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 ℋ(𝑈) be the set of functions which are regular in the unit disc 𝑈, 𝐴=𝑓∈ℋ(𝑈)∶𝑓(0)=ğ‘“î…žî€¾(0)−1=0(1.1) and 𝑆={𝑓∈𝐴∶𝑓isunivalentin𝑈}.

In [1], the subfamily 𝑇 of 𝑆 consisting of functions 𝑓 of the form 𝑓(𝑧)=ğ‘§âˆ’âˆžî“ğ‘—=2ğ‘Žğ‘—ğ‘§ğ‘—,ğ‘Žğ‘—â‰¥0,𝑗=2,3,…,𝑧∈𝑈,(1.2) was introduced, where 𝑇=𝑓∈𝑆∶𝑓(𝑧)=ğ‘§âˆ’âˆžî“ğ‘—=2ğ‘Žğ‘—ğ‘§ğ‘—,ğ‘Žğ‘—îƒ°â‰¥0,𝑗≥2,𝑧∈𝑈.(1.3)

For 𝑓(𝑧) belonging to 𝐴, Sălăgean [2] has introduced the following operator called the Sălăgean operator: 𝐷0𝑓(𝑧)=𝑓(𝑧),𝐷1𝑓(𝑧)=𝐷𝑓(𝑧)=ğ‘§ğ‘“î…ž(𝑧),𝐷𝑛𝐷𝑓(𝑧)=𝐷𝑛−1𝑓(𝑧),𝑛∈ℕ∗.(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): 𝐷𝛽𝜆∶𝐴⟶𝐴,𝐷𝛽𝜆𝑓(𝑧)=ğ‘§âˆ’âˆžî“ğ‘—=𝑛+1[]1+(𝑗−1)ğœ†ğ›½ğ‘Žğ‘—ğ‘§ğ‘—.(2.1)

Definition 2.2 (see [3]). Let 𝑓∈𝑇, ∑𝑓(𝑧)=ğ‘§âˆ’âˆžğ‘—=2ğ‘Žğ‘—ğ‘§ğ‘—, and ğ‘Žğ‘—â‰¥0, 𝑗=2,3,…, 𝑧∈𝑈. We say that 𝑓 is in the class 𝑇𝐿𝛽(𝛼) if 𝐷Re𝜆𝛽+1𝑓(𝑧)𝐷𝛽𝜆[𝑓(𝑧)>𝛼,𝛼∈0,1),𝜆≥0,𝛽≥0,𝑧∈𝑈.(2.2)

Theorem 2.3 (see [3]). Let 𝛼∈[0,1), 𝜆≥0,𝑛∈ℕ∗, and 𝛽≥0. The function 𝑓∈𝐴(𝑛) of the form (1.2) is in the class 𝑇∗𝐿𝛽,𝑛(𝛼) if and only if âˆžî“ğ‘—=𝑛+1(1+(𝑗−1)𝜆)𝛽(1+(𝑗−1)𝜆−𝛼)â‹…ğ‘Žğ‘—<1−𝛼.(2.3)

Remark 2.4. Using the condition (1.3), we can to prove that 𝑇∗𝐿𝛽+1,𝑛(𝛼)⊂𝑇∗𝐿𝛽,𝑛(𝛼),(2.4)  𝛽≥0, 𝛼∈[0,1), 𝜆≥0, and 𝑛∈ℕ∗.

Definition 2.5 (see [4]). Let 𝑓∈𝑇, ∑𝑓(𝑧)=ğ‘§âˆ’âˆžğ‘—=2ğ‘Žğ‘—ğ‘§ğ‘—, and ğ‘Žğ‘—â‰¥0, 𝑗=2,3,…, 𝑧∈𝑈. We say that 𝑓 is in the class 𝑇𝑐𝐿𝛽(𝛼) if 𝐷Re𝜆𝛽+2𝑓(𝑧)𝐷𝜆𝛽+1[𝑓(𝑧)>𝛼,𝛼∈0,1),𝜆≥0,𝛽≥0,𝑧∈𝑈.(2.5)

