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# On Bounded Boundary and Bounded Radius Rotations

*Journal of Inequalities and Applications*
**volume 2009**, Article number: 813687 (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.

Let . Then there exist such that

Proof.

It can easily be shown that if and only if there exists such that

From Brannan [7] representation form for functions with bounded boundary rotations, we have

Now, it is shown in [8] that for , we can write

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.

Let , and , with

If

then

where

denotes Gauss hypergeometric function. From (2.7), one can deduce the sharp result that with

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.

Since , we use Lemma 2.1, with relation (1.11) to have

where and ,

Therefore, from (2.4), we have

that is,

where we integrate along the straight line segment ,

Writing

and using (3.3) we have

where and hence by [11] we have

Therefore,

Let and , . For fixed and , we have from (2.4)

Now, using (3.8), we have, for a fixed ,

Let

with , , we have

By differentiating we note that

and therefore is a monotone increasing function of and hence

By letting

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

Sharpness can be shown by the function given by

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

### 3.2. Second Method

Theorem 3.2.

Let Then , where

Proof.

Let

is analytic in with Then

that is,

Since it implies that

We define

with By using (3.17) with convolution techniques, see [5], we have that

implies

Thus, from (3.20) and (3.23), we have

We now form the functional by choosing in (3.24) and note that the first two conditions of Lemma 2.2 are clearly satisfied. We check condition (iii) as follows:

where

The right-hand side of (3.25) is negative if and From , we have

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.

Let . Then , where

Proof.

Let

and let

Then are analytic in with

Logarithmic differentiation yields

Since it follows that , or for Consequently,

where , We use Lemma 2.3 with and in (3.32), to have where is given in (3.28) and this estimate is best possible, extremal function is given by

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.

Let and belong to . Then , defined by

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

Proof.

From (3.34), we can easily write

Since and belong to , then, by Theorem 3.3, and belong to , where is given by (1.12). Using

in (3.35), we have

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.

Let belongs to . Then , defined by

is in the class

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## 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.

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Noor, K.I., Ul-Haq, W., Arif, M. *et al.* On Bounded Boundary and Bounded Radius Rotations.
*J Inequal Appl* **2009, **813687 (2009). https://doi.org/10.1155/2009/813687

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DOI: https://doi.org/10.1155/2009/813687

### Keywords

- Analytic Function
- Representation Form
- Hypergeometric Function
- Interesting Application
- Differentiation Yield