- Research Article
- Open Access
On Bounded Boundary and Bounded Radius Rotations
© K. I. Noor et al. 2009
- 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.
- Analytic Function
- Representation Form
- Hypergeometric Function
- Interesting Application
- Differentiation Yield
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
where , and For we obtain the class introduced in . Also, for , we can write , We can also write, for ,
For (1.6) together with (1.7), see . 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
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  proved that implies that where
and this result is sharp.
In this paper, we prove the result of Goel  for the classes and by using three different methods. The first one is the same as done by Goel  while the second and third are the convolution and subordination techniques.
We need the following results to obtain our results.
Using (2.3) together with (2.4) in (2.2), we obtain the required result.
Lemma 2.2 (see ).
This result is a special case of the one given in [10, page 113].
By using the same method as that of Goel , we prove the following result. We include all the details for the sake of completeness.
3.1. First Method
3.2. Second Method
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 .
3.3. Third Method
3.4. Application of Theorem 3.3
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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