Theorem 2.6 (see [4]). Let 𝛼∈[0,1), 𝜆≥0, and 𝛽≥0. The function 𝑓∈𝑇 of the form (1.2) is in the class 𝑇𝑐𝐿𝛽(𝛼) if and only if âˆžî“ğ‘—=2(1+(𝑗−1)𝜆)𝛽+1(î€»ğ‘Ž1+(𝑗−1)𝜆−𝛼)𝑗<1−𝛼.(2.6)

Remark 2.7. Using the condition (2.2), we can prove that 𝑇𝑐𝐿𝛽+1,𝑛(𝛼)⊂𝑇𝑐𝐿𝛽,𝑛(𝛼),(2.7)  𝛽≥0, 𝛼∈[0,1), 𝜆≥0, and 𝑛∈ℕ∗.

Let 𝐴(𝑛) be the class of functions 𝑓(𝑧) of the form 𝑓(𝑧)=ğ‘§âˆ’âˆžî“ğ‘˜=𝑛+1ğ‘Žğ‘˜ğ‘§ğ‘˜,î€·ğ‘Žğ‘˜î€¸â‰¥0;𝑛∈ℕ−{0},(2.8) which are analytic in the open unit disk 𝑈={𝑧∶|𝑧|<1}. For any 𝑓(𝑧)∈𝐴(𝑛) and 𝛿≥0, we define 𝑁𝑛,𝛿=𝑔∈𝐴(𝑛)∶𝑔(𝑧)=ğ‘§âˆ’âˆžî“ğ‘˜=𝑛+1𝑏𝑘𝑧𝑘,âˆžî“ğ‘˜=𝑛+1||ğ‘Žğ‘˜â‹…ğ‘˜âˆ’ğ‘ğ‘˜||≤𝛿,(2.9) which was called (𝑛,𝛿)-neighborhood of 𝑓(𝑧). So, for 𝑒(𝑧)=𝑧, we observe that 𝑁𝑛,𝛿=𝑔∈𝐴(𝑛)∶𝑔(𝑧)=ğ‘§âˆ’âˆžî“ğ‘˜=𝑛+1𝑏𝑘𝑧𝑘,âˆžî“ğ‘˜=𝑛+1||𝑏𝑘⋅𝑘||≤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 𝑇∗𝐿𝛽,𝑛(𝛼) and 𝑇𝑐𝐿𝛽,𝑛(𝛼) of the class 𝐴(𝑛) of normalized analytic functions in 𝑈 with negative coefficients, where 𝑇∗𝐿𝛽,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 𝑇∗𝐿𝛽,𝑛(𝛼), and 𝑇𝑐𝐿𝛽,𝑛(𝛼) and from Theorem 2.3, we derive the corresponding results.

Theorem 3.1. Let 𝛿=(1−𝛼)(𝑛+1)(1+𝑛𝜆)𝛽(1+𝑛𝜆−𝛼),(3.1) where 𝛼∈[0,1), 𝜆≥0, 𝛽≥0, and 𝑛∈ℕ∗; then 𝑇∗𝐿𝛽,𝑛(𝛼)⊂𝑁𝑛,𝛿(𝑒).(3.2)

