# On Bounded Boundary and Bounded Radius Rotations

- K. I. Noor
^{1}, - W. Ul-Haq
^{1}, - M. Arif
^{1}Email author and - S. Mustafa
^{1}

**2009**:813687

**DOI: **10.1155/2009/813687

© K. I. Noor et al. 2009

**Received: **6 January 2009

**Accepted: **19 March 2009

**Published: **31 March 2009

## Abstract

We establish a relation between the functions of bounded boundary and bounded radius rotations by using three different techniques. A well-known result is observed as a special case from our main result. An interesting application of our work is also being investigated.

## 1. Introduction

Let be the class of functions of the form

which are analytic in the unit disc . We say that is subordinate to , written as , if there exists a Schwarz function , which (by definition) is analytic in with and , such that . In particular, when is univalent, then the above subordination is equivalent to and .

For any two analytic functions

the convolution (Hadamard product) of and is defined by

We denote by the classes of starlike and convex functions of order , respectively, defined by

For , we have the well-known classes of starlike and convex univalent functions denoted by and , respectively.

Let be the class of functions analytic in the unit disc satisfying the properties and

where , and For we obtain the class introduced in [1]. Also, for , we can write , We can also write, for ,

where is a function with bounded variation on such that

For (1.6) together with (1.7), see [2]. Since has a bounded variation on , we may write where and are two non-negative increasing functions on satisfying (1.7) Thus, if we set and then (1.6) becomes

Now, using Herglotz-Stieltjes formula for the class and (1.8), we obtain

where is the class of functions with real part greater than and , for , .

We define the following classes:

We note that

For we obtain the well-known classes and of analytic functions with bounded radius and bounded boundary rotations, respectively. These classes are studied by Noor [3–5] in more details. Also it can easily be seen that and

Goel [6] proved that implies that where

and this result is sharp.

In this paper, we prove the result of Goel [6] for the classes and by using three different methods. The first one is the same as done by Goel [6] while the second and third are the convolution and subordination techniques.

## 2. Preliminary Results

We need the following results to obtain our results.

Lemma 2.1.

Proof.

Using (2.3) together with (2.4) in (2.2), we obtain the required result.

Lemma 2.2 (see [9]).

Let , , and be a complex-valued function satisfying the conditions:

(i) is continuous in a domain

(ii) and

(iii) whenever and

If is a function analytic in such that and for then in

Lemma 2.3.

This result is a special case of the one given in [10, page 113].

## 3. Main Results

By using the same method as that of Goel [6], we prove the following result. We include all the details for the sake of completeness.

### 3.1. First Method

Theorem 3.1.

Let . Then , where is given by (1.12). This result is sharp.

Proof.

where and ,

where we integrate along the straight line segment ,

for all , we obtain the required result from (3.7), (3.13), and (3.14).

It is easy to check that where is the exact value given by (1.12).

### 3.2. Second Method

Theorem 3.2.

Proof.

and from it follows that

Since all the conditions of Lemma 2.2 are satisfied, it follows that in for and consequently and hence , where is given by (3.16). The case is discussed in [12].

### 3.3. Third Method

Theorem 3.3.

Proof.

Then are analytic in with

see [10]. MacGregor [13] conjectured the exact value given by (3.28). Thus and consequently where the exact value of is given by (3.28).

### 3.4. Application of Theorem 3.3

Theorem 3.4.

is in the class , where , , and is given by (1.12).

Proof.

Now by using the well-known fact that the class is a convex set together with (3.37), we obtain the required result.

For , , and , we have the following interesting corollary.

Corollary 3.5.

is in the class

## Declarations

### Acknowledgments

The authors are grateful to Dr. S. M. Junaid Zaidi, Rector, CIIT, for providing excellent research facilities and the referee for his/her useful suggestions on the earlier version of this paper. W. Ul-Haq and M. Arif greatly acknowledge the financial assistance by the HEC, Packistan, in the form of scholarship under indigenous Ph.D fellowship.

## Authors’ Affiliations

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