Subclass of Multivalent 𝛽-Uniformly Functions with Varying Arguments

Abstract

We introduce a class of multivalent 𝛽-uniformly functions with varying arguments, give some properties of functions in this class, and obtain sharp results including coefficient estimates and distortion theorems. Finally, we give the extreme points for this class.

1. Introduction

Let 𝐴(𝑝) denote the class of functions of the following form: 𝑓(𝑧)=𝑧^𝑝+^∞_𝑘=𝑝+1𝑎_𝑘𝑧^𝑘𝑝∈ℕ={1,2,3,…}, (1) which are analytic and 𝑝-valent in the open unit disc 𝑈={𝑧∈ℂ∶|𝑧|<1}. A function 𝑓(𝑧)∈𝐴(𝑝) is said to be 𝑝-valent starlike of order 𝛼 if it satisfies the following inequality: Re𝑧𝑓^′(𝑧)𝑓(𝑧)>𝛼0≤𝛼<𝑝;𝑝∈ℕ;𝑧∈𝑈. (2) We denote by 𝑆_𝑝(𝛼) the class of all 𝑝-valent starlike functions of order 𝛼. Also a function 𝑓(𝑧)∈𝐴(𝑝) is said to be 𝑝-valent convex of order 𝛼, if it satisfies the following inequality: Re1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)>𝛼0≤𝛼<𝑝;𝑝∈ℕ;𝑧∈𝑈. (3) We denote by 𝐾_𝑝(𝛼) the class of all 𝑝-valently convex functions of order 𝛼. The classes 𝑆_𝑝(𝛼) and 𝐾_𝑝(𝛼) were introduced by Patil and Thakare [1] and Owa [2]. Further from (2) and (3), we can see that 𝑓(𝑧)∈𝐾_𝑝(𝛼)⟺𝑧𝑓^′(𝑧)𝑝∈𝑆_𝑝(𝛼)0≤𝛼<𝑝;𝑝∈ℕ. (4) Also for a function 𝑓(𝑧)∈𝐴(𝑝), the differential operator 𝐷^𝑛_𝑝𝑓(𝑧) was defined by Kamali and Orhan [3], Orhan and Kiziltunç [4], and Aouf and Mostafa [5] as follows: 𝐷⁰_𝑝𝑓(𝑧)(𝑧),𝐷=𝑓_1𝑝𝑓(𝑧)=𝐷_𝑝𝑓(𝑧)=𝑧𝑝𝑓^′(𝑧)=𝑧^𝑝+^∞_𝑘=𝑝+1𝑘𝑝𝑎_𝑘𝑧^𝑘,𝐷²_𝑝𝑓(𝑧)=𝐷_𝑝𝐷¹_𝑝𝑓(𝑧)=𝑧^𝑝+^∞_𝑘=𝑝+1𝑘𝑝²𝑎_𝑘𝑧^𝑘, (5) and (in general) 𝐷^𝑛_𝑝𝑓(𝑧)=𝑧^𝑝+^∞_𝑘=𝑝+1𝑘𝑝^𝑛𝑎_𝑘𝑧^𝑘𝑝∈ℕ;𝑛∈ℕ₀=ℕ∪{0}. (6) We note that putting 𝑝=1 in (6), we have the operator 𝐷^𝑛₁=𝐷^𝑛 which was introduced by Sălăgean [6].

In [7], Eker and Şeker introduced the class 𝑁_𝑝(𝑚,𝑛;𝛼,𝛽) as follows.

