Table of Contents
ISRN Mathematical Analysis
VolumeΒ 2011Β (2011), Article IDΒ 610549, 7 pages
http://dx.doi.org/10.5402/2011/610549
Research Article

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

1Department of Mathematics, University of Piteşti, 110040 Argeş, Romania
2Department of Mathematics, β€œLucian Blaga” University of Sibiu, 550024 Sibiu, Romania
3Department of Mathematics, β€œ1 Decembrie 1918” University of Alba Iulia, 510009 Alba Iulia, Romania

Received 12 April 2011; Accepted 15 May 2011

Academic Editor: R.Β Avery

Copyright Β© 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 (𝑛,𝛿) neighborhoods of some subclasses of starlike and convex functions with negative coefficients.

1. Introduction

Let β„‹(π‘ˆ) be the set of functions which are regular in the unit disc π‘ˆ, 𝐴=π‘“βˆˆβ„‹(π‘ˆ)βˆΆπ‘“(0)=π‘“ξ…žξ€Ύ(0)βˆ’1=0(1.1) and 𝑆={π‘“βˆˆπ΄βˆΆπ‘“isunivalentinπ‘ˆ}.

In [1], the subfamily 𝑇 of 𝑆 consisting of functions 𝑓 of the form 𝑓(𝑧)=π‘§βˆ’βˆžξ“π‘—=2π‘Žπ‘—π‘§π‘—,π‘Žπ‘—β‰₯0,𝑗=2,3,…,π‘§βˆˆπ‘ˆ,(1.2) was introduced, where 𝑇=π‘“βˆˆπ‘†βˆΆπ‘“(𝑧)=π‘§βˆ’βˆžξ“π‘—=2π‘Žπ‘—π‘§π‘—,π‘Žπ‘—ξƒ°β‰₯0,𝑗β‰₯2,π‘§βˆˆπ‘ˆ.(1.3)

For 𝑓(𝑧) belonging to 𝐴, SΔƒlΔƒgean [2] has introduced the following operator called the SΔƒlΔƒgean operator: 𝐷0𝑓(𝑧)=𝑓(𝑧),𝐷1𝑓(𝑧)=𝐷𝑓(𝑧)=π‘§π‘“ξ…ž(𝑧),𝐷𝑛𝐷𝑓(𝑧)=π·π‘›βˆ’1𝑓(𝑧),π‘›βˆˆβ„•βˆ—.(1.4)

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): π·π›½πœ†βˆΆπ΄βŸΆπ΄,π·π›½πœ†π‘“(𝑧)=π‘§βˆ’βˆžξ“π‘—=𝑛+1[]1+(π‘—βˆ’1)πœ†π›½π‘Žπ‘—π‘§π‘—.(2.1)

Definition 2.2 (see [3]). Let π‘“βˆˆπ‘‡, βˆ‘π‘“(𝑧)=π‘§βˆ’βˆžπ‘—=2π‘Žπ‘—π‘§π‘—, and π‘Žπ‘—β‰₯0, 𝑗=2,3,…, π‘§βˆˆπ‘ˆ. We say that 𝑓 is in the class 𝑇𝐿𝛽(𝛼) if 𝐷Reπœ†π›½+1𝑓(𝑧)π·π›½πœ†[𝑓(𝑧)>𝛼,π›Όβˆˆ0,1),πœ†β‰₯0,𝛽β‰₯0,π‘§βˆˆπ‘ˆ.(2.2)

Theorem 2.3 (see [3]). Let π›Όβˆˆ[0,1), πœ†β‰₯0,π‘›βˆˆβ„•βˆ—, and 𝛽β‰₯0. The function π‘“βˆˆπ΄(𝑛) of the form (1.2) is in the class π‘‡βˆ—πΏπ›½,𝑛(𝛼) if and only if βˆžξ“π‘—=𝑛+1ξ€Ί(1+(π‘—βˆ’1)πœ†)𝛽(ξ€»1+(π‘—βˆ’1)πœ†βˆ’π›Ό)β‹…π‘Žπ‘—<1βˆ’π›Ό.(2.3)

Remark 2.4. Using the condition (1.3), we can to prove that π‘‡βˆ—πΏπ›½+1,𝑛(𝛼)βŠ‚π‘‡βˆ—πΏπ›½,𝑛(𝛼),(2.4)  𝛽β‰₯0, π›Όβˆˆ[0,1), πœ†β‰₯0, and π‘›βˆˆβ„•βˆ—.

