+λ+ℓ
1+ℓ
h
m
2 1+λ+ℓ
1+ℓ

for every function g in K, and

i (f ∗ g) (z) ≺ g(z) (z ∈ U )
+ |b|

Re (f (z)) > −

h

m

1+λ+ℓ
1+ℓ



m

1+λ+ℓ
1+ℓ

i
+ |b|
m
(z ∈ U ) .

(30)

(31)

( 1+λ+ℓ
1+ℓ )
in the subordination result (2.16) cannot be rem
2[( 1+λ+ℓ
+|b|]
1+ℓ )
placed by a larger one.
The constant factor

Putting µ = 1 in Theorem 1, we have
Corollary 8 Let f (z) ∈ R∗m (λ, ℓ, b). Then

m
1+λ+ℓ
1+ℓ
m

1+λ+ℓ
2 1+ℓ

(f ∗ g) (z) ≺ g(z) (z ∈ U )

+ |b|

30

(32)

M. K. Aouf, M. M. Hidan - Subordination properties for certain class of...

for every function g in K, and
i
m
h 
+
|b|
2 1+λ+ℓ
1+ℓ
m

(z ∈ U ) .
Re (f (z)) > −
1+λ+ℓ
2 1+ℓ

(33)

m

( 1+λ+ℓ
1+ℓ )
The constant factor 1+λ+ℓ
in the subordination result (2.18) cannot be rem
2( 1+ℓ ) +|b|
placed by a larger one.
Remark 1 Putting λ = 1 and ℓ = 0 in the above results we obtain the results
obtained by Aouf [2].

References
[1] F. M. Al-Oboudi, On univalent functions defined by a generalized Salagean
operator, Internat. J. Math. Math. Sci. 27(2004), 1429-1436.
[2] M. K. Aouf, Subordination properties for certain class of analytic functions
defined by Salagean operator, Applid. Math. Letters 22(2009), 1581-1585.
[3] A. A. Attiya, On some application of a subordination theorems, J. Math.
Anal. Appl. 311(2005), 489-494.
[4] A. Catas, A note on a certain subclass of analytic functions defined by multiplier transformations, in Proceedings of the Internat. Symposium on Geometric
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[5] N. E. Cho and H. M. Srivastava, Argument estimates of certain analytic functions defined by a class of multiplier transformations, Math. Comput. Modelling,
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Marcel Dekker, Inc. New York, 2000.
[8] G. S. Salagean, Subclasses of univalent functions, Lecture Notes in Math.(SpringerVerlag), 1013(1983), 362-372.
[9] H. M. Srivastava and A. A. Attiya, Some subordination rsults associated with
certain subclass of analytic functions, J. Inequal. Pure Appl. Math. 5(2004), Art.
82,1-6.
[10] B. A. Uralegaddi and C. Somanama, Certain classes of univalent functions,
In: Current Topics in Analytic Function Theory, (Edited by H. M. Srivastava and
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M. K. Aouf, M. M. Hidan - Subordination properties for certain class of...

[11] H. S. Wilf, Subordinating factor sequence for convex maps of the unit circle,
Proc. Amer. Math. Soc. 12(1961), 689-693.
M. K. Aouf
Department of Mathematics
Faculty of Science
Mansoura University
Mansoura 35516, Egypt.
email: mkaouf127@yahoo.com
M. M. Hidan
Abha, Box 1065,
Saudi Arabia.
email: majbh2001@yahoo.com

32