Research Article
Open Access
Convolution Properties of Classes of Analytic and Meromorphic Functions
- RosihanM Ali1Email author,
- MoradiNargesi Mahnaz1,
- V Ravichandran2 and
- KG Subramanian1
Journal of Inequalities and Applications20102010:385728
https://doi.org/10.1155/2010/385728
© Rosihan M. Ali et al. 2010
Received: 30 October 2009
Accepted: 13 May 2010
Published: 13 June 2010
Abstract
General classes of analytic functions defined by convolution with a fixed analytic function are introduced. Convolution properties of these classes which include the classical classes of starlike, convex, close-to-convex, and quasiconvex analytic functions are investigated. These classes are shown to be closed under convolution with prestarlike functions and the Bernardi-Libera integral operator. Similar results are also obtained for the classes consisting of meromorphic functions in the punctured unit disk.
1. Motivation and Definitions
Let
be the set of all analytic functions defined in the unit disk
. Denote by
the class of normalized analytic functions
defined in
. For two functions
and
in
, the convolution or Hadamard product of
and
is the function
defined by
. A function
is subordinate to an analytic function
, written
, if there exists a Schwarz function
, analytic in
with
and
satisfying
If the function
is univalent in
, then
is equivalent to
and
.
The classes of starlike and convex analytic functions and other related subclasses of analytic functions can be put in the form
(1.1)
where
is a fixed function and
is a suitably normalized function with positive real part. In particular, let
and
. For
,
,
and
are, respectively, the familiar classes
of starlike functions of order
and
consisting of convex functions of order
. Analogous to the class
, the class
is defined by
is a fixed function and
is a suitably normalized function with positive real part. In particular, let
and
. For
,
,
and
are, respectively, the familiar classes
of starlike functions of order
and
consisting of convex functions of order
. Analogous to the class
, the class
is defined by
(1.2)
Let
and
satisfy
and
satisfy
(1.3)
By adding the two inequalities, it is evident that the function
is starlike and hence both
and
are close-to-convex and univalent. This motivates us to consider the following classes of functions.
It is assumed in the sequel that
is a fixed integer,
is a fixed function in
, and
is a convex univalent function with positive real part in
satisfying
.
Definition 1.1.
The class
consists of
,
,
, satisfying
in
and the subordination
consists of
,
,
, satisfying
in
and the subordination
(1.4)
The class
consists of
for which
. The class
consists of
for which
, where
and
. Equivalently,
if
satisfies the condition
in
and the subordination
consists of
for which
. The class
consists of
for which
, where
and
. Equivalently,
if
satisfies the condition
in
and the subordination
(1.5)
The class
consists of
for which
.
Now let
and
. From (1.4), it follows that
and
. From (1.4), it follows that
(1.6)
The convexity of
implies that
implies that
(1.7)
which shows that the function
is starlike in
. Thus, it follows from (1.4) that the component function
of
is close-to-convex in
, and hence univalent. Similarly, the component function
of
is univalent.
If
, then the classes
and
are reduced, respectively, to
and
introduced and investigated in [1]; these classes were denoted there by
and
, respectively. If
, where
, then the classes
and
are reduced, respectively, to
and
introduced and investigated in [1]; these classes were denoted there by
and
, respectively. If
, where
(1.8)
then the class
coincides with the class studied in [2], which there was denoted by
, and
reduces to a class introduced in [3] which there was denoted by
. It is evident that the classes
and
extend the classical classes of starlike and convex functions, respectively.
Definition 1.2.
The class
consists of
,
,
, satisfying the subordination
consists of
,
,
, satisfying the subordination
(1.9)
for some
. In this case, we say that
with respect to
. The class
consists of
for which
. The class
consists of
for which
or equivalently satisfying the subordination
. In this case, we say that
with respect to
. The class
consists of
for which
. The class
consists of
for which
or equivalently satisfying the subordination
(1.10)
for some
with
,
. In this case, we say that
with respect to
. The class
consists of
for which
.
