Open Access

On Bounded Boundary and Bounded Radius Rotations

Journal of Inequalities and Applications20092009:813687

DOI: 10.1155/2009/813687

Received: 6 January 2009

Accepted: 19 March 2009

Published: 31 March 2009


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 [35] 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


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
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.


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 ,

and using (3.3) we have
where and hence by [11] we have
Let and , . For fixed and , we have from (2.4)
Now, using (3.8), we have, for a fixed ,
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


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
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:
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


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).


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



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

Department of Mathematics, COMSATS Institute of Information Technology


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© K. I. Noor et al. 2009

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