Definition 1. For 0≤𝛼<1,𝛽≥0,𝑚∈ℕ,𝑛∈ℕ₀,𝑚>𝑛, and 𝑧∈𝑈, let 𝑁_𝑝(𝑚,𝑛;𝛼,𝛽) be the class of functions 𝑓(𝑧)∈𝐴(𝑝) which satisfy 𝐷Re^𝑚𝑝𝑓(𝑧)𝐷^𝑛_𝑝𝑓(𝑧)𝐷−𝛼>𝛽^𝑚𝑝𝑓(𝑧)𝐷^𝑛_𝑝𝑓(𝑧)−1. (7) Specializing the parameters   𝛽,𝑚,𝑛, and 𝑝 we obtain the following subclasses studied by various authors:(i)𝑁_𝑝(1,0;𝛼,𝛽)=𝑈𝑆𝑇(𝛼,𝛽,𝑝) and 𝑁_𝑝(2,1;𝛼,𝛽)=𝑈𝐶𝑉(𝛼,𝛽,𝑝)(0≤𝛼<𝑝,𝛽≥0,𝑝∈ℕ) (see Khairnar and More [8]); (ii)𝑁_𝑝(1,0;𝛼,𝑜)=𝑆^∗(𝑝,𝛼) and 𝑁_𝑝(2,1;𝛼,0)=𝐾(𝑝,𝛼)(0≤𝛼<𝑝,𝑝∈ℕ) (see Patil and Thakare [1] and Owa [2]); (iii)𝑁₁(𝑚,𝑛;𝛼,𝛽)=𝑁_𝑚,𝑛(𝛼,𝛽)(0≤𝛼<1,𝛽≥0,𝑚∈ℕ,𝑛∈ℕ₀,𝑚>𝑛) (see Eker and Owa [9]);(iv)𝑁₁(𝑚,𝑛;𝛼,0)=𝐾_𝑚,𝑛(𝛼)(0≤𝛼<1,𝑚∈ℕ,𝑛∈ℕ₀,𝑚>𝑛) (see Eker and Owa [10]); (v)𝑁₁(𝑛+1,𝑛;𝛼,𝛽)=𝑆(𝑛,𝛼,𝛽)(0≤𝛼<1,𝛽≥0,𝑛∈ℕ₀) (see Rosy and Murugusundaramoorthy [11] and Aouf [12]); (vi)𝑁₁(1,0;𝛼,𝛽)=𝑆𝐷(𝛼,𝛽) and 𝑁₁(2,1;𝛼,𝛽)=𝐾𝐷(𝛼,𝛽)(0≤𝛼<1,𝛽≥0) (see Shams et al. [13] and Bharati et al. [14]).
Also we note that 𝑁_𝑝𝑛+1,𝑛;𝛼,𝛽=𝑁_𝑝=𝑝𝐷𝑛;𝛼,𝛽𝑓∈𝐴∶Re^𝑝𝑛+1𝑓(𝑧)𝐷^𝑛_𝑝𝑓(𝑧)𝐷−𝛼>𝛽^𝑝𝑛+1𝑓(𝑧)𝐷^𝑛_𝑝𝑓(𝑧)−10≤𝛼<1;𝛽≥0;𝑛∈ℕ₀;𝑧∈𝑈. (8) Silverman [15] defined the class of univalent functions 𝑓(𝑧)=𝑧+∑^∞_𝑘=2𝑎_𝑘𝑧^𝑘 for which arg(𝑎_𝑘) prescribed in such a way that 𝑓(𝑧) is univalent if and only if 𝑓(𝑧) is starlike. In this paper, we introduce the subclass of 𝑝-valent functions with varying arguments as follows.