When
, the classes
and
reduce, respectively, to
and
introduced and investigated in [1]. If
, where
is defined by (1.8), then the class
coincides with
studied in [2]. Clearly the classes
and
extend the classical classes of close-to-convex and quasiconvex functions, respectively.
For
, the class
of prestarlike functions of order
is defined by
, the class
of prestarlike functions of order
is defined by
(1.11)
while
consists of
satisfying
.
The well-known result that the classes of starlike functions of order
and convex functions of order
are closed under convolution with prestarlike functions of order
follows from the following.
Theorem 1.3 (see [4, Theorem
]).
Let
,
, and
. Then
,
, and
. Then
(1.12)
for any analytic function
, where
denotes the closed convex hull of
.
In the following section, by using the methods of convex hull and differential subordination, convolution properties of functions belonging to the four classes
,
,
and
, are investigated. It would be evident that various earlier works, see, for example, [5–10], are special instances of our work.
In Section 3, new subclasses of meromorphic functions are introduced. These subclasses extend the classical subclasses of meromorphic starlike, convex, close-to-convex, and quasiconvex functions. Convolution properties of these newly defined subclasses will be investigated. Simple consequences of the results obtained will include the work of Bharati and Rajagopal [6] involving the function
,
, as well as the work of Al-Oboudi and Al-Zkeri [5] on the modified Salagean operator.
2. Convolution of Analytic Functions
Our first result shows that the classes
and
are closed under convolution with prestarlike functions.
Theorem 2.1.
Let
be a fixed integer and
a fixed function in
. Let
be a convex univalent function satisfying
,
, and
.
(1)If
, then
.
(2)If
, then
.
Proof.
- (1)It is sufficient to prove that
whenever
. Once this is established, the general result for
follows from the fact that 
(2.1)
For
, define the functions
and
by
, define the functions
and
by
(2.2)
It will first be proved that
belongs to
. For
and
, clearly
belongs to
. For
and
, clearly
(2.3)
Since
is a convex domain, it follows that
is a convex domain, it follows that
(2.4)
or
(2.5)
Since
, the subordination (2.5) yields
, the subordination (2.5) yields
(2.6)
and hence
A computation shows that
(2.7)
Since
and
, Theorem 1.3 yields
and
, Theorem 1.3 yields
(2.8)
and because
, we deduce that
, we deduce that
(2.9)
Thus
.
.- (2)
The function
is in
if and only if
is in
and by the first part above, it follows that
. Hence
.
Remark 2.2.
The above theorem can be expressed in the following equivalent forms:
(2.10)
When
, various known results are easily obtained as special cases of Theorem 2.1. For instance, [1, Theorem
, page 336] is easily deduced from Theorem 2.1(1), while [1, Corollary
, page 336] follows from Theorem 2.1(2). If
is defined by (1.8), then [3, Theorem
, page 110] follows from Theorem 2.1(1), and [3, Corollary
, page 111] follows from Theorem 2.1(2).
Corollary 2.3.
Let
be a fixed integer and
a fixed function in
. Let
be a convex univalent function satisfying
,
. Define
be a fixed integer and
a fixed function in
. Let
be a convex univalent function satisfying
,
. Define
(2.11)
If
, then
. Similarly, if
, then
.
Proof.
Define the function
by
by
(2.12)
For
, the function
is a convex function [11], and hence
([4, Theorem
, page 49]). It is clear from the definition of
that
, the function
is a convex function [11], and hence
([4, Theorem
, page 49]). It is clear from the definition of
that
(2.13)
so that
. By Theorem 2.1(1), it follows that
.
The second result is proved in a similar manner.
Remark 2.4.
If
is defined by (1.8), then Corollary 2.3 reduces to [2, Theorem
, page 324].
Theorem 2.5.
Let
be a fixed integer and
a fixed function in
. Let
be a convex univalent function satisfying
, and
.