Proof. For 𝑓(𝑧)∈𝑇∗𝐿𝛽,𝑛(𝛼) and making use of the condition (2.2), we obtain [(1+𝑛𝜆)𝛽∑(1+𝑛𝜆−𝛼)]âˆžğ‘—=𝑛+1ğ‘Žğ‘—<1−𝛼, so that âˆžî“ğ‘—=𝑛+1ğ‘Žğ‘—<1−𝛼(1+𝑛𝜆)𝛽(1+𝑛𝜆−𝛼).(3.3) On the other hand, we also find from (2.2) and (3.3) that âˆžî“ğ‘—=𝑛+1(1+(𝑗−1)𝜆)𝛽(î€»ğ‘Ž1+(𝑗−1)𝜆−𝛼)𝑗(<1−𝛼,1+𝑛𝜆)ğ›½âˆžî“ğ‘—=𝑛+1(1+(𝑗−1)𝜆−𝛼)ğ‘Žğ‘—<1−𝛼,𝜆(1+𝑛𝜆)ğ›½âˆžî“ğ‘—=𝑛+1ğ‘—â‹…ğ‘Žğ‘—î€º(<1−𝛼−1−𝜆−𝛼)(1+𝑛𝜆)ğ›½î€»â‹…âˆžî“ğ‘—=𝑛+1ğ‘Žğ‘—.(3.4)
Thus, âˆžî“ğ‘—=𝑛+1ğ‘—â‹…ğ‘Žğ‘—<(1−𝛼)(𝑛+1)(1+𝑛𝜆)𝛽(1+𝑛𝜆−𝛼)=𝛿,(3.5) which in view of definition (2.10), proves Theorem 3.1.

Remark 3.2. If 𝜆=1 and 𝛽=1, we obtain that âˆ‘âˆžğ‘—=𝑛+1ğ‘—â‹…ğ‘Žğ‘—<(1−𝛼)/(1+𝑛−𝛼)=𝛿 for 𝐷11𝑓(𝑧)=ğ‘§ğ‘“î…ž(𝑧), 𝑓(𝑧)∈𝑇∗𝑛(𝛼).

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

Theorem 3.3. Let 𝛿=(1−𝛼)(𝑛+1)(1+𝑛𝜆)𝛽+1(1+𝑛𝜆−𝛼),(3.6) where 𝛼∈[0,1), 𝜆≥0, 𝛽≥0, and 𝑛∈ℕ∗; then 𝑇𝑐𝐿𝛽,𝑛(𝛼)⊂𝑁𝑛,𝛿(𝑒).(3.7)

Consequently, we determine the neighborhood for each of the classes 𝑇∗(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 𝑇∗(𝜈)𝐿𝛽,𝑛(𝛼) if there exists a function 𝑔(𝑧)∈𝑇∗𝐿𝛽,𝑛(𝛼) such that ||||𝑓(𝑧)||||[𝑔(𝑧)−1<1−𝛼,𝑧∈𝑈,𝛼∈0,1).(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)∈𝑇𝑐𝐿𝛽,𝑛(𝛼) such the inequality (3.8) holds.

Remark 3.4. If 𝜆=1and𝛽=1, we obtain that âˆ‘âˆžğ‘—=𝑛+1ğ‘—â‹…ğ‘Žğ‘—<(1−𝛼)/((1+𝑛)(1+𝑛−𝛼))=𝛿 for 𝐷11𝑓(𝑧)=ğ‘§ğ‘“î…ž(𝑧), 𝑓(𝑧)∈𝑇𝑐𝑛(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)∈𝑇∗𝐿𝛽,𝑛(𝛼) and 𝜈=1−𝛿⋅(1+𝑛𝜆)𝛽(1+𝑛𝜆−𝛼)(𝑛+1)⋅(1+𝑛𝜆)𝛽(1+𝑛𝜆−𝛼)−(1−𝛼),(3.9) then 𝑁𝑛,𝛿(𝑔)⊂𝑇∗(𝜈)𝐿𝛽,𝑛(𝛼),(3.10)𝛼∈[0,1), 𝜆≥0, 𝛽≥0, and 𝑛∈ℕ∗.