Definition 2. A function 𝑓(𝑧) of the form (1) is said to be in the class 𝑉_𝑝(𝜃_𝑘) if 𝑓∈𝐴(𝑝) and arg(𝑎_𝑘)=𝜃_𝑘 for all 𝑘≥𝑝+1. If furthermore there exists a real number 𝛿 such that 𝜃_𝑘+(𝑘−𝑝)𝛿≡𝜋(mod2𝜋) for all 𝑘≥𝑝+1, then 𝑓(𝑧) is said to be in the class 𝑉_𝑝(𝜃_𝑘,𝛿). The union of 𝑉_𝑝(𝜃_𝑘,𝛿) taken over all possible sequences {𝜃_𝑘} and all possible real numbers 𝛿 is denoted by 𝑉_𝑝.
Let 𝑉_𝑝(𝑚,𝑛;𝛼,𝛽) denote the subclass of 𝑉_𝑝 consisting of functions 𝑓(𝑧)∈𝑁_𝑝(𝑚,𝑛;𝛼,𝛽).
We note that 𝑉₁(1,0;𝛼,0)=𝑉^∗(𝛼), which was introduced and studied by Silverman [15].
Specializing the parameters 𝛽,𝑚,𝑛, and 𝑝, we obtain the following classes with varying arguments which are associated to the subclasses mentioned before:(i)𝑉_𝑝(1,0;𝛼,𝛽)=𝑉𝑆𝑇(𝛼,𝛽,𝑝) and 𝑉_𝑝(2,1;𝛼,𝛽)=𝑉𝐶𝑉(𝛼,𝛽,𝑝)(0≤𝛼<𝑝,𝛽≥0,𝑝∈ℕ);(ii)𝑉_𝑝(1,0;𝛼,𝑜)=𝑉𝑆^∗(𝑝,𝛼) and 𝑉_𝑝(2,1;𝛼,0)=𝑉𝐾(𝑝,𝛼)(0≤𝛼<𝑝,𝑝∈ℕ);(iii)𝑉₁(𝑚,𝑛;𝛼,𝛽)=𝑉_𝑚,𝑛(𝛼,𝛽)(0≤𝛼<1,𝛽≥0,𝑚∈ℕ,𝑛∈ℕ₀,𝑚>𝑛); (iv)𝑉₁(𝑚,𝑛;𝛼,0)=𝑉_𝑚,𝑛(𝛼)(0≤𝛼<1,𝑚∈ℕ,𝑛∈ℕ₀,𝑚>𝑛); (v)𝑉₁(𝑛+1,𝑛;𝛼,𝛽)=𝑉(𝑛,𝛼,𝛽)(0≤𝛼<1,𝛽≥0,𝑛∈ℕ₀); (vi)𝑉₁(1,0;𝛼,𝛽)=𝑉𝑆𝐷(𝛼,𝛽) and 𝑉₁(2,1;𝛼,𝛽)=𝑉𝐾𝐷(𝛼,𝛽)(0≤𝛼<1,𝛽≥0);(vii)𝑉_𝑝(𝑛+1,𝑛;𝛼,𝛽)=𝑉_𝑝(𝑛;𝛼,𝛽).

In this paper, we obtain coefficient bounds for functions in the class 𝑉_𝑝(𝑚,𝑛;𝛼,𝛽), further we obtain distortion theorems and the extreme points for functions in this class.

2. Coefficient Estimates

Unless otherwise mentioned, we assume in the reminder of this paper that0≤𝛼<1,𝛽≥0,𝑚∈ℕ,𝑛∈ℕ₀,𝑚>𝑛, and 𝑧∈𝑈.

We will need the following lemma.

Lemma 3. The sufficient condition for 𝑓(𝑧) given by (1) to be in the class 𝑁_𝑝(𝑚,𝑛;𝛼,𝛽) is that ^∞_𝑘=𝑝+1𝑘1+𝛽𝑝^𝑚−𝑘𝛼+𝛽𝑝^𝑛𝑎_𝑘≤1−𝛼. (9)