(1)If
with respect to
, then
with respect to
.
(2)If
with respect to
, then
with respect to
.
Proof.
- (1)In view of the fact that
(2.14)
we well only prove that
when
. Let
. For
, define the functions
and
by
when
. Let
. For
, define the functions
and
by
(2.15)
Since
, it is evident from (2.6) that
.
That
follows from Theorem 2.1(1). Now a computation shows that
follows from Theorem 2.1(1). Now a computation shows that
(2.16)
Since
and
, Theorem 1.3 yields
and
, Theorem 1.3 yields
(2.17)
and because
, it follows that
, it follows that
(2.18)
Thus
.
.- (2)
The function
is in
if and only if
is in
and by the first part, clearly
. Hence
.
Remark 2.6.
Again when
, known results are easily obtained as special cases of Theorem 2.5. For instance, [1, Theorem
, page 337] follows from Theorem 2.5(1), and [1, Theorem
, page 339] is a special case of Theorem 2.5(2).
Corollary 2.7.
Let
be a fixed integer and
a fixed function in
. Let
be a convex univalent function satisfying
,
. Let
be the Bernardi-Libera integral transform of
defined by (2.11). If
, then
.
The proof is similar to the proof of Corollary 2.3 and is therefore omitted.
Remark 2.8.
If
is defined by (1.8), then Corollary 2.7 reduces to [2, Theorem
, page 326].
3. Convolution of Meromorphic Functions
Let
denote the class of functions
of the form
denote the class of functions
of the form
(3.1)
that are analytic in the punctured unit disk
. The convolution of two meromorphic functions
and
, where
is given by (3.1) and
, is given by
. The convolution of two meromorphic functions
and
, where
is given by (3.1) and
, is given by
(3.2)
In this section, several subclasses of meromorphic functions in the punctured unit disk are introduced by means of convolution with a given fixed meromorphic function. First we take note that the familiar classes of meromorphic starlike and convex functions and other related subclasses of meromorphic functions can be put in the form
(3.3)
where
is a fixed function in
and
is a suitably normalized analytic function with positive real part. For instance, the class of meromorphic starlike functions of order
,
, defined by
is a fixed function in
and
is a suitably normalized analytic function with positive real part. For instance, the class of meromorphic starlike functions of order
,
, defined by
(3.4)
is a particular case of
with
and
.
Here four classes
,
,
, and
of meromorphic functions are introduced and the convolution properties of these new subclasses are investigated. As before, it is assumed that
is a fixed integer,
is a fixed function in
, and
is a convex univalent function with positive real part in
satisfying
.
Definition 3.1.
The class
c onsists of
,
,
, satisfying
in
and the subordination
c onsists of
,
,
, satisfying
in
and the subordination
(3.5)
The class
consists of
for which
. The class
consists of
for which
or equivalently satisfying the condition
in
and the subordination
consists of
for which
. The class
consists of
for which
or equivalently satisfying the condition
in
and the subordination
(3.6)
The class
consists of
for which
.
Various subclasses of meromorphic functions investigated in earlier works are special instances of the above defined classes. For instance, if
, then
coincides with
. By putting
, where
, then
coincides with
. By putting
, where
(3.7)
then the class
coincides with
investigated in [6].
Definition 3.2.
The class
consists of
,
,
, satisfying the subordination
consists of
,
,
, satisfying the subordination
(3.10)
for some
. In this case, we say that
with respect to
. The class
consists of
for which
. The class
consists of
for which
or equivalently satisfying the subordination
. In this case, we say that
with respect to
. The class
consists of
for which
. The class
consists of
for which
or equivalently satisfying the subordination
(3.11)
for some
with
and
. The class
consists of
for which
.
If
, then
coincides with
. If
is defined by (3.8), then
reduces to
investigated in [5]. If
is defined by (3.9), then the class
is the class
studied in [6].
We shall require the theorem below which is a simple modification of Theorem 1.3.
