Inclusion and Neighborhood Properties for Certain Classes of Multivalently Analytic Functions

Abstract

We introduce and investigate two new general subclasses of multivalently analytic functions of complex order by making use of the familiar convolution structure of analytic functions. Among the various results obtained here for each of these function classes, we derive the coefficient inequalities and other interesting properties and characteristics for functions belonging to the classes introduced here.

1. Introduction and Definitions

Let ℝ=(−∞,∞) be the set of real numbers, let ℂ be the set of complex numbers, let ℕ={1,2,…} be the set of positive integers, and let ℕ₀=ℕ∪{0}.

Let 𝒯_𝑝 denote the class of functions of the form 𝑓(𝑧)=𝑧^𝑝−^∞_𝑗=𝑘𝑎_𝑗𝑧^𝑗𝑝<𝑘;𝑎_𝑗≥0𝑗≥𝑘;𝑘,𝑝∈ℕ, (1) which are analytic and 𝑝-valent in the open unit disk 𝕌={𝑧∈ℂ∶|𝑧|<1}. (2)

Denote by 𝑓∗𝑔 the Hadamard product (or convolution) of the functions 𝑓 and 𝑔; that is, if 𝑓 is given by (1) and 𝑔 is given by 𝑔(𝑧)=𝑧^𝑝+^∞_𝑗=𝑘𝑏_𝑗𝑧^𝑗𝑝<𝑘;𝑏_𝑗≥0𝑗≥𝑘;𝑘,𝑝∈ℕ, (3) then 𝑓∗𝑔(𝑧)∶=𝑧^𝑝−^∞_𝑗=𝑘𝑎_𝑗𝑏_𝑗𝑧^𝑗=∶𝑔∗𝑓(𝑧). (4)

Definition 1. Let the function 𝑓∈𝒯_𝑝. Then one says that 𝑓 is in the class 𝒮_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛) if it satisfies the condition 1𝑏𝑧^𝑛ℱ_𝜆,𝜇∗𝑔^(𝑚+𝑛)(𝑧)ℱ_𝜆,𝜇∗𝑔^(𝑚)(𝑧)−𝑝−𝑚_𝑛<𝛽𝑚+𝑛<𝑝<𝑘;𝑝,𝑛∈ℕ;𝑚∈ℕ₀{0};,;𝑏∈ℂ⧵0≤𝜇≤𝜆≤1;0<𝛽≤1;𝑧∈𝕌 (5) where ℱ_𝜆,𝜇(𝑧)=𝜆𝜇𝑧²𝑓^′′(𝑧)+𝜆−𝜇𝑧𝑓^′(𝑧)+1−𝜆+𝜇𝑓(𝑧), (6)𝑔 is given by (3), and (𝜈)_𝑛 denotes the falling factorial defined as follows: (𝜈)₀𝜈0,(𝜈)=1=∶_𝑛()⋯()𝜈𝑛=𝜈𝜈−1𝜈−𝑛+1=∶𝑛!(𝑛∈ℕ). (7)

Various special cases of the class 𝒮_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛) were considered by many earlier researchers on this topic of Geometric Function Theory. For example, 𝒮_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛) reduces to the function class(i) for 𝑚=0,𝑛=1, and 𝜇=0, studied by Mostafa and Aouf [1];(ii)for 𝜆=𝜇=0 and 𝛽=1, studied by Srivastava et al. [2];(iii) for 𝑛=1,𝜆=𝜇=0, and 𝛽=1, studied by Prajapat et al. [3];(iv) for 𝑚=0,𝑛=1,𝛽=1, and 𝑏=𝑝(1−𝛼)(0≤𝛼<1), studied by Srivastava and Bulut [4];(v)for 𝑚=0, 𝑛=1, 𝜆=𝜇=0, 𝛽=1 and 𝑏=𝑝(1−𝛼)(0≤𝛼<1), studied by Ali et al. [5].

Definition 2. Let the function 𝑓∈𝒯_𝑝. Then one says that 𝑓 is in the class 𝒫_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛;𝛾) if it satisfies the condition 1𝑏𝑝_𝑛ℱ1−𝛾^𝜆,𝜇∗𝑔(𝑧)𝑧^𝑛(𝑚)+𝛾ℱ_𝜆,𝜇∗𝑔^(𝑚+𝑛)(𝑧)−𝑝−𝑚_𝑛<𝑝−𝑚_𝑛𝑚+𝑛<𝑝<𝑘;𝑝,𝑛∈ℕ;𝑚∈ℕ₀{0};,;𝑏∈ℂ⧵0≤𝜇≤𝜆≤1;𝛾≥0;𝑧∈𝕌 (8) where 𝑔 and ℱ_𝜆,𝜇 are defined by (3) and (6), respectively.

