On Generalized Bazilevic Functions Related with Conic Regions
Khalida Inayat Noor1and Kamran Yousaf1
Academic Editor: Yonghong Yao
Received12 Mar 2012
Accepted18 Mar 2012
Published20 May 2012
Abstract
We define and study some generalized classes of Bazilevic functions associated with convex domains. These convex domains are formed by conic regions which are included in the right half plane. Such results as inclusion relationships and integral-preserving properties are proved. Some interesting special cases of the main results are also pointed out.
1. Introduction
Let denote the class of analytic functions defined in the unit disc and satisfying the conditions ,. Let denote the subclass of consisting of univalent functions in , and let and be the subclasses of which contains, respectively, star-like and convex in BazileviΔ [1] introduced the class as follows.
Let . Then, real and if
for some and .
The powers appearing in (1.1) are meant as principle values. The functions in the class are shown to be analytic and univalent, see [1]. is the largest known subclass of univalent functions defined by an explicit formula and contains many of the heavily researched subclasses of . We note the following:(i),
(ii),
(iii), where is the class of close-to-convex functions introduced by Kaplan [2],(iv) is the class of -spiral like functions which are univalent for .
For analytic functions and , by we denote the Hadamard product (convolution) of and , defined by
For , the conic domain is defined in [3] as follows:
For fixed represents the conic region bounded successively by the imaginary axis , the right branch of hyperbola , a parabola and an ellipse .
The following univalent functions, defined by with and , map the unit disc onto
where is the Jacobi elliptic integral of the first kind:and is chosen such that , where is the complete elliptic integral of the first kind, .
It is known that are continuous as regards to and have real coefficients for .
Let be the subclass of the class of Caratheodory functions , analytic in with and such that is subordinate to , written as in .
We define the following.
Definition 1.1. Let be analytic in with . Then, if and only if, for , we can write
We note that , and , see [4].
Definition 1.2. Let . Then, is said to belong to the class if and only if for , and . For , the class coincides with the class of starlike functions, and consists of analytic functions with bounded radius rotation, see [5, 6]. Also is the class studied by several authors, see [7, 8].
Definition 1.3. Let . Then, if and only if is as given by (1.1) for some in with and real. When and , we obtain the class of Bazilevic functions.
We shall assume throughout, unless otherwise stated, that , real and .
2. Preliminary Results
Lemma 2.1 (see [3]). Let , and let be any complex numbers with and . If is analytic in , and satisfies
and is analytic solution of
then is univalent,
and is the best dominant of (2.1).
Lemma 2.2 (see [9]). Let be convex in and with . If is analytic in with and satisfies , then .
Lemma 2.3 (see [9]). Let , and let 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 .
3. Main Results
Theorem 3.1. Let , for and . Define
Then, in . In particular in .
Proof. Let
where is analytic in with , and let
From (3.1) and (3.2), we have
Logarithmic differentiation of (3.4) and some computation yield
That is
Let Then,
Using convolution technique (3.7) with , we obtain, from (3.3) and (3.6),
Since , we apply Lemma 2.1 with to obtain , where is the best dominant and is given as
Consequently, in , and this completes the result. As a special case, we prove the following.
Corollary 3.2. Let and let in . Then, for defined by (3.1), in where
Proof. We can write
where in . Now proceeding as before, we have, with
Using convolution technique together with (3.11), we obtain
for . We construct the functional by taking as
The first two conditions of Lemma 2.3 are clearly satisfied. We verify condition (iii) as follows.
where , . if and only if . From , we obtain as given by (3.10) and ensures that . Now proceeding as before, it follows from (3.12) that , and this proves our result.
By assigning certain permissible values to different parameters, we obtain several new and some known result.
Corollary 3.3. Let . Then, it is known that , and, form Corollary 3.2, it follows that where is given by (3.10). Also a starlike function is -uniformly convex for ,
Therefore, for , it follows that for , where is given by (3.10). As special cases we note the following. (i)For , we have and implies that , with
(ii)When , we have and .
Theorem 3.4. Let . Define, for ,
Then, in , where is given by (3.1), and is analytic in with .
Proof. Set
We note that is analytic in with . From (3.20), we have
using (3.1), we note that
From (3.21) and (3.22), it follows that
where since by Theorem 3.1. It can easily be seen that and . Now, using (3.8), we can easily derive
where and . Applying Lemma 2.2, it follows from (3.24) in and therefore in . This completes the proof.
Theorem 3.5. Let be given by (1.1) with ,. Then, for (i),
(ii)For ,
Proof. (i) From (1.1), we have
Define a function analytic in by
We can easily check that . Now, from (3.26) and (3.27), we have
That is
and, with , we apply convolution technique used before to have
Applying Lemma, it follows that
where is the best dominant and is given by
From (3.31), we have in , and this proves part (i). (ii) From part (i), we have
Now,
, since is convex set, see [8]. Therefore, for . This completes the proof.
As a special case, with ,, we obtain a result proved in [10].
By assigning certain permissible values to the parameters and , we have several other new results.
Acknowledgment
The authors are grateful to Dr. S. M. Junaid Zaidi, Rector, COMSATS Institute of Information Technology, Islamabad, Pakistan, for providing excellent research facilities and environment
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