Abstract and Applied Analysis

VolumeÂ 2012, Article IDÂ 383592, 10 pages

http://dx.doi.org/10.1155/2012/383592

## -Bazilevic Functions

Department of Mathematical Sciences, College of Science, Princess Nora bint Abdul Rahman University, P.O. Box 4384, Riyadh 11491, Saudi Arabia

Received 31 December 2011; Revised 2 March 2012; Accepted 3 March 2012

Academic Editor: Khalida InayatÂ Noor

Copyright Â© 2012 F. M. Al-Oboudi. 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

The aim of this paper is to define and study a class of Bazilevic functions using the generalized Salagean operator. Some properties of this class are investigated: inclusion relation, some convolution properties, coefficient bounds, and other interesting results.

#### 1. Introduction

Let be the set of analytic functions in the open unit disc . Let be the set of functions , with , and let be the set of functions , with . Let be the class of functions , which are univalent in . Denote by , , the class of starlike (convex)(close-to-convex) functions of order . Note that when , then , let . A function , where belongs to the Kaplan class , , , [1] if for and .

The Dual of is defined as where denotes Hadamard product (convolution).

A set is called a test set for (denoted by ) if . Note that if , then .

Denote by the class of functions , such that , in , and let . Note that for , and that , , if, and only if, .

For and , define the class as Note that , , .

A function , is called prestarlike of order , , (denoted by ) if and only if , or if and only if

Let , , , denote the class of Bazilevic functions in , introduced by Bazilevic [2], , , , if and only if there exists a , such that for where at . We denote by . Bazilevic shows that , for , . Note that

For further information, see [3â€“7].

The generalized Salagean operator , , , is defined [8] as where

The Operator satisfies the following identity: Not that for , , Salagean differential operator [9].

Let we mean by , the solution of . Hence where . It is known [10] that , hence , and that

The class , is defined as if and only if . For , we get Salagean-type -starlike functions [9].

The operator is now called â€śAl-Oboudi Operatorâ€ť and has been extensively studied latly, [5, 11, 12].

In this paper we define and study a class of Bazilevic functions using the operator and study some of its basic properties, inclusion relation, convolution properties coefficient bounds, and other interesting results.

#### 2. Definition and Preliminaries

In this section, the class of -Bazilevic functions , , where , is defined and some preliminary lemmas are given.

##### 2.1. Definition

Let . Then , , , if and only if there exists a , such that where the power is chosen as a principal one.

Denote by the class of functions , where .

Using (1.10), we see that , from which the following special cases are clear.

###### 2.1.1. Special Cases

(1)For , , , Bazilevic [2].(2)For , , Salagean type close to convex functions, Blezu [13].(3)For , , , Kaplan [14].(4)For , , Abdul Halim [15] and , Opoola [16].##### 2.2. Lemmas

The following lemmas are needed to prove our results.

Lemma 2.1 (see [10]). *Let and , . Then for *

Lemma 2.2. *If , , then .*

*Proof. *Since , we will show that . Now , implies
Hence
From (1.13), we get the required result.

Lemma 2.3 (see [1]). *Let and , . Then .*

For , let denote the largest positive number so that every is convex in . The following result is due to Al-Amiri [17].

Lemma 2.4. *One has
*

Lemma 2.5 (see [18]). *Let , with and . Let in , where
**
Then for each ,
**
where stands for closed convex hull. *

*Remark 2.6. *In [1], it was shown that condition (2.6) is satisfied for all in whenever is in and is in .

Lemma 2.7 (see [10]). *Let be such that
**
and let , , . Then
*

From (1.12) and (1.13), we immediately have;

Lemma 2.8. *One has
*

Lemma 2.9 (see [10]). *Let , . Then
**
where stands for coefficient majorization.*

#### 3. Main Results

Theorem 3.1. *One has
*

*Proof. *Let . Then there exists , such that
Hence
Since , and , application of Lemma 2.3 gives
hence
Using Lemma 2.2 we deduce that .

As a consequence of (3.1) we immediately have the following.Corollary 3.2. *One has
*

Corollary 3.3. *If , , , then, for *

*Proof. *Since , there exists a or such that
using (1.3). From (3.6), we conclude that
which implies that
Applying Lemma 2.3 to (3.9), we get the result.

Theorem 3.4. *Let . Then
*

*Proof. *From (2.1), we see that
Since , and , then
which is the required result.

In the following we prove the converse of Theorem 3.1, for .

Theorem 3.5. *Let , . Then in , where is given by (2.5)*

*Proof. * implies (2.1), where .

Now
Using Lemma 2.4, we see that in , for , where is given by (2.5).

From Remark 2.6, we conclude

Applying Lemma 2.5, we deduce
hence in , as required.

Corollary 3.3 can be improved for , as follows.

Theorem 3.6. *Let , . Then
*

*Proof. *We will use Ruscheweyhâ€™s.method of proof [10]. implies (3.8), where , .

Let , where and .

Then and . This implies

Using Lemma 2.7, we get

Hence
To prove that , we have to show that , or equivalently .

Since , then from (1.5)
From Lemma 2.8, (1.13), and (3.22), we see that . Using (3.21), we obtain . From (1.1) we get the required result.