Theorem 3.3.
Let
,
,
, and
. Then, for any analytic function
,
,
,
, and
. Then, for any analytic function
,
(3.12)
Theorem 3.4.
Assume that
is a fixed integer and
is a fixed function in
. Let
be a convex univalent function satisfying
,
, and
with
.
(1)If
, then
.
(2)If
, then
.
Proof.
- (1)It is enough to prove the result for
For
, define the functions
and
by 
(3.13)
We show that
satisfies the condition
. For
and
, clearly
satisfies the condition
. For
and
, clearly
(3.14)
Since
is a convex domain, it follows that
is a convex domain, it follows that
(3.15)
or
(3.16)
Since
, the subordination (3.16) yields
, the subordination (3.16) yields
(3.17)
and thus
(3.18)
Inequality (3.18) shows that
.
A routine computation now gives
(3.19)
Since
and
, Theorem 3.3 yields
and
, Theorem 3.3 yields
(3.20)
and because
, it is clear that
, it is clear that
(3.21)
Thus
.
.- (2)
The function
is in
if and only if
is in
and the result of part (1) shows that
. Hence
.
Remark 3.5.
- (1)The above theorem can be written in the following equivalent forms:
(3.22)
(
) When
, various known results are easily obtained as special cases of Theorem 3.4. For instance, if
is defined by (3.7), then [9, Theorem
, page 1265] follows from Theorem 3.4(1).
Corollary 3.6.
Assume that
is a fixed integer and
is a fixed function in
. Let
be a convex univalent function satisfying
,
. Define
is a fixed integer and
is a fixed function in
. Let
be a convex univalent function satisfying
,
. Define
(3.23)
If
, then
. Similarly, if
, then
.
Proof.
Define the function
by
by
(3.24)
For
, the function
is a convex function [11], and hence
([4, Theorem
, page 49]). It is clear from the definition of
that
, the function
is a convex function [11], and hence
([4, Theorem
, page 49]). It is clear from the definition of
that
(3.25)
so that
. By Theorem 3.4, it follows that
.
The second result is established analogously.
Remark 3.7.
Again we take note of how our results extend various earlier works. If
is defined by (3.7), then [7, Proposition
, page 512] follows from Corollary 3.6. If
is defined by (3.8), then Corollary 3.6 yields [5, Theorem
, page 4]. If
is defined by (3.9), then Corollary 3.6 reduces to [6, Theorem
, page 11].
Theorem 3.8.
Assume that
is a fixed integer and
is a fixed function in
. Let
be a convex univalent function satisfying
,
, and
with
.
(1)If
with respect to
, then
with respect to
.
(2)If
with respect to
, then
with respect to
.
Proof.
- (1)In view of the fact that
(3.26)
it is sufficient to prove that
when
. Let
. For
, define the functions
and
by
when
. Let
. For
, define the functions
and
by
(3.27)
Inequality (3.18) shows that
.
It is evident that
(3.28)
Since
and
, Theorem 3.3 yields
and
, Theorem 3.3 yields
(3.29)
and because
, it follows that
, it follows that
(3.30)
Thus
.
(
) The function
is in
if and only if
is in
and from the first part above, it follows that
. Hence
.
Corollary 3.9.
Assume that
is a fixed integer and
is a fixed function in
. Let
be a convex univalent function satisfying
,
. Let
be defined by (3.23). If
, then
.
The proof is analogous to Corollary 2.3 and is omitted.
Remark 3.10.
If
is defined by (3.8), then Corollary 3.9 yields [5, Theorem
, page 9]. If
is defined by (3.9), then Corollary 3.9 reduces to [6, Theorem
, page 14].
Declarations
Acknowledgment
The work presented here was supported in part by research grants from Universiti Sains Malaysia and University of Delhi.
Authors’ Affiliations
(1)
School of Mathematical Sciences, Universiti Sains Malaysia
(2)
Department of Mathematics, University of Delhi
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