ISRN Mathematical AnalysisVolume 2011 (2011), Article ID 610549, 7 pagesdoi:10.5402/2011/610549

Research Article

Note on Neighborhoods of Some Classes of Analytic Functions with Negative Coefficients

Irina Dorca,¹ Mugur Acu,² and Daniel Breaz³

¹Department of Mathematics, University of Piteşti, 110040 Argeş, Romania²Department of Mathematics, “Lucian Blaga” University of Sibiu, 550024 Sibiu, Romania³Department of Mathematics, “1 Decembrie 1918” University of Alba Iulia, 510009 Alba Iulia, Romania

Received 12 April 2011; Accepted 15 May 2011

Academic Editor: R. Avery

Copyright © 2011 Irina Dorca et al. 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.

In this paper, we prove several inclusion relations associated with the neighborhoods of some subclasses of starlike and convex functions with negative coefficients.

1. Introduction

Let be the set of functions which are regular in the unit disc , and .

In [1], the subfamily of consisting of functions of the form was introduced, where

For belonging to , Sălăgean [2] has introduced the following operator called the Sălăgean operator:

The object of the present paper is to derive some properties of neighborhoods for some subclasses of analytic functions with negative coefficients, which we have already studied.

2. Preliminary Results

Remark 2.1. In [3], we have introduced the following operator concerning the functions of form (1.2):

Definition 2.2 (see [3]). Let , , and , , . We say that is in the class if

Theorem 2.3 (see [3]). Let , , and . The function of the form (1.2) is in the class if and only if

Remark 2.4. Using the condition (1.3), we can to prove that  , , , and .

Definition 2.5 (see [4]). Let , , and , , . We say that is in the class if

Theorem 2.6 (see [4]). Let , , and . The function of the form (1.2) is in the class if and only if

Remark 2.7. Using the condition (2.2), we can prove that  , , , and .

Let be the class of functions of the form which are analytic in the open unit disk . For any and , we define which was called -neighborhood of . So, for , we observe that

The concept of neighborhoods was first introduced by Goodman in [5] and then generalized by Ruscheweyh in [6].

We propose to investigate the -neighborhoods of the subclasses and of the class of normalized analytic functions in with negative coefficients, where is the subclass of -starlike functions with negative coefficients of order and type introduced in [3] and is the subclass of -convex functions with negative coefficients of order and type studied in [4].

3. Main Results

We start by considering the linear operator (2.1) and conclude the study with several general inclusion relations associated with the neighborhoods for some subclasses of starlike and convex functions with negative coefficients.

Using the inequality (1.3) from Definition 2.5 and the inequality (2.1) from Definition  2.2, we obtain the subclasses , and and from Theorem 2.3, we derive the corresponding results.

Theorem 3.1. Let where , , , and ; then

Proof. For and making use of the condition (2.2), we obtain , so that On the other hand, we also find from (2.2) and (3.3) that Thus, which in view of definition (2.10), proves Theorem 3.1.

Remark 3.2. If and , we obtain that for , .

In a similar way, applying (2.5) instead of (2.2), we can prove the following.

Theorem 3.3. Let where , , , and ; then

Consequently, we determine the neighborhood for each of the classes and , which we define as follows. A function defined by (2.8) is said to be in the class if there exists a function such that

Analogously, a function defined by (2.8) is said to be in the class if there exists a function such the inequality (3.8) holds.

Remark 3.4. If , we obtain that for , .

Further, we consider the inclusion relations just studied and generalize them by taking into account the relation (2.9).

Theorem 3.5. If and then , , , and .

Proof. Let . Making use of (2.9), we find that which readily implies the coefficients of inequality Furthermore, since , we have so that provided that is given precisely by (3.9), which evidently completes our proof of Theorem 3.5.

Example 3.6. For a given , , , we consider with , , , (for , ), where is given by (3.9).Then we have that where .Therefore, .

In a similar way, we can prove Theorem 3.7.

Theorem 3.7. If and then , , , and .

Example 3.8. For a given , , , we consider with , , , (for , ), where is given by (3.16).Then we have that where .Therefore, .

Acknowledgment

This work was partially supported by the strategic project POSDRU 107/1.5/S/77265, inside POSDRU Romania 2007-2013 cofinanced by the European Social Fund-Investing in People.

References

H. Silverman, “Univalent functions with negative coefficients,” Proceedings of the American Mathematical Society, vol. 51, pp. 109–116, 1975.

G. S. Sălăgean, “Subclasses of univalent functions,” in Complex Analysis—Fifth Romanian-Finnish Seminar, Part 1 (Bucharest, 1981), vol. 1013 of Lecture Notes in Mathematics, pp. 362–372, Springer, Berlin, Germany, 1983.

M. Acu, I. Dorca, and S. Owa, “On some starlike functions with negative coefficients,” Acta Universitatis Apulensis, Special Issue, Alba Iulia. In press.

M. Acu, I. Dorca, and D. Breaz, “About some convex functions with negative coefficients,” Acta Universitatis Apulensis, no. 14, 2007.

A. W. Goodman, “Univalent functions and nonanalytic curves,” Proceedings of the American Mathematical Society, vol. 8, pp. 598–601, 1957.

S. Ruscheweyh, “Neighborhoods of univalent functions,” Proceedings of the American Mathematical Society, vol. 81, no. 4, pp. 521–527, 1981.