Definition 2.5 (see [4]). Let π‘“βˆˆπ‘‡, βˆ‘π‘“(𝑧)=π‘§βˆ’βˆžπ‘—=2π‘Žπ‘—π‘§π‘—, and π‘Žπ‘—β‰₯0, 𝑗=2,3,…, π‘§βˆˆπ‘ˆ. We say that 𝑓 is in the class 𝑇𝑐𝐿𝛽(𝛼) if 𝐷Reπœ†π›½+2𝑓(𝑧)π·πœ†π›½+1[𝑓(𝑧)>𝛼,π›Όβˆˆ0,1),πœ†β‰₯0,𝛽β‰₯0,π‘§βˆˆπ‘ˆ.(2.5)

Theorem 2.6 (see [4]). Let π›Όβˆˆ[0,1), πœ†β‰₯0, and 𝛽β‰₯0. The function π‘“βˆˆπ‘‡ of the form (1.2) is in the class 𝑇𝑐𝐿𝛽(𝛼) if and only if βˆžξ“π‘—=2ξ€Ί(1+(π‘—βˆ’1)πœ†)𝛽+1(ξ€»π‘Ž1+(π‘—βˆ’1)πœ†βˆ’π›Ό)𝑗<1βˆ’π›Ό.(2.6)

Remark 2.7. Using the condition (2.2), we can prove that 𝑇𝑐𝐿𝛽+1,𝑛(𝛼)βŠ‚π‘‡π‘πΏπ›½,𝑛(𝛼),(2.7)  𝛽β‰₯0, π›Όβˆˆ[0,1), πœ†β‰₯0, and π‘›βˆˆβ„•βˆ—.

Let 𝐴(𝑛) be the class of functions 𝑓(𝑧) of the form 𝑓(𝑧)=π‘§βˆ’βˆžξ“π‘˜=𝑛+1π‘Žπ‘˜π‘§π‘˜,ξ€·π‘Žπ‘˜ξ€Έβ‰₯0;π‘›βˆˆβ„•βˆ’{0},(2.8) which are analytic in the open unit disk π‘ˆ={π‘§βˆΆ|𝑧|<1}. For any 𝑓(𝑧)∈𝐴(𝑛) and 𝛿β‰₯0, we define 𝑁𝑛,𝛿=ξƒ―π‘”βˆˆπ΄(𝑛)βˆΆπ‘”(𝑧)=π‘§βˆ’βˆžξ“π‘˜=𝑛+1π‘π‘˜π‘§π‘˜,βˆžξ“π‘˜=𝑛+1||π‘Žπ‘˜β‹…π‘˜βˆ’π‘π‘˜||≀𝛿,(2.9) which was called (𝑛,𝛿)-neighborhood of 𝑓(𝑧). So, for 𝑒(𝑧)=𝑧, we observe that 𝑁𝑛,𝛿=ξƒ―π‘”βˆˆπ΄(𝑛)βˆΆπ‘”(𝑧)=π‘§βˆ’βˆžξ“π‘˜=𝑛+1π‘π‘˜π‘§π‘˜,βˆžξ“π‘˜=𝑛+1||π‘π‘˜β‹…π‘˜||≀𝛿.(2.10)