Setting 𝜆=𝜇=0, 𝛽=1 in Definition 2, we have the special class (which generalizes the class defined by Prajapat et al. [3]) introduced by Srivastava et al. [2].

Following a recent investigation by Frasin and Darus [6], if 𝑓∈𝒯_𝑝 and 𝛿≥0, then we define the (𝑞,𝛿)-neighborhood of the function 𝑓 by 𝒩^𝑞_𝑘,𝛿𝑓=ℎ∈𝒯_𝑝∶ℎ(𝑧)=𝑧^𝑝−^∞_𝑗=𝑘𝑐_𝑗𝑧^𝑗,_∞𝑗=𝑘𝑗^𝑞+1𝑎_𝑗−𝑐_𝑗≤𝛿. (9)

It follows from the definition (9) that if 𝑒(𝑧)=𝑧^𝑝𝑝∈ℕ, (10) then 𝒩^𝑞_𝑘,𝛿(𝑒)=ℎ∈𝒯_𝑝∶ℎ(𝑧)=𝑧^𝑝−^∞_𝑗=𝑘𝑐_𝑗𝑧^𝑗,_∞𝑗=𝑘𝑗^𝑞+1𝑐_𝑗≤𝛿. (11)

The main object of this paper is to investigate the various properties and characteristics of functions belonging to the above-defined classes 𝒮_𝑔𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛,𝒫_𝑔𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛;𝛾. (12) Apart from deriving coefficient bounds and coefficient inequalities for each of these classes, we establish several inclusion relationships involving the (𝑞,𝛿)-neighborhoods of functions belonging to the general classes which are introduced above.

2. Coefficient Bounds and Coefficient Inequalities

We begin by proving a necessary and sufficient condition for the function 𝑓∈𝒯_𝑝 to be in each of the classes 𝒮_𝑔𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛,𝒫_𝑔𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛;𝛾. (13)

Theorem 3. Let the function 𝑓∈𝒯_𝑝 be given by (1). Then 𝑓 is in the class 𝒮_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛) if and only if ^∞_𝑗=𝑘𝑗_𝑚𝑗−𝑚_𝑛−𝑝−𝑚_𝑛+𝛽|𝑏|𝜓𝑗𝑎_𝑗𝑏_𝑗≤𝛽|𝑏|𝑝_𝑚𝜓𝑝𝑚+𝑛<𝑝<𝑘;𝑝,𝑛∈ℕ;𝑚∈ℕ₀;𝑏∈ℂ⧵{0};0<𝛽≤1;𝑧∈𝕌, (14) where 𝜓(𝑠)=(𝑠−1)𝜆𝜇𝑠+𝜆−𝜇+10≤𝜇≤𝜆≤1. (15)

Proof. We first suppose that the function 𝑓 given by (1) is in the class 𝒮_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛). Then, in view of (3)–(6), we have 𝑧Re^𝑛ℱ_𝜆,𝜇∗𝑔^(𝑚+𝑛)(𝑧)ℱ_𝜆,𝜇∗𝑔^(𝑚)(𝑧)−𝑝−𝑚_𝑛>−𝛽|𝑏| (16) or equivalently Re−∑^∞_𝑗=𝑘𝑗_𝑚𝑗−𝑚_𝑛−𝑝−𝑚_𝑛𝜓𝑗𝑎_𝑗𝑏_𝑗𝑧^𝑗−𝑝𝑝_𝑚𝜓𝑝−∑^∞_𝑗=𝑘𝑗_𝑚𝜓𝑗𝑎_𝑗𝑏_𝑗𝑧^𝑗−𝑝>−𝛽|𝑏|. (17) If we choose 𝑧 to be real and let 𝑧→1⁻, we arrive easily at the inequality (14).
Conversely, we suppose that the inequality (14) holds true and let 𝑧∈𝜕𝕌={𝑧∈ℂ∶|𝑧|=1}. (18) Then, we find that 𝑧^𝑛ℱ_𝜆,𝜇∗𝑔^(𝑚+𝑛)(𝑧)ℱ_𝜆,𝜇∗𝑔^(𝑚)(𝑧)−𝑝−𝑚_𝑛=∑^∞_𝑗=𝑘𝑗_𝑚𝑗−𝑚_𝑛−𝑝−𝑚_𝑛𝜓𝑗𝑎_𝑗𝑏_𝑗𝑧^𝑗−𝑝𝑝_𝑚𝜓𝑝−∑^∞_𝑗=𝑘𝑗_𝑚𝜓𝑗𝑎_𝑗𝑏_𝑗𝑧^𝑗−𝑝≤∑^∞_𝑗=𝑘𝑗_𝑚𝑗−𝑚_𝑛−𝑝−𝑚_𝑛𝜓𝑗𝑎_𝑗𝑏_𝑗𝑝_𝑚𝜓𝑝−∑^∞_𝑗=𝑘𝑗_𝑚𝜓𝑗𝑎_𝑗𝑏_𝑗<𝛽|𝑏|𝑝_𝑚𝜓𝑝−∑^∞_𝑗=𝑘𝑗_𝑚𝜓𝑗𝑎_𝑗𝑏_𝑗𝑝_𝑚𝜓𝑝−∑^∞_𝑗=𝑘𝑗_𝑚𝜓𝑗𝑎_𝑗𝑏_𝑗=𝛽|𝑏|. (19) Hence, by the Maximum Modulus Theorem, we have 𝑓∈𝒮_𝑔𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛 (20) which evidently completes the proof of Theorem 3.

