Abstract

Making use of the concept of k-uniformly bounded boundary rotation and Ruscheweyh differential operator, we introduce some new classes of meromorphic functions in the punctured unit disc. Convolution technique and principle of subordination are used to investigate these classes. Inclusion results, generalized Bernardi integral operator, and rate of growth of coefficients are studied. Some interesting consequences are also derived from the main results.

1. Introduction

Let denote the class of functions of the form which are analytic in punctured unit disc . At , the function has a simple pole.

Let, for , and be well-known subclasses of consisting of functions meromorphic starlike and meromorphic convex of order , respectively, see [1].

Let , and be given by (1.1). Then convolution (Hadamard product) of and is defined by Robertson [2] showed that also belongs to .

Let and define, for as It can easily be seen that, for , We note that and so Equation (1.6) can be verified as follows.

Since we have From (1.3), we can readily obtain the following identity for and :  , is known as generalized Ruscheweyh derivative for meromorphic functions.

For , define the domain as follows, see [3]: For fixed , represents the conic region bounded successively, by the imaginary axis , the right branch of hyperbola , and a parabola. Related with , the domain can be defined as below, see [4]: The functions with , univalent in , map onto and are given as in the following:

The functions are continuous as regard to , have real coefficients for , and play the part of extremal ones for many problems related to .

Let be the class of analytic functions with positive real part, and let be the class of functions which are analytic in , such that for , where “” denotes subordination, and are given by (1.12).

We define the following.

Definition 1.1. Let be analytic in with . Then is said to belong to the class , for , if and only if there exist such that We note that(i), and with coincides with and implies in ;(ii)when , , we have the class introduced in [5].

Definition 1.2. Let . Then is said to belong to the class if and only if in .
For , we obtain the class of meromorphic starlike functions of order .
We can define the class by the following relation: When , , , we obtain of meromorphic convex functions.

Definition 1.3. Let . Then if and only if for .
Similarly if and only if . We note that the classes and are related by relation (1.14).
For , , we have , the class of meromorphic functions of bounded boundary rotation which was studied in [6]. The functions have integral representation of the form where is a real-valued function of bounded variation on satisfying the conditions

With simple computations, it can easily be seen that the third of conditions (1.16) guarantees that the singularity of at is a simple pole with no logarithm term.

Also it is known that if and only if is a domain containing infinity with boundary rotation at most , see [6]. The class is wellknown [1] and consists of analytic functions with boundary rotation at most . Noonan [6] established the relation between the classes and as follows.

A function if and only if there exists of the form with such that

It is also shown [6] that, for , there exist given by , , such that and

We note that and therefore of analytic functions, in (1.18). This give us by distortion results and subordination for the class .

We can easily extend the relations (1.17) and (1.18) by noting that implies that there exists such that , , see [7]. For , , we can write relation (1.18) as Throughout this paper, we will assume , , and unless otherwise stated.

We also note that all the results proved in this paper hold for in general.

2. Preliminary Results

The following lemma is a generalized version of a result proved in [3].

Lemma 2.1 (see [4]). Let and let be any complex numbers with and . If is analytic in and satisfies and is an analytic solution of then is univalent, and is the best dominant of (2.1).

Lemma 2.2 (see [8]). Let , and let be complex-valued function satisfying the following conditions:(i) is continuous in a domain ,(ii) and ,(iii) whenever and .If is a function analytic in such that , and for , then for .

Lemma 2.3 (see [9]). Let and be analytic in and and for . Further let and be complex constants such that . Then where This result is sharp for real and nonnegative constant.

3. Main Results

Theorem 3.1. One has This result is best possible and sharpness follows from the best dominant property.

Proof. Let , and set with Then is single-valued in .
Using identity (1.9), it follows that is analytic in and .
Now, from (1.9) and (3.4), we obtain With , , we can write (3.5) as where , .
Since , it follows from (3.6) that Define and let Then using convolution technique, we have Thus, from (3.7) and (3.10), we obtain It can easily be seen that , so we apply Lemma 2.1 to have from (3.11) where is the best dominant and is given as Since , , , we have and from this it follows that for .
Now from (3.4) we have in and the proof is complete.

As a special case, we have the following.

Corollary 3.2. Let in Theorem 3.1. Then where and

Proof. From (3.6), we have Proceeding as in Theorem 3.1, it follows that Let , .
Then We construct a functional by taking , . Then The first two conditions of Lemma 2.2 are easily verified. For condition (iii), we proceed as follows: By putting , we have where From , we obtain as given by (3.15) and ensures .
Applying Lemma 2.2, we now have , and therefore in , consequently in and the proof is complete.

We note that, for , we have Also, for , and .

We will now investigate the rate of growth of coefficients for and the corresponding result for the class will follow from the relation (1.14).

Theorem 3.3. Let and be given by (1.1). Then, for , , , one has and depends only on and .
The exponent in (3.24) is best possible for the class as can be seen from the function given by

Proof. Since , and implies , we use (1.20) to write and .
Now, with and , we have Pommerenke [10] has shown that Thus we use (1.19), (3.26), and (3.28) to have from (3.27) We take , , denotes gamma function, and have where is as given in (3.24) and is a constant depending only on and .
This completes the proof.

Next we will show that the class is preserved under an integral operator.

For , the generalized Bernardi operator for the class is defined in [11] as below.

Let and be given by (1.1). Then the integral transform is defined as Also It easily follows from (3.31) that and as in (1.9), From (3.33) and (3.34), we have We now prove the following.

Theorem 3.4. Let . Then , defined by (3.31), also belong to the same class in .

Proof. We put Then is single valued and analytic in and defined by is analytic in .
Form (3.33), (3.34), and (3.37), we obtain Since , it follows that and with the convolution technique used before, we have, for Since , we apply Lemma 2.1 to have , where is the best dominant. The required result now follows from (3.37).

Corollary 3.5. Let . Then and , defined by (3.31), belongs to , where is given as

The proof follows on the similar lines of Corollary 3.2.

Remark 3.6. We note that , , and therefore , , .

We prove a partial converse of Theorem 3.4 as following.

Theorem 3.7. Let be defined by (3.31) and let, for , , . Then for , where

Proof. We write Since , in with , and , .
Proceeding on the similar lines as before, we obtain form (3.42) and (3.43) and with convolution technique as previously used, we get from (3.44) where , , , , .
Then, by using, Lemma 2.3, we have where is given by (3.42).
Now, from (3.44) and (3.46), we have the required result that in .
From Lemma 2.3 it follows that this result is sharp for .

Acknowledgment

The authors are grateful to Dr. S. M. Junaid Zaidi, Rector, COMSATS Institute of Information Technology, Park Road, Islamabad, Pakistan for providing excellent research facilities.