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

We propose to investigate the (𝑛,𝛿)-neighborhoods of the subclasses π‘‡βˆ—πΏπ›½,𝑛(𝛼) and 𝑇𝑐𝐿𝛽,𝑛(𝛼) of the class 𝐴(𝑛) of normalized analytic functions in π‘ˆ with negative coefficients, where π‘‡βˆ—πΏπ›½,𝑛(𝛼) is the subclass of 𝑛-starlike functions with negative coefficients of order 𝛼 and type 𝛽 introduced in [3] and 𝑇𝑐𝐿𝛽,𝑛(𝛼) is the subclass of 𝑛-convex functions with negative coefficients of order 𝛼 and type 𝛽 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 (𝑛,𝛿) 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 π‘‡βˆ—πΏπ›½,𝑛(𝛼), and 𝑇𝑐𝐿𝛽,𝑛(𝛼) and from Theorem 2.3, we derive the corresponding results.

Theorem 3.1. Let 𝛿=(1βˆ’π›Ό)(𝑛+1)(1+π‘›πœ†)𝛽(1+π‘›πœ†βˆ’π›Ό),(3.1) where π›Όβˆˆ[0,1), πœ†β‰₯0, 𝛽β‰₯0, and π‘›βˆˆβ„•βˆ—; then π‘‡βˆ—πΏπ›½,𝑛(𝛼)βŠ‚π‘π‘›,𝛿(𝑒).(3.2)

Proof. For 𝑓(𝑧)βˆˆπ‘‡βˆ—πΏπ›½,𝑛(𝛼) and making use of the condition (2.2), we obtain [(1+π‘›πœ†)π›½βˆ‘(1+π‘›πœ†βˆ’π›Ό)]βˆžπ‘—=𝑛+1π‘Žπ‘—<1βˆ’π›Ό, so that βˆžξ“π‘—=𝑛+1π‘Žπ‘—<1βˆ’π›Ό(1+π‘›πœ†)𝛽(1+π‘›πœ†βˆ’π›Ό).(3.3) On the other hand, we also find from (2.2) and (3.3) that βˆžξ“π‘—=𝑛+1ξ€Ί(1+(π‘—βˆ’1)πœ†)𝛽(ξ€»π‘Ž1+(π‘—βˆ’1)πœ†βˆ’π›Ό)𝑗(<1βˆ’π›Ό,1+π‘›πœ†)π›½βˆžξ“π‘—=𝑛+1(1+(π‘—βˆ’1)πœ†βˆ’π›Ό)π‘Žπ‘—<1βˆ’π›Ό,πœ†(1+π‘›πœ†)π›½βˆžξ“π‘—=𝑛+1π‘—β‹…π‘Žπ‘—ξ€Ί(<1βˆ’π›Όβˆ’1βˆ’πœ†βˆ’π›Ό)(1+π‘›πœ†)π›½ξ€»β‹…βˆžξ“π‘—=𝑛+1π‘Žπ‘—.(3.4)
Thus, βˆžξ“π‘—=𝑛+1π‘—β‹…π‘Žπ‘—<(1βˆ’π›Ό)(𝑛+1)(1+π‘›πœ†)𝛽(1+π‘›πœ†βˆ’π›Ό)=𝛿,(3.5) which in view of definition (2.10), proves Theorem 3.1.

Remark 3.2. If πœ†=1 and 𝛽=1, we obtain that βˆ‘βˆžπ‘—=𝑛+1π‘—β‹…π‘Žπ‘—<(1βˆ’π›Ό)/(1+π‘›βˆ’π›Ό)=𝛿 for 𝐷11𝑓(𝑧)=π‘§π‘“ξ…ž(𝑧), 𝑓(𝑧)βˆˆπ‘‡βˆ—π‘›(𝛼).

