Let
denote the class of functions
of the form
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
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
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 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
The class
consists of
for which
. The class
consists of
for which
or equivalently satisfying the condition
in
and the subordination
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
the class
reduces to the class
investigated in [9]. If
, where
then the class of
is the class
studied in [5]. If
, where
then the class
coincides with
investigated in [6].
Definition 3.2.
The class
consists of
,
,
, satisfying the subordination
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
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
,
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
We show that
satisfies the condition
. For
and
, clearly
Since
is a convex domain, it follows that
or
Since
, the subordination (3.16) yields
and thus
Inequality (3.18) shows that
.
A routine computation now gives
Since
and
, Theorem 3.3 yields
and because
, it is clear that
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:
(
) 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
If
, then
. Similarly, if
, then
.
Proof.
Define the function
by
For
, the function
is a convex function [11], and hence
([4, Theorem
, page 49]). It is clear from the definition of
that
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
it is sufficient to prove that
when
. Let
. For
, define the functions
and
by
Inequality (3.18) shows that
.
It is evident that
Since
and
, Theorem 3.3 yields
and because
, it follows that
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].