*Remark 3.7. *For , Theorem 3.6 and other stronger results depending on , are proved by Sheil-Small [7].

For the coefficient bounds of , Theorem 3.6 is not strong enough to settle this problem for , In 1962, Zamorski [19] proved the Bieberbach conjecture for , when , in the following we prove this result for , using the extreme points of Kaplan class .

Theorem 3.8. *For , ,
*

*Proof. *From (3.9) and Lemma 2.9, we get
using (2.10). Raising both sides of (3.24) to the th power, where , we get the required result.

*Remark 3.9. *For , we get the result of Zamorski [19], and the result of Sheil-Small [7], from which we get the idea of proof.

#### References

- T. Sheil-Small, â€śThe Hadamard product and linear transformations of classes of analytic functions,â€ť
*Journal d'Analyse Mathematique*, vol. 34, pp. 204â€“239, 1978. View at Publisher Â· View at Google Scholar - I. E. Bazilevic, â€śOn a case of integrability in quadratures of the Loewner-Kufarev equation,â€ť
*Matematicheskii Sbornik*, vol. 37, pp. 471â€“476, 1955. View at Google Scholar - M. Arif, K. I. Noor, and M. Raza, â€śOn a class of analytic functions related with generalized Bazilevic type functions,â€ť
*Computers and Mathematics with Applications*, vol. 61, no. 9, pp. 2456â€“2462, 2011. View at Publisher Â· View at Google Scholar - Y. C. Kim, â€śA note on growth theorem of Bazilevic functions,â€ť
*Applied Mathematics and Computation*, vol. 208, no. 2, pp. 542â€“546, 2009. View at Publisher Â· View at Google Scholar - A. T. Oladipo, â€śOn a new subfamilies of Bazilevic functions,â€ť
*Acta Universitatis Apulensis*, no. 29, pp. 165â€“185, 2012. View at Google Scholar - Q. Deng, â€śOn the coefficients of Bazilevic functions and circularly symmetric functions,â€ť
*Applied Mathematics Letters*, vol. 24, no. 6, pp. 991â€“995, 2011. View at Publisher Â· View at Google Scholar - T. Sheil-Small, â€śSome remarks on Bazilevic functions,â€ť
*Journal d'Analyse Mathematique*, vol. 43, pp. 1â€“11, 1983. View at Publisher Â· View at Google Scholar - F. M. Al-Oboudi, â€śOn univalent functions defined by a generalized Salagean operator,â€ť
*International Journal of Mathematics and Mathematical Sciences*, no. 25–28, pp. 1429â€“1436, 2004. View at Publisher Â· View at Google Scholar - G. S. Salagean, â€śSubclasses of univalent functions,â€ť in
*Complex Analysis, Fifth Romanian-Finnish Seminar, Part 1 (Bucharest, 1981)*, vol. 1013 of*Lecture Notes in Mathematics*, pp. 362â€“372, Springer, Berlin, Germany, 1983. View at Publisher Â· View at Google Scholar - St. Ruscheweyh,
*Convolutions in Geometric Function Theory*, vol. 83 of*Seminaire de Mathematiques Superieures*, Presses de l'Universite de Montreal, Montreal, Canada, 1982. - A. Cătaş, G. I. Oros, and G. Oros, â€śDifferential subordinations associated with multiplier transformations,â€ť
*Abstract and Applied Analysis*, vol. 2008, Article ID 845724, 11 pages, 2008. View at Publisher Â· View at Google Scholar - M. Darus and I. Faisal, â€śA study on Becker's univalence criteria,â€ť
*Abstract and Applied Analysis*, vol. 2011, Article ID 759175, 13 pages, 2011. View at Publisher Â· View at Google Scholar - D. Blezu, â€śOn the n-close-to-convex functions with respect to a convex set. I,â€ť
*Mathematica Revue d'Analyse Numérique et de Théorie de l'Approximation*, vol. 28(51), no. 1, pp. 9â€“19, 1986. View at Google Scholar - W. Kaplan, â€śClose-to-convex schlicht functions,â€ť
*The Michigan Mathematical Journal*, vol. 1, pp. 169â€“185, 1952. View at Google Scholar - S. Abdul Halim, â€śOn a class of analytic functions involving the Salagean differential operator,â€ť
*Tamkang Journal of Mathematics*, vol. 23, no. 1, pp. 51â€“58, 1992. View at Google Scholar - T. O. Opoola, â€śOn a new subclass of univalent functions,â€ť
*Mathematica*, vol. 36, no. 2, pp. 195â€“200, 1994. View at Google Scholar - H. S. Al-Amiri, â€śOn the Hadamard products of schlicht functions and applications,â€ť
*International Journal of Mathematics and Mathematical Sciences*, vol. 8, no. 1, pp. 173â€“177, 1985. View at Publisher Â· View at Google Scholar - R. W. Barnard and C. Kellogg, â€śApplications of convolution operators to problems in univalent function theory,â€ť
*The Michigan Mathematical Journal*, vol. 27, no. 1, pp. 81â€“94, 1980. View at Google Scholar - J. Zamorski, â€śOn Bazilevic schlicht functions,â€ť
*Annales Polonici Mathematici*, vol. 12, pp. 83â€“90, 1962. View at Google Scholar