Research Article | Open Access
Note on Neighborhoods of Some Classes of Analytic Functions with Negative Coefficients
In this paper, we prove several inclusion relations associated with the neighborhoods of some subclasses of starlike and convex functions with negative coefficients.
Let be the set of functions which are regular in the unit disc , and .
In , the subfamily of consisting of functions of the form was introduced, where
For belonging to , Sălăgean  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
Definition 2.2 (see ). Let , , and , , . We say that is in the class if
Remark 2.4. Using the condition (1.3), we can to prove that , , , and .
Definition 2.5 (see ). Let , , and , , . We say that is in the class 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
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  and is the subclass of -convex functions with negative coefficients of order and type studied in .
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.
Theorem 3.1. Let where , , , and ; then
Remark 3.2. If and , we obtain that for , .
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
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 .
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 .
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.
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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.