Proof. It suffices to show that inequality (7) holds true. Then it suffices to show that 𝛽𝐷^𝑚_𝑝𝑓(𝑧)𝐷^𝑛_𝑝𝑓(𝑧)𝐷−1−Re^𝑚𝑝𝑓(𝑧)𝐷^𝑛_𝑝𝑓(𝑧)−1≤1−𝛼. (10) We have 𝛽𝐷^𝑚_𝑝𝑓(𝑧)𝐷^𝑛_𝑝𝑓(𝑧)𝐷−1−Re^𝑚𝑝𝑓(𝑧)𝐷^𝑛_𝑝𝑓(𝑧)≤𝐷−11+𝛽^𝑚𝑝𝑓(𝑧)𝐷^𝑛_𝑝𝑓(𝑧)−1, (11) or, equivalently, 𝛽𝐷^𝑚_𝑝𝑓(𝑧)𝐷^𝑛_𝑝𝑓(𝑧)𝐷−1−Re^𝑚𝑝𝑓(𝑧)𝐷^𝑛_𝑝𝑓(𝑧)≤∑−11+𝛽^{∞𝑘=𝑝+1}𝑘/𝑝^𝑚−𝑘/𝑝^𝑛𝑎_𝑘𝑧^𝑘𝑧^𝑝+∑^∞_𝑘=𝑝+1𝑘/𝑝^𝑛𝑎_𝑘𝑧^𝑘≤∑1+𝛽^{∞𝑘=𝑝+1}𝑘/𝑝^𝑚−𝑘/𝑝^𝑛𝑎_𝑘𝑧^𝑘−𝑝1+∑^∞_𝑘=𝑝+1𝑘/𝑝^𝑛𝑎_𝑘𝑧^𝑘−𝑝, (12) then we have 𝛽𝐷^𝑚_𝑝𝑓(𝑧)𝐷^𝑛_𝑝𝑓(𝑧)𝐷−1−Re^𝑚𝑝𝑓(𝑧)𝐷^𝑛_𝑝𝑓(𝑧)≤∑−11+𝛽^{∞𝑘=𝑝+1}𝑘/𝑝^𝑚−𝑘/𝑝^𝑛𝑎_𝑘|𝑧|^𝑘−𝑝1−∑^∞_𝑘=𝑝+1𝑘/𝑝^𝑛𝑎_𝑘|𝑧|^𝑘−𝑝. (13) Letting 𝑧→1⁻ through real values, then we have 𝛽𝐷^𝑚_𝑝𝑓(𝑧)𝐷^𝑛_𝑝𝑓(𝑧)𝐷−1−Re^𝑚𝑝𝑓(𝑧)𝐷^𝑛_𝑝𝑓(𝑧)≤∑−11+𝛽^{∞𝑘=𝑝+1}𝑘/𝑝^𝑚−𝑘/𝑝^𝑛𝑎_𝑘1−∑^∞_𝑘=𝑝+1𝑘/𝑝^𝑛𝑎_𝑘. (14) The last expression is bounded above by (1−𝛼) if ^∞_𝑘=𝑝+1𝑘1+𝛽𝑝^𝑚−𝑘𝛼+𝛽𝑝^𝑛𝑎_𝑘≤1−𝛼. (15) And hence the proof of Lemma 3 is completed.

Remark 4. (i) The result obtained from Lemma 3 corrects the result obtained by Eker and Şeker [7, Theorem 1].
(ii) Putting 𝑝=1 in Lemma 3, we correct the result obtained by Eker and Owa [9, Theorem 2.1].

Corollary 5. The sufficient condition for 𝑓(𝑧) given by (1) to be in the class 𝑁_𝑝(𝑛;𝛼,𝛽), which is defined by (8) is that ^∞_𝑘=𝑝+1𝑘𝑝^𝑛𝑘𝑝1+𝛽−𝛼+𝛽𝑎_𝑘≤1−𝛼. (16)

Theorem 6. Let 𝑓(𝑧) of the form (1), then 𝑓(𝑧)∈𝑉_𝑝(𝑚,𝑛;𝛼,𝛽) if and only if ^∞_𝑘=𝑝+1𝑘1+𝛽𝑝^𝑚−𝑘𝛼+𝛽𝑝^𝑛𝑎_𝑘≤1−𝛼. (17)