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

Theorem 3.3. Let 𝛿=(1βˆ’π›Ό)(𝑛+1)(1+π‘›πœ†)𝛽+1(1+π‘›πœ†βˆ’π›Ό),(3.6) where π›Όβˆˆ[0,1), πœ†β‰₯0, 𝛽β‰₯0, and π‘›βˆˆβ„•βˆ—; then 𝑇𝑐𝐿𝛽,𝑛(𝛼)βŠ‚π‘π‘›,𝛿(𝑒).(3.7)

Consequently, we determine the neighborhood for each of the classes π‘‡βˆ—(𝜈)𝐿𝛽,𝑛(𝛼) and 𝑇𝑐(𝜈)𝐿𝛽,𝑛(𝛼), which we define as follows. A function 𝑓(𝑧)∈𝐴(𝑛) defined by (2.8) is said to be in the class π‘‡βˆ—(𝜈)𝐿𝛽,𝑛(𝛼) if there exists a function 𝑔(𝑧)βˆˆπ‘‡βˆ—πΏπ›½,𝑛(𝛼) such that ||||𝑓(𝑧)||||[𝑔(𝑧)βˆ’1<1βˆ’π›Ό,π‘§βˆˆπ‘ˆ,π›Όβˆˆ0,1).(3.8)

Analogously, a function 𝑓(𝑧)∈𝐴(𝑛) defined by (2.8) is said to be in the class 𝑇𝑐(𝜈)𝐿𝛽,𝑛(𝛼) if there exists a function 𝑔(𝑧)βˆˆπ‘‡π‘πΏπ›½,𝑛(𝛼) such the inequality (3.8) holds.

Remark 3.4. If πœ†=1and𝛽=1, we obtain that βˆ‘βˆžπ‘—=𝑛+1π‘—β‹…π‘Žπ‘—<(1βˆ’π›Ό)/((1+𝑛)(1+π‘›βˆ’π›Ό))=𝛿 for 𝐷11𝑓(𝑧)=π‘§π‘“ξ…ž(𝑧), 𝑓(𝑧)βˆˆπ‘‡π‘π‘›(𝛼).

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

Theorem 3.5. If 𝑔(𝑧)βˆˆπ‘‡βˆ—πΏπ›½,𝑛(𝛼) and ξ€Ίπœˆ=1βˆ’π›Ώβ‹…(1+π‘›πœ†)𝛽(1+π‘›πœ†βˆ’π›Ό)ξ€Ί(𝑛+1)β‹…(1+π‘›πœ†)𝛽(1+π‘›πœ†βˆ’π›Ό)βˆ’(1βˆ’π›Ό),(3.9) then 𝑁𝑛,𝛿(𝑔)βŠ‚π‘‡βˆ—(𝜈)𝐿𝛽,𝑛(𝛼),(3.10)π›Όβˆˆ[0,1), πœ†β‰₯0, 𝛽β‰₯0, and π‘›βˆˆβ„•βˆ—.

Proof. Let 𝑓(𝑧)βˆˆπ‘π‘›,𝛿(𝑔). Making use of (2.9), we find that 𝑗β‰₯𝑛+1||π‘Žπ‘—β‹…π‘—βˆ’π‘π‘—||<𝛿,π‘›βˆˆβ„•βˆ—,(3.11) which readily implies the coefficients of inequality 𝑗β‰₯𝑛+1||π‘Žπ‘—βˆ’π‘π‘—||<𝛿𝑛+1,π‘›βˆˆβ„•.(3.12) Furthermore, since 𝑔(𝑧)βˆˆπ‘‡βˆ—πΏπ›½,𝑛(𝛼), we have 𝑗β‰₯𝑛+1𝑏𝑗<1βˆ’π›Ό(1+π‘›πœ†)𝛽(1+π‘›πœ†βˆ’π›Ό),(3.13) so that ||||𝑓(𝑧)||||<βˆ‘π‘”(𝑧)βˆ’1𝑗β‰₯𝑛+1||π‘Žπ‘—βˆ’π‘π‘—||βˆ‘1βˆ’π‘—β‰₯𝑛+1𝑏𝑗<𝛿⋅(1+π‘›πœ†)𝛽(1+π‘›πœ†βˆ’π›Ό)ξ€Ί(𝑛+1)β‹…(1+π‘›πœ†)𝛽(1+π‘›πœ†βˆ’π›Ό)βˆ’(1βˆ’π›Ό)=1βˆ’πœˆ,(3.14) provided that 𝜈 is given precisely by (3.9), which evidently completes our proof of Theorem 3.5.

