# 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

https://doi.org/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:

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 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.

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.

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.

## 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

## References

- Pinchuk B:
**Functions of bounded boundary rotation.***Israel Journal of Mathematics*1971,**10**(1):6–16. 10.1007/BF02771515MathSciNetView ArticleMATHGoogle Scholar - Padmanabhan KS, Parvatham R:
**Properties of a class of functions with bounded boundary rotation.***Annales Polonici Mathematici*1975,**31**(3):311–323.MathSciNetMATHGoogle Scholar - Noor KI:
**Some properties of certain analytic functions.***Journal of Natural Geometry*1995,**7**(1):11–20.MathSciNetMATHGoogle Scholar - Noor KI:
**On some subclasses of functions with bounded radius and bounded boundary rotation.***Panamerican Mathematical Journal*1996,**6**(1):75–81.MathSciNetMATHGoogle Scholar - Noor KI: On analytic functions related to certain family of integral operators. Journal of Inequalities in Pure and Applied Mathematics 2006,7(2, article 69):-6.Google Scholar
- Goel RM:
**Functions starlike and convex of order A.***Journal of the London Mathematical Society. Second Series*1974,**s2–9**(1):128–130.MathSciNetView ArticleMATHGoogle Scholar - Brannan DA:
**On functions of bounded boundary rotation. I.***Proceedings of the Edinburgh Mathematical Society. Series II*1969,**16:**339–347. 10.1017/S001309150001302XMathSciNetView ArticleMATHGoogle Scholar - Pinchuk B:
**On starlike and convex functions of order .***Duke Mathematical Journal*1968,**35:**721–734. 10.1215/S0012-7094-68-03575-8MathSciNetView ArticleMATHGoogle Scholar - Miller SS:
**Differential inequalities and Carathéodory functions.***Bulletin of the American Mathematical Society*1975,**81:**79–81. 10.1090/S0002-9904-1975-13643-3MathSciNetView ArticleMATHGoogle Scholar - Miller SS, Mocanu PT:
*Differential Subordinations: Theory and Application, Monographs and Textbooks in Pure and Applied Mathematics*.*Volume 225*. Marcel Dekker, New York, NY, USA; 2000:xii+459.Google Scholar - Nahari Z:
*Conformal Mappings*. Dover, New York, NY, USA; 1952.Google Scholar - Jack IS:
**Functions starlike and convex of order .***Journal of the London Mathematical Society. Second Series*1971,**3:**469–474. 10.1112/jlms/s2-3.3.469MathSciNetView ArticleMATHGoogle Scholar - MacGregor TH:
**A subordination for convex functions of order .***Journal of the London Mathematical Society. Second Series*1975,**9:**530–536. 10.1112/jlms/s2-9.4.530MathSciNetView ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.