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Abstract and Applied Analysis

Volume 2008 (2008), Article ID 845724, 11 pages

http://dx.doi.org/10.1155/2008/845724

## Differential Subordinations Associated with Multiplier Transformations

Department of Mathematics, Faculty of Science, University of Oradea, 1 University Street, 410087 Oradea, Romania

Received 26 February 2008; Accepted 15 May 2008

Academic Editor: Ferhan Atici

Copyright © 2008 Adriana Cătaş et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The authors introduce new classes of analytic functions in the open unit disc which are defined by using multiplier transformations. The properties of these classes will be studied by using techniques involving the Briot-Bouquet differential subordinations. Also an integral transform is established.

#### 1. Introduction and Definitions

Let be the class of analytic functions in the open unit discand let be the subclass of consisting of functions of the form . Let denote the class of functions normalized bywhich are analytic in the open unit disc. In particular, we set If a function belongs to the class it has the form For two functions given by (1.4) and for given bythe Hadamard product (or convolution) is defined, as usual, by If and are analytic in , we say that is subordinate to , written symbolically asif there exists a Schwarz function in which is analytic in with and such that , .

We consider the following multiplier transformations.

*Definition 1.1 (See [1]). * Let .
For , , , ,
define the multiplier transformations on by the following infinite
series:

It follows from (1.8) that

*Remark 1.2 (See [1]). * For , , , , , ,
the operator was introduced and studied by Al-Oboudi
[2] which
is reduced to the Sălăgean differential operator [3] for .
The operator was studied recently by Cho and Srivastava
[4] and by Cho and Kim
[5]. The operator was studied by Uralegaddi and Somanatha
[6], the operator was introduced by Acu and Owa [7] and the operator was investigated recently by
Sivaprasad Kumar et al. [8].

If is given by (1.2), then we havewhere In particular, we set In order to prove our main results, we will make use of the following lemmas.

Lemma 1.3 (See [9]). * For real
or complex numbers , ,
and
,
the following hold:
*

Lemma 1.4 (See [10]). * Let , and let be convex in ,
with**
If the function ,
then*

Lemma 1.5 (See [11]). * Let be a positive measure on the unit interval .
Let be a function analytic in ,
for each and integrable in ,
for each and for almost all .
Suppose also that** is real for real and**
If**then*

Lemma 1.6 (See [12]). * Let be univalent in the unit disc and let and be analytic in a domain with ,
when .
Set**
Suppose that*

(1)* is starlike in and*(2)* for .**
If is analytic in ,
with , ,
and**then and is the best dominant.*

Lemma 1.7 (See [12, Theorem 3.3d]). * Let ,
with and let satisfy either**when ,
or**when .
If satisfies**then**where is the univalent solution of the differential
equation**In addition, the function ,
is the best -dominant and the function is given by**where**and the univalent function is given by*

Now we define new classes of analytic functions by using the multiplier transformations defined by (1.8) as follows.

#### 2. Main Results

*Definition 2.1. *Let , , , .
A function is said to be in the class if it satisfies the following
subordination:

*Remark 2.2. *We note thatwhere and
denote the subclasses of functions in which are, respectively, starlike of order and convex of order in .
Also we have the classstudied by Patel [13].

Let be analytic in and . We introduce the following definition.

*Definition 2.3. *A function is said to be in the class if it satisfies the following
subordination:

*Remark 2.4. *We note that the classes were investigated recently by Sivaprasad Kumar et al. [8].

Theorem 2.5. *Let , , ,
and**
(i) Then**
Further, for
the following
hold:
**where**and is the best dominant of (2.7).**(ii) Furthermore, in addition to (2.5), one consider
the inequality**where ,
then**where**
The result is the best possible.*

*Proof. *Settingwe note that is analytic in and
Using the identityin definition of and carrying out logarithmic differentiation
in the resulting equation, one obtains
Since ,
we get
By applying Lemma 1.4, we obtain that
Hence we have shown the inclusion (2.6). Also, making
use of Lemma 1.7 with and , is the best dominant of (2.7) and is defined by (2.8). This proves part (i) of
Theorem 2.5.

To establish (2.10), we need to show
thatThe proof of the assertion
(2.18) will be deduced on the same lines as in [14] making use of
Lemma 1.5. If we set , ,
then by using (1.13), (1.14), and
(1.15), we find from (2.8) thatwhereBy using (1.13), the above
equality yieldswhereis a positive measure on the
closed interval .

For ,
we note that , is real for and andTherefore, by using Lemma 1.5,
one obtainswhich, upon letting ,
yieldsNow, the assertion (2.18)
follows by using Lemma 1.5. The result is the best possible and is the best dominant of (2.7). This completes
the proof of Theorem 2.5.

Taking , , , , , in Theorem 2.5, we get the following result due to MacGregor [15].

Corollary 2.6. *For ,
one obtains**where**
The result is the best possible.*

Theorem 2.7. *Let be univalent in with , ,
and let
be starlike in .
Let be defined by**Then*

*Proof. *Settingwe note that is analytic in .

By a simple computation, we observe from (2.30) that
Making use of the identity (2.14), one obtains from
(2.31)
By the hypothesis of Theorem 2.7 that belongs to the class and in view of (2.32), we have
If we letwhereand since is starlike, our theorem is an immediate
consequence of Lemma 1.6.

Theorem 2.8. *Let be univalent in , and let
be a complex number. Suppose that*

(1)*
and*(2)* is starlike in U.**Let the function be defined by**and the function**then implies .*

*Proof. *From the definition of andif we letthen we note that is analytic in .
Using (2.38) and (2.39), one obtains
Differentiating this equality, we
obtain
For ,
we have from (2.41)
If we letwhereand since is starlike in ,
our theorem is an immediate consequence of Lemma 1.6.

Theorem 2.9. *Let .
Then belongs to the class if and only if defined by**belongs to the class .*

*Proof. *From the definition of ,
we have
By convoluting (2.46) with the functionand using a convolution propertyone obtainsthat is,
By using identity (2.14), we get
Also, we obtain
From (2.51) and
(2.52), we get
By the hypothesis of Theorem 2.9 thatand using (2.53), the desired
result follows at once.

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