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].