Proof. In view of Lemma 3, we need only to show that function 𝑓(𝑧)∈𝑉_𝑝(𝑚,𝑛;𝛼,𝛽) satisfies the coefficient inequality (17). If 𝑓(𝑧)∈𝑉_𝑝(𝑚,𝑛;𝛼,𝛽), then from (7), we have 𝐷Re^𝑚𝑝𝑓(𝑧)𝐷^𝑛_𝑝𝑓(𝑧)𝐷−𝛼>𝛽^𝑚𝑝𝑓(𝑧)𝐷^𝑛_𝑝𝑓(𝑧)−1, (18) then we have ()𝑧Re1−𝛼^𝑝+∑^∞_𝑘=𝑝+1𝑘/𝑝^𝑚−𝛼𝑘/𝑝^𝑛𝑎_𝑘𝑧^𝑘𝑧^𝑝+∑^∞_𝑘=𝑝+1𝑘/𝑝^𝑛𝑎_𝑘𝑧^𝑘∑>𝛽^{∞𝑘=𝑝+1}𝑘/𝑝^𝑚−𝑘/𝑝^𝑛𝑎_𝑘𝑧^𝑘𝑧^𝑝+∑^∞_𝑘=𝑝+1𝑘/𝑝^𝑛𝑎_𝑘𝑧^𝑘 (19) or ()Re1−𝛼+∑^∞_𝑘=𝑝+1𝑘/𝑝^𝑚−𝛼𝑘/𝑝^𝑛𝑎_𝑘𝑧^𝑘−𝑝1+∑^∞_𝑘=𝑝+1𝑘/𝑝^𝑛𝑎_𝑘𝑧^𝑘−𝑝∑>𝛽^{∞𝑘=𝑝+1}𝑘/𝑝^𝑚−𝑘/𝑝^𝑛𝑎_𝑘𝑧^𝑘−𝑝1+∑^∞_𝑘=𝑝+1𝑘/𝑝^𝑛𝑎_𝑘𝑧^𝑘−𝑝. (20) Since 𝑓(𝑧)∈𝑉, 𝑓(𝑧)lies in the class 𝑉(𝜃_𝑘,𝛿) for some sequence {𝜃_𝑘} and a real number 𝛿 such that 𝜃_𝑘+𝑘−𝑝𝛿≡𝜋(mod2𝜋)𝑘≥𝑝+1. (21) Set 𝑧=𝑟𝑒^𝑖𝛿 in (20), we get (1−𝛼)−∑^∞_𝑘=𝑝+1𝑘/𝑝^𝑚−𝛼𝑘/𝑝^𝑛𝑎_𝑘𝑟^𝑘−𝑝1−∑^∞_𝑘=𝑝+1𝑘/𝑝^𝑛𝑎_𝑘𝑟^𝑘−𝑝∑>𝛽^{∞𝑘=𝑝+1}𝑘/𝑝^𝑚−𝑘/𝑝^𝑛𝑎_𝑘𝑟^𝑘−𝑝1−∑^∞_𝑘=𝑝+1𝑘/𝑝^𝑛𝑎_𝑘𝑟^𝑘−𝑝. (22) Letting 𝑟→1, then we have the inequality (17).
Hence the proof of Theorem 6 is completed.

Corollary 7. If 𝑓(𝑧)∈𝑉_𝑝(𝑚,𝑛;𝛼,𝛽), then 𝑎_𝑘≤1−𝛼1+𝛽𝑘/𝑝^𝑚−𝛼+𝛽𝑘/𝑝^𝑛𝑘≥𝑝+1;𝑝∈ℕ. (23) The result is sharp for the function 𝑓(𝑧) given by 𝑓(𝑧)=𝑧^𝑝+1−𝛼1+𝛽𝑘/𝑝^𝑚−𝛼+𝛽𝑘/𝑝^𝑛𝑒^𝑖𝜃𝑘𝑧^𝑘𝑘≥𝑝+1;𝑝∈ℕ;𝑧∈𝑈. (24)

3. Distortion Theorems

Theorem 8. Let the function 𝑓(𝑧) defined by (1) be in the class 𝑉_𝑝(𝑚,𝑛;𝛼,𝛽). Then |𝑧|^𝑝−1−𝛼Φ|𝑧|𝑝+1^𝑝+1≤𝑓(𝑧)≤|𝑧|^𝑝+1−𝛼Φ|𝑧|𝑝+1^𝑝+1, (25) where Φ𝑝+1=1+𝛽𝑝+1𝑝^𝑚−𝛼+𝛽𝑝+1𝑝^𝑛. (26) The result is sharp.

