`Abstract and Applied AnalysisVolume 2012, Article ID 345261, 15 pageshttp://dx.doi.org/10.1155/2012/345261`
Research Article

On Uniformly Bazilevic and Related Functions

Received 28 November 2011; Accepted 10 December 2011

Copyright © 2012 Khalida Inayat Noor. 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.

Abstract

We introduce a new class of functions analytic in the open unit disc, which contains the class of Bazilevic functions and also generalizes the concept of uniform convexity. We establish univalence criterion for the functions in this class and investigate rate of growth of coefficients, arc length problem, inclusion results, and distortion bounds. Some interesting results are derived as special cases.

1. Introduction

Let be the class functions analytic in the open unit disc and satisfying the conditions . Let be the class of functions which are univalent, and also let be the subclasses of which consists of starlike and convex functions of order , respectively.

Kanas and Wisniowska [1, 2] studied the classes of -uniformly convex functions, denoted by - and the corresponding class related with the Alexander-type relation. In [3], the domain is defined as follows: For fixed represents the conic region bounded, successively, by the imaginary axis , the right branch of hyperbola , a parabola , and an ellipse . Also, we note that, for no choice of reduces to a disc.

In this paper, we will choose . Related with , we define the domain , see [4], as follows: For , the following functions denoted by are univalent in , continuous as regards to and , have real coefficients, and map onto , such that : Let denote the class of the Caratheodory functions of positive real part. We define a subclass of as follows.

Definition 1.1. Let be the class consisting of functions which are analytic in with and which are subordinate to in . We write implies , where is the function, given by (1.3), and maps onto . That is . We note that and implies that . It is easy to verify that is a convex set, and .

The class is extended as follows.

Definition 1.2. Let be analytic in with . Then, if and only if, for , we have When , we obtain the class which reduces to the class with , introduced and studied in [5]. Also .

We now define the following.

Definition 1.3. Let with in . Let, for real and , Then if and only if For any real number and , we note that the identity function belongs to so that is not empty.

Throughout this paper, we assume that , unless otherwise specified.

We note the following special cases.(i)For , we have a subclass of a class introduced by Mocanu [6]. Also see [7, 8].(ii)It is well known that contains the Bazilevic functions with and .(iii), where is the well-known class of functions with bounded boundary rotation, see [9].(iv), where denotes the class of functions with bounded radius rotation, see [9].(v) is the class of uniformly convex functions of order , and .

Also , and .

Remark 1.4. (i)From Definition 1.3, it can easily be seen that if and only if, for , there exists a function such that where A simple computation shows that (1.7) can be written as with .(ii)Also, for , it can be verified from (1.5) that belongs to for all .

2. Preliminary Results

The following lemma is an easy generalization of a result due to Kanas [3].

Lemma 2.1 (see [10]). Let , and let be any complex numbers with and . If is analytic in and satisfies and is an analytic solution of then is univalent, and is the best dominant of (2.1), where is given as

Lemma 2.2 (see [11]). Let and , and let be a complex-valued function satisfying the following conditions:(i) is continuous in ,(ii) and ,(iii), whenever and .If is a function analytic in such that and for , then in .

Lemma 2.3 (see [12]). Let with in . Then is a Bazilevic function (hence univalent) in if and only if, for , one has where real.

3. Main Results

In the following, we establish the criterion of univalence for the class with certain restriction on the upper bound of the value of .

Theorem 3.1. Let . Then for in .

Proof. Since , we note that, in (1.9), . It is known [13] that there exists such that, for , Now from (1.9) and (3.1), we have, for , We use Lemma 2.3 with to have the required result.

As special cases, we note that(i)for is univalent for . We observe that, when , we obtain a well-known result that the class of functions with bounded boundary rotation contains univalent functions for , see [9];(ii)for is univalent for .

Let denote the Gamma function, and let be the hypergeometric function which is analytic in and is defined by where .

Define where Also, for , let We now consider the distortion problem.

Theorem 3.2. Let . Then, for ,(i), for ,(ii), for .

Proof. Suppose is a point on the circumference such that and let denote the preimage under on the segment .
Consider first case .
Distortion results for are given as follows: see [9]. In view of (1.7), (3.1), and (3.7), we have Hence where is defined in (3.4).
The proof of (i) for the upper bound of can be obtained in the similar manner.
The proof of (ii) for the case is analogous.

Remark 3.3. The bounds in Theorem 3.2 are sharp for , and the equality occurs for the function given by (3.6) with suitably chosen.
As an application of Theorem 3.2, we derive bound for initial Taylor coefficient of as follows.

