A New Subclass of Salagean-Type Harmonic Univalent Functions

Abstract

We define and investigate a new subclass of Salagean-type harmonic univalent functions. We obtain coefficient conditions, extreme points, distortion bounds, convolution, and convex combination for the above subclass of harmonic functions.

1. Introduction

Let 𝒜 denote the class of functions of the form:𝑓(𝑧)=𝑧+^∞_𝑘=2𝑎_𝑘𝑧^𝑘 (1.1) which are analytic in the open unit disk 𝕌={𝑧∈ℂ∶|𝑧|<1}.

We denote the subclass of A consisting of analytic and univalent functions 𝑓(𝑧) in the unit disk 𝕌 by 𝑆.

The following classes of functions and many others are well known and have been studied repeatedly by many authors, namely, Sălăgean [1], Abdul Halim [2], and Darus [3] to mention but a few. (i)𝑆₀={𝑓(𝑧)∈𝒜∶Re{𝑓(𝑧)/𝑧}>0,𝑧∈𝕌}. (ii)𝐵(𝛼)={𝑓(𝑧)∈𝒜∶Re{𝑓(𝑧)/𝑧}>𝛼,0≤𝛼<1,𝑧∈𝕌}. (iii)𝛿(𝛼)={𝑓(𝑧)∈𝒜∶Re{𝑓^′(𝑧)}>𝛼,0≤𝛼<1,𝑧∈𝕌}. (iv)𝐵_𝑛(𝛽)={𝑓(𝑧)∈𝒜∶Re{𝐷^𝑛𝑓(𝑧)^𝛽/𝑧^𝛽}>0,𝑧∈U,𝑛∈ℕ₀=N∪{0},𝛽>0}.

In 1994, Opoola defined the class 𝑇^𝛽_𝑛(𝛼) to be a subclass of 𝐴 consisting of analytic functions satisfying the condition𝐷Re^𝑛𝑓(𝑧)^𝛽𝑧^𝛽>𝛼,𝑧∈𝕌,𝑛∈ℕ₀,0≤𝛼<1,𝛽>0, (1.2) where 𝐷^𝑛 is the Salagean differential operator defined as follows: 𝐷⁰𝑓(𝑧)(𝑧),𝐷=𝑓₁𝑓(𝑧)=𝐷𝑓(𝑧)=𝑧𝑓^′(𝑧),𝐷^𝑛𝑓(𝑧)=𝐷𝐷^𝑛−1𝑓(𝑧)=𝑧𝐷^𝑛−1𝑓(𝑧). (1.3) We note that 𝑇^𝛽_𝑛(𝛼) is a generalization of the classes of functions 𝑆₀,𝐵(𝛼),𝛿(𝛼), and 𝐵_𝑛(𝛽).

Some properties of this class of functions were established by Opoola [4] namely, (i)𝑇^𝛽_𝑛(𝛼) is a subclass of univalent functions;(ii)𝑇^𝛽_𝑛+1(𝛼)⊂𝑇^𝛽_𝑛(𝛼);(iii)if 𝑓(𝑧)∈𝑇^𝛽_𝑛(𝛼), then the integral operator𝐹_𝑐(𝑧)^𝛽=𝛽+𝑐𝑧^𝛽𝑐₀𝑡^𝛽−1𝑓(𝑧)^𝛽𝑑𝑡(𝑐≥0) (1.4) is also in 𝑇^𝛽_𝑛(𝛼).