Remark 4. If we set 𝜆=𝜇=0 and 𝛽=1 in Theorem 3, then we have [2, Theorem 1].

Lemma 5. Let the function 𝑓∈𝒯_𝑝 given by (1) be in the class 𝒮_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛). Then, for 𝑏_𝑗≥𝑏_𝑘(𝑗≥𝑘), one has ^∞_𝑗=𝑘𝑎_𝑗≤𝛽|𝑏|𝑝_𝑚𝜓𝑝(𝑘)_𝑚(𝑘−𝑚)_𝑛−𝑝−𝑚_𝑛+𝛽|𝑏|𝜓(𝑘)𝑏_{𝑘∞𝑗=𝑘},𝑗𝑎_𝑗≤𝛽|𝑏|𝑝_𝑚𝜓𝑝(𝑘−𝑚)!(𝑘−1)!(𝑘−𝑚)_𝑛−𝑝−𝑚_𝑛+𝛽|𝑏|𝜓(𝑘)𝑏_𝑘𝑝>|𝑏|, (21)(22) where 𝜓 is defined by (15).

Proof. Let 𝑓∈𝒮_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛). Then, in view of the assertion (14), we have ^∞_𝑗=𝑘𝑎_𝑗≤𝛽|𝑏|𝑝_𝑚𝜓𝑝(𝑘)_𝑚(𝑘−𝑚)_𝑛−𝑝−𝑚_𝑛+𝛽|𝑏|𝜓(𝑘)𝑏_𝑘. (23) Furthermore, by rewriting the assertion (14) as follows: ^∞_𝑗=𝑘𝑗!𝑗−𝑚!𝑗−𝑚_𝑛−𝑝−𝑚_𝑛+𝛽|𝑏|𝜓𝑗𝑎_𝑗𝑏_𝑗≤𝛽|𝑏|𝑝_𝑚𝜓𝑝, (24) we obtain ^∞_𝑗=𝑘𝑗𝑎_𝑗≤𝛽|𝑏|𝑝_𝑚𝜓𝑝(𝑘−𝑚)!(𝑘−1)!(𝑘−𝑚)_𝑛−𝑝−𝑚_𝑛+𝛽|𝑏|𝜓(𝑘)𝑏_𝑘. (25)

Similar to Theorem 3, we can prove the following result.

Theorem 6. Let the function 𝑓∈𝒯_𝑝 be given by (1). Then 𝑓 is in the class 𝒫_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛;𝛾) if and only if ^∞_𝑗=𝑘𝑗−𝑛_𝑚𝛾𝑗_𝑛−𝛾−1𝑝_𝑛𝜓𝑗𝑎_𝑗𝑏_𝑗≤𝑝−𝑚_𝑛|𝑏|−1+𝑝_𝑚𝜓𝑝𝑚+𝑛<𝑝<𝑘;𝑝,𝑛∈ℕ;𝑚∈ℕ₀{0};,;𝑏∈ℂ⧵0≤𝜇≤𝜆≤1;𝛾≥0;𝑧∈𝕌 (26) where 𝜓 is defined by (15).

