Inclusion Properties for Certain Subclasses of Analytic Functions Defined by a Linear Operator
Nak Eun Cho1
Academic Editor: John Michael Rassias
Received08 Aug 2007
Revised29 Oct 2007
Accepted23 Nov 2007
Published04 Feb 2008
Abstract
The purpose of the present paper is to investigate some inclusion properties of certain subclasses of analytic functions associated with a family of linear operators, which are defined by means of the Hadamard product (or convolution). Some integral preserving properties are also considered.
1. Introduction
Let denote the
class of functions of the form which are
analytic in the open unit disk . If and are analytic in , we say that is subordinate
to , written or if there exists
an analytic function in with and for such that . We denote by , , and the subclasses
of consisting of
all analytic functions which are, respectively, starlike, convex, and
close-to-convex in .
Let be the class of
all functions which are
analytic and univalent in and for which is convex with and for .
Making use of the principle of subordination between
analytic functions, many authors investigated the subclasses , , and of the class for (cf. [1, 2]),
which are defined by For in the
definitions defined above, we have the well-known classes , , and , respectively. Furthermore, for the function classes and investigated by
Janowski [3] (also see [4]), it is easily seen that
We now define the function by where is the
Pochhammer symbol (or the shifted factorial) defined (in terms of the Gamma
function) by
We also denote by : the operator
defined by where the
symbol () stands for
the Hadamard product (or convolution). Then it is easily observed from
definitions (1.4) and (1.6) that and Furthermore, we
note that where the
symbol denotes the
familiar Ruscheweyh derivative [5] (also, see [6]) for . The operator was introduced
and studied by Carlson and Shaffer [7] which has been used widely on the space of
analytic and univalent functions in (see also
[8]).
By using the operator , we introduce the following classes of analytic
functions for , and : We also note
that In particular,
we set
In this paper, we investigate several inclusion
properties of the classes , , and . The integral preserving properties in connection
with the operator are also
considered. Furthermore, relevant connections of the results presented here
with those obtained in earlier works are pointed out.
2. Inclusion Properties Involving the Operator
The following
lemmas will be required in our investigation.
Lemma 2.1 (See [9, Pages 60-61]). Let . If or , then the function defined by (1.4) belongs to the class .
Lemma 2.2 (See [10]). Let and . Then for every analytic function in , where denote the
closed convex hull of .
Theorem 2.3. Let , , and . If or , then
Proof. Let . Then there exists an analytic function in with and such that By using (1.6)
and (2.3), we have It follows from
(2.3) and Lemma 2.1 that and , respectively. Then by applying Lemma 2.2 to (2.4),
we obtain since is convex
univalent. Therefore, from the definition of subordination and (2.5), we have or,
equivalently, , which completes the proof of Theorem 2.3.
Theorem 2.4. Let , and . If or , then
Proof. (). Using a similar argument as in the proof of Theorem
2.3, we obtain where is an analytic
function in with and . Applying Lemma 2.1 and the fact that , we see that since is convex
univalent. Thus the proof of Theorem 2.3 is completed.
Corollary 2.5. Let , , and . If and , then
Theorem 2.6. Let , and . If and , then
Proof. Applying (1.9) and Corollary 2.5,
we observe that which evidently
proves Theorem 2.6.
Taking in Corollary
2.5 and Theorem 2.6, we have the following corollary.
Corollary 2.7. Let and . If and , then
To prove the
theorems below, we need the following lemma.
Lemma 2.8. Let . If and , then .
Proof. Let . Then where is an analytic
function in with and . Thus we have By using
similar arguments to those used in the proof of Theorem 2.3, we conclude that
(2.15) is subordinated to in and so .
Theorem 2.9. Let , and . If and , then
Proof.
First of all, we show that Let . Then there exists a function such that From (2.18), we
obtain where is an analytic
function in with and . By virtue of Lemmas 2.1 and 2.8, we see that belongs to . Then we have which implies
that . Moreover, the proof of the second part is similar to that
of the first part and so we omit the details involved.
3. Inclusion Properties Involving Various Operators
The next
theorem shows that the classes , , and are invariant
under convolution with convex functions.
Theorem 3.1. Let , , and let . Then (i)(ii)(iii)
Proof. (i) Let . Then we have By using the
same techniques as in the proof of Theorem 2.3, we obtain (i). (ii) Let . Then, by (1.9), and hence from
(i), . Since we have (ii)
applying (1.9) once again. (iii) Let . Then there exists a function such that where is an analytic
function in with and . From Lemma 2.8, we have that . Since we obtain
(iii).
Now we
consider the following operators [5, 11] defined by It is well
known ([12], see also [5]) that the operators and are convex
univalent in . Therefore, we have the following result, which can
be obtained from Theorem 3.1 immediately.
Corollary 3.2. Let , , and let be defined by (3.5). Then (i)(ii)(iii)
Acknowledgments
The author would like to express his
gratitude to the referees for their valuable suggestions. This work was
supported by Pukyong National University Research Fund in 2007 (PK-2007-013).
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