Now, by Binomial expansion, we have 𝑓(𝑧)^𝛽=𝑧^𝛽+𝛽𝑎₂𝑧^𝛽+1+𝛽𝑎₃+𝛽𝛽−1𝑎2!³²𝑧^𝛽+2+𝛽𝑎₄+𝛽𝛽−12!2𝑎₂𝑎₃+𝛽𝛽−1𝛽−2𝑎3!³²𝑧^𝛽+3+⋯. (1.5) Hence, we define𝑓(𝑧)^𝛽=𝑧^𝛽+^∞_𝑘=2𝛽𝑎_𝑘𝑧^{𝛽+𝑘−1}𝐷,𝛽>0,_𝑛𝑓(𝑧)^𝛽=𝑧^𝛽+^∞_𝑘=2𝛽𝑘^𝑛𝑎_𝑘𝑧^{𝛽+𝑘−1},𝑛∈ℕ₀. (1.6)

2. Preliminaries

A continuous function 𝑓=𝑢+𝑖𝑣 is a complex-valued harmonic function in a domain 𝐷⊂ℂ if both 𝑢 and 𝑣 are real harmonic in 𝐷. In any simply connected domain, we can write𝑓=ℎ+𝑔, (2.1) where ℎ and 𝑔 are analytic in 𝐷. We call ℎ the analytic part and 𝑔 the coanalytic part of 𝑓. A necessary and sufficient condition for 𝑓 to be locally univalent and sense-preserving in 𝐷 is that |ℎ^′|>|𝑔^′| in 𝐷.

Denote by 𝑆_ℋ the class of functions 𝑓 of the form (2.1) that are harmonic univalent and sense-preserving in the unit disk 𝕌. The subclasses of harmonic univalent functions have been studied by some authors for different purposes and different properties (see examples [5–12]). In this work, we may express the analytic functions ℎ and 𝑔 asℎ(𝑧)^𝛽=𝑧^𝛽+^∞_𝑘=2𝛽𝑎_𝑘𝑧^{𝛽+𝑘−1},𝑔(𝑧)^𝛽=^∞_𝑘=1𝛽𝑏_𝑘𝑧^{𝛽+𝑘−1},𝑏₁<1. (2.2) Therefore,𝑓(𝑧)^𝛽=ℎ(𝑧)^𝛽+𝑔(𝑧)^𝛽. (2.3) We define the modified Salagean operator of 𝑓 as 𝐷^𝑛𝑓(𝑧)^𝛽=𝐷^𝑛ℎ(𝑧)^𝛽+(−1)^𝑛𝐷^𝑛𝑔(𝑧)^𝛽, (2.4) where𝐷^𝑛ℎ(𝑧)^𝛽=𝑧^𝛽+^∞_𝑘=2𝛽𝑘^𝑛𝑎_𝑘𝑧^{𝛽+𝑘−1},𝐷^𝑛𝑔(𝑧)^𝛽=^∞_𝑘=1𝛽𝑘^𝑛𝑏_𝑘𝑧^{𝛽+𝑘−1}. (2.5) We let ℋ(𝑛,𝛽,𝛼) be the family of harmonic functions of the form (2.3) such that𝐷Re^𝑛+1𝑓(𝑧)^𝛽𝐷^𝑛𝑓(𝑧)^𝛽>𝛼,𝛽≥1,0≤𝛼<1,𝑛∈ℕ₀, (2.6) where 𝐷^𝑛𝑓(𝑧)^𝛽 is defined by (2.4).

It is clear that the class ℋ(𝑛,𝛽,𝛼) includes a variety of well-known subclasses of 𝑆_ℋ. For example, ℋ(0,1,𝛼)≡𝑆^∗_ℋ(𝛼) is the class of sense-preserving, harmonic univalent functions 𝑓 which are starlike of order 𝛼 in 𝕌, that is, 𝜕/𝜕𝜃{arg(𝑓(𝑟𝑒^𝑖𝜃))}>𝛼, and ℋ(1,1,𝛼)≡ℋ𝒦(𝛼) is the class of sense-preserving, harmonic univalent functions 𝑓 which are convex of order 𝛼 in 𝕌, that is 𝜕/𝜕𝜃{arg(𝜕/𝜕𝜃𝑓(𝑟𝑒^𝑖𝜃))}>𝛼. Note that the classes 𝑆^∗_ℋ(𝛼) and ℋ𝒦(𝛼) were introduced and studied by Jahangiri [5]. Also note that the class ℋ(𝑛,1,𝛼)≡ℋ𝒦(𝛼) is the class of Salagean-type harmonic univalent functions introduced by Jahangiri et al. [13].