Corollary 3.4. Let satisfy the conditions of Theorem 3.2. Then where and are given in (3.4).

Proof. First consider the case and assume to be real.
We find In view of Theorem 3.2(i), we have For , we proceed in a similar manner and use Theorem 3.2(ii) to complete the proof.

We note that and are increasing functions of . Thus letting in the left hand side of (i) and (ii) in Theorem 3.2, we have the following.

Corollary 3.5. Let satisfy the conditions of Theorem 3.2. Then where

Using Theorem 3.1 and Corollary 3.4, we have the following covering result.

Corollary 3.6. Let , and . If is the boundary of the image of under , then every point of is at distance at least from the origin.

Proof. Let . Then given by is univalent, since is univalent by Theorem 3.1. Writing , we have and since , it follows that Now using Corollary 3.4, and this proves the result.

As a special case, for , and is convex, and in this case, .

We now proceed to investigate some inclusion properties.

Theorem 3.7. Let . Then for . In particular, for , one has

Proof. Since , we can write from (1.7) or Let . Then is analytic in with . Then we can write (3.21) as or where .
Since and is a convex set, see [4], it follows that , with , belong to in .
Define Writing we use convolution technique, see [4], to have where denotes convolution (Hadamard product).
Therefore, from (3.23), (3.25), and (3.26), we have for We now use Lemma 2.1 with to obtain , and consequently in . This proves that in .

For , we can improve this result by restricting suitably as follows.

Corollary 3.8. Let . Then, , where

Proof. Writing and proceeding as in Theorem 3.7, it follows from (3.25) and (3.27) that for Constructing the functional with , we note that the first two conditions of Lemma 2.2 are easily verified. For condition (iii), we proceed as follows: where ,  .
The right-hand side of (3.30) is less than or equal to zero if and , and condition (iii) is satisfied. Form , we obtain as given by (3.28), and ensures that .
We now use Lemma 2.2 to have , and this implies .
Consequently in , and this completes the proof.

We note that with . Thus the above result can be restated in a general form as follows.

Corollary 3.9. Let . Then in , where

By taking , we have the following.

Corollary 3.10. Let . Then , where with When , one has , and this value of is sharp, see [14].

Proof. In Theorem 3.7, with , we have and therefore, from (3.27), it follows that By taking , it implies that That is, , and it has been proved in [14] that every function in the class is starlike of order where this order is exact and is given by (3.32).
Now, using the argument given in Theorem 3.7, we have , and the proof is complete.

We now prove the following.

Theorem 3.11. Let . Then

Proof. Let . Then, for Also, by Theorem 3.7, we have Now and since is convex set, in . This proves the result.

Let the length of the curve , and the area of the region bounded by .

We will now study the arc length problem for the class as follows.

Theorem 3.12. For , let . Then, for ,(i), where is given by (3.28).(ii).

Proof. We prove (i), and proof of (ii) follows on similar lines.
Solving (1.7) for , we obtain a formal representation as with . Integration by parts gives us where we have used Logarithmic differentiation of (3.39).
Now, from Corollary 3.8, it follows that , where is given by (3.28). Also, since , we have Using these observations in (3.41), we prove part (i).

Remark 3.13. In Theorem 3.12, we have, for , where is a constant depending only on , and . This can be improved by using the univalence criterion of as follows.

Corollary 3.14. Let , and . Then

Proof. In view of the univalence of by Theorem 3.1, we have , see [9], and Making now use of the area theorem and the Schwarz inequality, we obtain the required result.

Corollary 3.15. Let satisfy the condition of Theorem 3.12. Then

Proof is immediately since by Cauchy’s Theorem.

We can write Now taking in Theorem 3.12, we prove the result.

We study the arc length problem and corresponding rate of growth of coefficient for the class .

Theorem 3.16. Let, for . Then for , one has where depends only on , and and are as given by (3.4).

Proof. From (1.7), we can write Now, for , Since , we use a result proved in [4] and write Since , see [4].
We can write Using (3.51), (3.52), and distortion results for the class of starlike functions, we obtain from (3.50) where .

Remark 3.17. (i)For the case , we can solve the arc length problem in a similar manner. We define and proceed as before to obtain (ii)For , we have

We can derive the following result of order of growth of coefficients from Theorem 3.16 with the same method used before.

Corollary 3.18. Let . Then where depends only on and .
The exponent is best possible for . The extremal function satisfies the equation with

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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