Proof. We employ the same technique as used by Silverman [15]. In view of Theorem 6, since Φ(𝑘)=𝑘1+𝛽𝑝^𝑚−𝑘𝛼+𝛽𝑝^𝑛, (27) is an increasing function of 𝑘(𝑘≥𝑝+1;𝑝∈ℕ), we have Φ𝑝+1^∞_𝑘=𝑝+1𝑎_𝑘≤^∞_𝑘=𝑝+1Φ(𝑘)𝑎_𝑘≤1−𝛼, (28) that is ^∞_𝑘=𝑝+1𝑎_𝑘≤1−𝛼Φ.𝑝+1 (29) Thus, we have 𝑓(𝑧)≤|𝑧|^𝑝+|𝑧|^∞𝑝+1_𝑘=𝑝+1𝑎_𝑘,𝑓(𝑧≤|𝑧|)_𝑝+1−𝛼1+𝛽𝑝+1/𝑝^𝑚−𝛼+𝛽𝑝+1/𝑝^𝑛|𝑧|^𝑝+1. (30) Similarly, we get 𝑓(𝑧)≥|𝑧|^𝑝−|𝑧|^∞𝑝+1_𝑘=𝑝+1𝑎_𝑘≥|𝑧|^𝑝−1−𝛼1+𝛽𝑝+1/𝑝^𝑚−𝛼+𝛽𝑝+1/𝑝^𝑛×|𝑧|^𝑝+1. (31)
This completes the proof of Theorem 8. Finally the result is sharp for the following function: 𝑓(𝑧)=𝑧^𝑝+1−𝛼1+𝛽𝑝+1/𝑝^𝑚−𝛼+𝛽𝑝+1/𝑝^𝑛×𝑒^{𝑖𝜃𝑝+1}𝑧^𝑝+1, (32) at 𝑧=±|𝑧|𝑒^{−𝑖𝜃𝑝+1}.

Corollary 9. Under the hypotheses of Theorem 8, 𝑓(𝑧) is included in a disc with center at the origin and radius 𝑟₁ given by 𝑟₁=1+1−𝛼1+𝛽𝑝+1/𝑝^𝑚−𝛼+𝛽𝑝+1/𝑝^𝑛. (33)

Theorem 10. Let the function 𝑓(𝑧) defined by (1) be in the class 𝑉_𝑝(𝑚,𝑛;𝛼,𝛽). Then ()𝑝−𝑝+11−𝛼Φ|𝑧𝑧|𝑝+1||^𝑝−1≤𝑓^′(𝑧≤())𝑝+𝑝+11−𝛼Φ|𝑧𝑧|𝑝+1||^𝑝−1, (34) where Φ(𝑝+1) is defined by (26). Also the result of Theorem 10 is sharp.

Proof. Similarly for Φ(𝑘) defined by (27) it is clear that Φ(𝑘)/𝑘 is an increasing function of 𝑘(𝑘≥𝑝+1;𝑝∈ℕ), in view of Theorem 6, we have Φ𝑝+1𝑝+1^∞_𝑘=𝑝+1𝑘𝑎_𝑘≤^∞_𝑘=𝑝+1Φ(𝑘)𝑘𝑘𝑎_𝑘=^∞_𝑘=𝑝+1Φ(𝑘)𝑎_𝑘≤1−𝛼, (35) that is ^∞_𝑘=𝑝+1𝑘𝑎_𝑘≤𝑝+1(1−𝛼)Φ.𝑝+1 (36) Thus, we have 𝑓^′(𝑧)≤𝑝|𝑧|^𝑝−1+|𝑧|^𝑝∞_𝑘=𝑝+1𝑘𝑎_𝑘≤𝑝+|𝑧|^∞_𝑘=𝑝+1𝑘𝑎_𝑘|𝑧|^𝑝−1≤()𝑝+𝑝+11−𝛼Φ|𝑧𝑧|𝑝+1||^𝑝−1. (37) Similarly, 𝑓^′(𝑧)≥𝑝|𝑧|^𝑝−1−|𝑧|^𝑝∞_𝑘=𝑝+1𝑘𝑎_𝑘≥𝑝−|𝑧|^∞_𝑘=𝑝+1𝑘𝑎_𝑘|𝑧|^𝑝−1≤()𝑝−𝑝+11−𝛼Φ|𝑧𝑧|𝑝+1||^𝑝−1. (38) Finally, we can see that the assertions of Theorem 10 are sharp for the function 𝑓(𝑧) defined by (32). This completes the proof of Theorem 10.