Proof. Let 𝑓(𝑧)∈𝑁𝑛,𝛿(𝑔). Making use of (2.9), we find that 𝑗≥𝑛+1||ğ‘Žğ‘—â‹…ğ‘—âˆ’ğ‘ğ‘—||<𝛿,𝑛∈ℕ∗,(3.11) which readily implies the coefficients of inequality 𝑗≥𝑛+1||ğ‘Žğ‘—âˆ’ğ‘ğ‘—||<𝛿𝑛+1,𝑛∈ℕ.(3.12) Furthermore, since 𝑔(𝑧)∈𝑇∗𝐿𝛽,𝑛(𝛼), we have 𝑗≥𝑛+1𝑏𝑗<1−𝛼(1+𝑛𝜆)𝛽(1+𝑛𝜆−𝛼),(3.13) so that ||||𝑓(𝑧)||||<∑𝑔(𝑧)−1𝑗≥𝑛+1||ğ‘Žğ‘—âˆ’ğ‘ğ‘—||∑1−𝑗≥𝑛+1𝑏𝑗<𝛿⋅(1+𝑛𝜆)𝛽(1+𝑛𝜆−𝛼)(𝑛+1)⋅(1+𝑛𝜆)𝛽(1+𝑛𝜆−𝛼)−(1−𝛼)=1−𝜈,(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 ∑𝑔(𝑧)=𝑧−𝑗≥𝑛+1𝑏𝑗𝑧𝑗∈𝑇∗𝐿𝛽,𝑛(𝛼), 𝑛∈ℕ, 𝑛>1, we consider ∑𝑓(𝑧)=𝑧−𝑗≥𝑛+1ğ‘Žğ‘—ğ‘§ğ‘—âˆˆğ‘‡ with ğ‘Žğ‘—=(1−𝜈)/(𝑗2(𝑗−1))+𝑏𝑗, 𝑗≥𝑛+1, 𝑛∈ℕ⧵{0}, (for 𝛽=0, 𝜆=1), where 1−𝜈 is given by (3.9).
Then we have that 𝑗≥𝑛+1𝑗||ğ‘Žğ‘—âˆ’ğ‘ğ‘—||=𝑗≥𝑛+1𝑗||||1−𝜈𝑗2||||(𝑗−1)=(1−𝜈)𝑗≥𝑛+11−1𝑗−1𝑗=1−𝜈,(3.15) where 1−𝜈=𝛿(1+𝑛−𝛼)/(𝑛(𝑛+1)).
Therefore, 𝑓(𝑧)∈𝑁𝑛,𝛿(𝑔).

In a similar way, we can prove Theorem 3.7.

Theorem 3.7. If 𝑔(𝑧)∈𝑇𝑐𝐿𝛽,𝑛(𝛼) and 𝜈=1−𝛿⋅(1+𝑛𝜆)𝛽+1(1+𝑛𝜆−𝛼)(𝑛+1)⋅(1+𝑛𝜆)𝛽+1(1+𝑛𝜆−𝛼)−(1−𝛼),(3.16) then 𝑁𝑛,𝛿(𝑔)⊂𝑇𝑐(𝜈)𝐿𝛽,𝑛(𝛼),(3.17)𝛼∈[0,1), 𝜆≥0, 𝛽≥0, and 𝑛∈ℕ∗.

Example 3.8. For a given ∑𝑔(𝑧)=𝑧−𝑗≥𝑛+1𝑏𝑗𝑧𝑗∈𝑇𝑐𝐿𝛽,𝑛(𝛼), 𝑛∈ℕ, 𝑛>1, we consider ∑𝑓(𝑧)=𝑧−𝑗≥𝑛+1ğ‘Žğ‘—ğ‘§ğ‘—âˆˆğ‘‡ with ğ‘Žğ‘—=(1−𝜈)/(𝑗2(𝑗−1))+𝑏𝑗, 𝑗≥𝑛+1, 𝑛∈ℕ⧵{0}, (for 𝛽=0, 𝜆=1), where 1−𝜈 is given by (3.16).
Then we have that 𝑗≥𝑛+1𝑗||ğ‘Žğ‘—âˆ’ğ‘ğ‘—||=𝑗≥𝑛+1𝑗||||1−𝜈𝑗2||||(𝑗−1)=(1−𝜈)𝑗≥𝑛+11−1𝑗−1𝑗=1−𝜈,(3.18) where 1−𝜈=𝛿(1+𝑛−𝛼)/(𝑛(𝑛+1)).
Therefore, 𝑓(𝑧)∈𝑁𝑛,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.