Example 3.6. For a given βˆ‘π‘”(𝑧)=π‘§βˆ’π‘—β‰₯𝑛+1π‘π‘—π‘§π‘—βˆˆπ‘‡βˆ—πΏπ›½,𝑛(𝛼), π‘›βˆˆβ„•, 𝑛>1, we consider βˆ‘π‘“(𝑧)=π‘§βˆ’π‘—β‰₯𝑛+1π‘Žπ‘—π‘§π‘—βˆˆπ‘‡ with π‘Žπ‘—=(1βˆ’πœˆ)/(𝑗2(π‘—βˆ’1))+𝑏𝑗, 𝑗β‰₯𝑛+1, π‘›βˆˆβ„•β§΅{0}, (for 𝛽=0, πœ†=1), where 1βˆ’πœˆ is given by (3.9).
Then we have that 𝑗β‰₯𝑛+1𝑗||π‘Žπ‘—βˆ’π‘π‘—||=𝑗β‰₯𝑛+1𝑗||||1βˆ’πœˆπ‘—2||||(π‘—βˆ’1)=(1βˆ’πœˆ)𝑗β‰₯𝑛+1ξ‚΅1βˆ’1π‘—βˆ’1𝑗=1βˆ’πœˆ,(3.15) where 1βˆ’πœˆ=𝛿(1+π‘›βˆ’π›Ό)/(𝑛(𝑛+1)).
Therefore, 𝑓(𝑧)βˆˆπ‘π‘›,𝛿(𝑔).

In a similar way, we can prove Theorem 3.7.

Theorem 3.7. If 𝑔(𝑧)βˆˆπ‘‡π‘πΏπ›½,𝑛(𝛼) and ξ€Ίπœˆ=1βˆ’π›Ώβ‹…(1+π‘›πœ†)𝛽+1ξ€»(1+π‘›πœ†βˆ’π›Ό)ξ€Ί(𝑛+1)β‹…(1+π‘›πœ†)𝛽+1ξ€»(1+π‘›πœ†βˆ’π›Ό)βˆ’(1βˆ’π›Ό),(3.16) then 𝑁𝑛,𝛿(𝑔)βŠ‚π‘‡π‘(𝜈)𝐿𝛽,𝑛(𝛼),(3.17)π›Όβˆˆ[0,1), πœ†β‰₯0, 𝛽β‰₯0, and π‘›βˆˆβ„•βˆ—.

Example 3.8. For a given βˆ‘π‘”(𝑧)=π‘§βˆ’π‘—β‰₯𝑛+1π‘π‘—π‘§π‘—βˆˆπ‘‡π‘πΏπ›½,𝑛(𝛼), π‘›βˆˆβ„•, 𝑛>1, we consider βˆ‘π‘“(𝑧)=π‘§βˆ’π‘—β‰₯𝑛+1π‘Žπ‘—π‘§π‘—βˆˆπ‘‡ with π‘Žπ‘—=(1βˆ’πœˆ)/(𝑗2(π‘—βˆ’1))+𝑏𝑗, 𝑗β‰₯𝑛+1, π‘›βˆˆβ„•β§΅{0}, (for 𝛽=0, πœ†=1), where 1βˆ’πœˆ is given by (3.16).
Then we have that 𝑗β‰₯𝑛+1𝑗||π‘Žπ‘—βˆ’π‘π‘—||=𝑗β‰₯𝑛+1𝑗||||1βˆ’πœˆπ‘—2||||(π‘—βˆ’1)=(1βˆ’πœˆ)𝑗β‰₯𝑛+1ξ‚΅1βˆ’1π‘—βˆ’1𝑗=1βˆ’πœˆ,(3.18) where 1βˆ’πœˆ=𝛿(1+π‘›βˆ’π›Ό)/(𝑛(𝑛+1)).
Therefore, 𝑓(𝑧)βˆˆπ‘π‘›,𝛿(𝑔).

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.

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