Corollary 11. Under the hypotheses of Theorem 10, 𝑓^′(𝑧) is included in a disc with center at the origin and radius 𝑟₂ given by 𝑟₂()=𝑝+𝑝+11−𝛼Φ.𝑝+1 (39)

4. Extreme Points

Theorem 12. Let the function 𝑓(𝑧) defined by (1) be in the class 𝑉_𝑝(𝑚,𝑛;𝛼,𝛽), with arg(𝑎_𝑘)=𝜃_𝑘, where 𝜃_𝑘+(𝑘−𝑝)𝛿≡𝜋(mod2𝜋)(𝑘≥𝑝+1). Define 𝑓_𝑝(𝑧)=𝑧^𝑝,𝑓_𝑘(𝑧)=𝑧^𝑝+1−𝛼1+𝛽𝑘/𝑝^𝑚−𝛼+𝛽𝑘/𝑝^𝑛𝑒^𝑖𝜃𝑘𝑧^𝑘𝑘≥𝑝+1;𝑧∈𝑈. (40) Then 𝑓(𝑧)∈𝑉_𝑝(𝑚,𝑛;𝛼,𝛽) if and only if 𝑓(𝑧) can be expressed in the form 𝑓(𝑧)=∑^∞_𝑘=𝑝𝜇_𝑘𝑓_𝑘(𝑧), where 𝜇_𝑘≥0 and ∑^∞_𝑘=𝑝𝜇_𝑘=1.

Proof. If 𝑓(𝑧)=∑^∞_𝑘=𝑝𝜇_𝑘𝑓_𝑘(𝑧) with 𝜇_𝑘≥0 and ∑^∞_𝑘=𝑝𝜇_𝑘=1, then ^∞_𝑘=𝑝+1𝑘1+𝛽𝑝^𝑚−𝑘𝛼+𝛽𝑝^𝑛×1−𝛼1+𝛽𝑘/𝑝^𝑚−𝛼+𝛽𝑘/𝑝^𝑛𝜇_𝑘=^∞_𝑘=𝑝+1(1−𝛼)𝜇_𝑘=(1−𝛼)1−𝜇_𝑝≤(1−𝛼). (41) Hence, 𝑓(𝑧)∈𝑉_𝑝(𝑚,𝑛;𝛼,𝛽).
Conversely, let the function 𝑓(𝑧) defined by (1) be in the class 𝑉_𝑝(𝑚,𝑛;𝛼,𝛽). Define 𝜇_𝑘=1+𝛽𝑘/𝑝^𝑚−𝛼+𝛽𝑘/𝑝^𝑛𝑎1−𝛼^𝑘,𝜇𝑘≥𝑝+1;𝑝∈ℕ_𝑝=1−^∞_𝑘=𝑝+1𝜇_𝑘. (42) From Theorem 6, ∑^∞_𝑘=𝑝+1𝜇_𝑘≤1 and so 𝜇_𝑝≥0. Since 𝜇_𝑘𝑓_𝑘(𝑧)=𝜇_𝑘𝑧^𝑝+𝑎_𝑘𝑧^𝑘, then ^∞_𝑘=𝑝𝜇_𝑘𝑓_𝑘(𝑧)=𝑧^𝑝+^∞_𝑘=𝑝+1𝑎_𝑘𝑧^𝑘=𝑓(𝑧). (43) This completes the proof of Theorem 12.

Remark 13. (i) Putting 𝛽=𝑛=0 and 𝑝=𝑚=1 in all the above results, we obtain the corresponding results obtained by Silverman [15].
(ii) Putting (i) 𝛽=𝑛=0and 𝑚=1 and (ii) 𝛽=0,𝑛=1and 𝑚=2 in all the above results, we obtain the corresponding results obtained by Aouf et al. [16].
(iii) Putting 𝑚=𝑛+1(𝑛∈ℕ₀) in all the above results, we obtain the corresponding results of the classes 𝑉_𝑝(𝑛;𝛼,𝛽), defined by (8).

Acknowledgment

The authors thank the referees for their valuable suggestions which led to improvement of this study.