Remark 7. If we set 𝜆=𝜇=0 and 𝛽=1 in Theorem 6, then we have [2, Theorem 2].

Lemma 8. Let the function 𝑓∈𝒯_𝑝 given by (1) be in the class 𝒫_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛;𝛾). Then, for 𝑏_𝑗≥𝑏_𝑘(𝑗≥𝑘), one has ^∞_𝑗=𝑘𝑎_𝑗≤𝑝−𝑚_𝑛|𝑏|−1+𝑝_𝑚𝜓𝑝(𝑘−𝑛)_𝑚𝛾(𝑘)_𝑛−𝛾−1𝑝_𝑛𝜓(𝑘)𝑏_{𝑘∞𝑗=𝑘},𝑗𝑎_𝑗≤𝑘𝑝−𝑚_𝑛|𝑏|−1+𝑝_𝑚𝜓𝑝(𝑘−𝑛−𝑚)!(𝑘−𝑛)!𝛾(𝑘)_𝑛−𝛾−1𝑝_𝑛𝜓(𝑘)𝑏_𝑘𝛾>1, (27)(28) where 𝜓 is defined by (15).

Proof. Let 𝑓∈𝒫_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛;𝛾). Then, in view of the assertion (26), we have ^∞_𝑗=𝑘𝑎_𝑗≤𝑝−𝑚_𝑛|𝑏|−1+𝑝_𝑚𝜓𝑝(𝑘−𝑛)_𝑚𝛾(𝑘)_𝑛−𝛾−1𝑝_𝑛𝜓(𝑘)𝑏_𝑘. (29) Furthermore, we also have from the assertion (26) ^∞_𝑗=𝑘𝑗𝑎_𝑗≤𝑘𝑝−𝑚_𝑛|𝑏|−1+𝑝_𝑚𝜓𝑝(𝑘−𝑛−𝑚)!(𝑘−𝑛)!𝛾(𝑘)_𝑛−𝛾−1𝑝_𝑛𝜓(𝑘)𝑏_𝑘. (30)

3. A Set of Inclusion Relationships

In this section, we determine inclusion relations for the classes 𝒮_𝑔𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛,𝒫_𝑔𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛;𝛾 (31) involving (𝑞,𝛿)-neighborhoods defined by (9) and (11).

Theorem 9. If 𝑏_𝑗≥𝑏_𝑘(𝑗≥𝑘) and 𝛽|𝑏|𝑝𝛿=^𝑚𝜓𝑝(𝑘−𝑚)!(𝑘−1)!(𝑘−𝑚)_𝑛−𝑝−𝑚_𝑛+𝛽|𝑏|𝜓(𝑘)𝑏_𝑘𝑝>|𝑏|, (32) then 𝒮_𝑔𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛⊂𝒩⁰_𝑘,𝛿(𝑒), (33) where 𝑒 and 𝜓 are given by (10) and (15), respectively.

Proof. The inclusion relation (33) would follow readily from the definition (11) and the assertion (22).

Remark 10. If we set 𝜆=𝜇=0 and 𝛽=1 in Theorem 9, then we have [2, Theorem 3].

Theorem 11. If 𝑏_𝑗≥𝑏_𝑘(𝑗≥𝑘) and 𝑘𝛿=𝑝−𝑚_𝑛|𝑏|−1+𝑝_𝑚𝜓𝑝(𝑘−𝑛−𝑚)!(𝑘−𝑛)!𝛾(𝑘)_𝑛−𝛾−1𝑝_𝑛𝜓(𝑘)𝑏_𝑘𝛾>1, (34) then 𝒫_𝑔𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛;𝛾⊂𝒩⁰_𝑘,𝛿(𝑒), (35) where 𝑒 and 𝜓 are given by (10) and (15), respectively.

Proof. The inclusion relation (35) would follow readily from the definition (11) and the assertion (28).

Remark 12. If we set 𝜆=𝜇=0 and 𝛽=1 in Theorem 11, then we have [2, Theorem 4].

4. Neighborhood Properties

In this section, we determine the neighborhood properties for each of the function classes 𝒮_𝑔^(𝛼)𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛,𝒫_𝑔^(𝛼)𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛;𝛾 (36) which are defined as follows.

