Convolution Properties of Classes of Analytic and Meromorphic Functions

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

(1.2)

Let 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

(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

(1.5)

The class consists of for which .

Now let and . From (1.4), it follows that

(1.6)

The convexity of 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

(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

(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

(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

(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

(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

(2.2)

It will first be proved that belongs to . For and , clearly

(2.3)

Since is a convex domain, it follows that

(2.4)

or

(2.5)

Since , the subordination (2.5) yields

(2.6)

and hence

A computation shows that

(2.7)

Since and , Theorem 1.3 yields

(2.8)

and because , 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

(2.11)

If , then . Similarly, if , then .

Proof.

Define the function 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

(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

(2.15)

Since , it is evident from (2.6) that .

That follows from Theorem 2.1(1). Now a computation shows that

(2.16)

Since and , Theorem 1.3 yields

(2.17)

and because , 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

(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

(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

(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

(3.5)

The class 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

(3.7)

the class reduces to the class investigated in [9]. If , where

(3.8)

then the class of is the class studied in [5]. If , where

(3.9)

then the class coincides with investigated in [6].

Definition 3.2.

The class 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

(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 ,

(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

(3.14)

Since is a convex domain, it follows that

(3.15)

or

(3.16)

Since , 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

(3.20)

and because , 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

(3.23)

If , then . Similarly, if , then .

Proof.

Define the function 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

(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

(3.27)

Inequality (3.18) shows that .

It is evident that

(3.28)

Since and , Theorem 3.3 yields

(3.29)

and because , 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].