On some inequalities of a certain class of analytic functions
© Raza et al.; licensee Springer 2012
Received: 11 March 2012
Accepted: 15 October 2012
Published: 29 October 2012
The aim of this paper is to study the properties of a subclass of analytic functions related to p-valent Bazilevic functions by using the concept of differential subordination. We investigate some results concerned with coefficient bounds, inclusion results, radius problem, covering theorem, angular estimation of a certain integral operator, and some other interesting properties.
1 Introduction and preliminaries
Using this concept, we generalize and define a subclass of p-valent Bazilevic functions of type as follows.
where , , , and β is any real.
For , , we have the following subclass of analytic functions.
We need the following definition and lemmas which will be used in our main results.
The univalent function q is called dominant of the differential subordination (1.4) if for all p satisfies (1.4). If for all dominants of (1.4), then we say that is the best dominant of (1.4).
Lemma 1.4 ()
and is the best dominant.
Lemma 1.5 ()
Let ε be a positive measure on . Let g be a complex-valued function defined on such that is analytic in E for each and is ε-integrable on for all . In addition, suppose that , is real and for and . If , then .
Lemma 1.6 ([, Chapter 14])
Lemma 1.7 ()
Lemma 1.8 ()
Lemma 1.9 ()
Lemma 1.10 ()
belongs to for .
Proof is straightforward by using Lemma 1.4.
Throughout this paper, , , , and unless otherwise stated.
2 Main results
This result is best possible.
Therefore, . □
For , we have the following result proved in .
This result is best possible.
by , where such that . Now, we derive the following result for the class .
Using the coefficient bound for the class , we have the required result. □
For and , we have the following result proved in .
For , , , and , we have the following result proved in .
Using Lemma 1.8, we have the required result. □
For , we have the following result.
This result is proved in .
for . Now have the following result for the class proved in .
Let . Since and , therefore, and . It follows that the root lies in . This implies that if , where is given by (2.10). □
For and , we have the following result which is proved in .
Let . Then for , and , and . It shows that the root lies in . This implies that if , where is given by (2.11). □
For , , , , we have the following result proved in .
for some , then
which is a contradiction to the assumption of our theorem. Hence, we have the proof. □
Remark 2.16 By using the suitable choices of parameters c, α, A and B, we can find many results proved in the literature.
The third author would like to express gratitude to Dr. S.M. Junaid Zaidi, Rector CIIT for his support and for providing excellent research facilities and to the Higher Education Commission of Pakistan for financial assistance.
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