Definition 13. A function 𝑓∈𝒯_𝑝 is said to be in the class 𝒮_𝑔^(𝛼)(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛) if there exists a function ℎ∈𝒮_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛) such that 𝑓(𝑧)ℎ(𝑧)−1<𝑝−𝛼0≤𝛼<𝑝;𝑧∈𝕌. (37)

Definition 14. A function 𝑓∈𝒯_𝑝 is said to be in the class 𝒫_𝑔^(𝛼)(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛;𝛾) if there exists a function ℎ∈𝒫_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛;𝛾) such that the inequality (37) holds true.

Setting 𝜆=𝜇=0,𝛽=1 in Definitions 13 and 14, we have the special classes 𝒮_𝑔^(𝛼)𝑝,𝑘,𝑏,𝑚,𝑛,𝒫_𝑔^(𝛼)𝑝,𝑘,𝑏,𝑚,𝑛;𝛾 (38) introduced by Srivastava et al. [2], respectively.

Theorem 15. If ℎ∈𝒮_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛) and 𝛿𝛼=𝑝−𝑘^𝑞+1×(𝑘)_𝑚(𝑘−𝑚)_𝑛−𝑝−𝑚_𝑛+𝛽|𝑏|𝜓(𝑘)𝑏_𝑘×(𝑘)_𝑚(𝑘−𝑚)_𝑛−𝑝−𝑚_𝑛|𝑏(𝑘)𝑏+𝛽|×𝜓_𝑘−𝛽|𝑏|𝑝_𝑚𝜓𝑝⁻¹, (39) then 𝒩^𝑞_𝑘,𝛿(ℎ)⊂𝒮_𝑔^(𝛼)𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛, (40) where 𝜓 is defined by (15).

Proof. Suppose that 𝑓∈𝒩^𝑞_𝑘,𝛿(ℎ). Then we find from (9) that ^∞_𝑗=𝑘𝑗^𝑞+1𝑎_𝑗−𝑐_𝑗≤𝛿, (41) which readily implies that ^∞_𝑗=𝑘𝑎_𝑗−𝑐_𝑗≤𝛿𝑘^𝑞+1. (42) Since ℎ∈𝒮_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛), we find from (21) that ^∞_𝑗=𝑘𝑐_𝑗≤𝛽|𝑏|𝑝_𝑚𝜓𝑝(𝑘)_𝑚(𝑘−𝑚)_𝑛−𝑝−𝑚_𝑛+𝛽|𝑏|𝜓(𝑘)𝑏_𝑘, (43) so that 𝑓(𝑧)ℎ(𝑧)≤∑−1_∞𝑗=𝑘𝑎_𝑗−𝑐_𝑗1−∑^∞_𝑗=𝑘𝑐_𝑗≤𝛿𝑘^𝑞+1(𝑘)_𝑚(𝑘−𝑚)_𝑛−𝑝−𝑚_𝑛+𝛽|𝑏|𝜓(𝑘)𝑏_𝑘×(𝑘)_𝑚(𝑘−𝑚)_𝑛−𝑝−𝑚_𝑛|𝑏(𝑘)𝑏+𝛽|×𝜓_𝑘−𝛽|𝑏|𝑝_𝑚𝜓𝑝⁻¹=𝑝−𝛼, (44) where 𝛼 is given by (39). Thus, by Definition 13, 𝑓∈𝒮_𝑔^(𝛼)(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛). This completes the proof of Theorem 15.

Remark 16. If we set 𝜆=𝜇=0 and 𝛽=1 in Theorem 15, then we have [2, Theorem 5].

The proof of Theorem 17 (based upon Definition 14) is similar to that of Theorem 15. Therefore we omit the details involved.

Theorem 17. If ℎ∈𝒫_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛;𝛾) and 𝛿𝛼=𝑝−𝑘^𝑞+1×(𝑘−𝑛)_𝑚𝛾(𝑘)_𝑛−𝛾−1𝑝_𝑛𝜓(𝑘)𝑏_𝑘×(𝑘−𝑛)_𝑚𝛾(𝑘)_𝑛−𝛾−1𝑝_𝑛𝜓(𝑘)𝑏_𝑘−𝑝−𝑚_𝑛|𝑏|−1+𝑝_𝑚𝜓𝑝⁻¹, (45) then 𝒩^𝑞_𝑘,𝛿(ℎ)⊂𝒫_𝑔^(𝛼)𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛;𝛾, (46) where 𝜓 is defined by (15).

Remark 18. If we set 𝜆=𝜇=0 and 𝛽=1 in Theorem 17, then we have [2, Theorem 6].

Acknowledgment

The present investigation was supported by the Kocaeli University under Grant HD 2011/22.