We let the subclass ℋ(𝑛,𝛽,𝛼) consist of harmonic functions 𝑓_𝑛=ℎ+𝑔_𝑛 in ℋ(𝑛,𝛽,𝛼) so ℎ and 𝑔 are of the formℎ^𝛽(𝑧)=𝑧^𝛽−^∞_𝑘=2𝑎_𝑘𝑧^{𝛽+𝑘−1},𝑔^𝛽_𝑛(𝑧)=(−1)^𝑛∞_𝑘=1𝑏_𝑘𝑧^{𝛽+𝑘−1}. (2.7)

In 1984, Clunie and Sheil-Small [14] investigated the class 𝑆_ℋ as well as its geometric subclasses and obtained some coefficient bounds. Since then, there have been several related papers on 𝑆_ℋ and its subclasses such that Silverman [15], Silverman and Silvia [16], and Jahangiri [5, 17] studied the harmonic univalent functions. Jahangiri [5] proved the following theorem.

Theorem 2.1. Let 𝑓=ℎ+𝑔 where ℎ=𝑧+∑^∞_𝑘=2𝑎_𝑘𝑧^𝑘 and 𝑔=∑^∞_𝑘=1𝑏_𝑘𝑧^𝑘. If ^∞_𝑘=1𝑘−𝛼𝑎1−𝛼^𝑘+𝑘+𝛼𝑏1−𝛼^𝑘≤2,(0≤𝛼<1), (2.8) then 𝑓 is sense-preserving, harmonic, and univalent in 𝕌 and 𝑓∈𝑆^∗_ℋ(𝛼). The condition (2.8) is also necessary if 𝑓∈𝒯𝐻(𝛼)≡ℋ(0,1,𝛼).

In this paper, we will give the sufficient condition for functions 𝑓^𝛽=ℎ^𝛽+𝑔^𝛽 where ℎ^𝛽 and 𝑔^𝛽 are given by (2.2) to be in the class ℋ(𝑛,𝛽,𝛼) and it is shown that these coefficient conditions are also necessary for functions in the class ℋ(𝑛,𝛽,𝛼). Also, we obtain distortion theorems and characterize the extreme points for functions in ℋ(𝑛,𝛽,𝛼). Convolution and convex combination are also obtained.

3. Main Results

In this section, we prove the main results.

3.1. Coefficient Estimates

Theorem 3.1. Let 𝑓^𝛽=ℎ^𝛽+𝑔^𝛽, where ℎ^𝛽 and 𝑔^𝛽 are given by (2.2). If (𝑘−𝛼)𝑎^∞_𝑘=1𝑘+(𝑘+𝛼)𝑏_𝑘𝛽𝑘^𝑛≤1+𝛽(1−𝛼), (3.1) where 𝑎₁=1, 𝑛∈ℕ₀, 𝛽≥1, and 0≤𝛼<1, then 𝑓^𝛽 is sense-preserving, harmonic univalent in U, and 𝑓∈ℋ(𝑛,𝛽,𝛼).

Proof. If 𝑧^𝛽₁≠𝑧^𝛽₂, then 𝑓𝑧₁^𝛽−𝑓𝑧₂^𝛽ℎ𝑧₁^𝛽−ℎ𝑧₂^𝛽𝑔𝑧≥1−^1𝛽−𝑔𝑧₂^𝛽ℎ𝑧₁^𝛽−ℎ𝑧₂^𝛽∑=1−^∞𝑘=1𝛽𝑏_𝑘𝑧₁^{𝑘+𝛽−1}−𝑧₂^{𝑘+𝛽−1}𝑧^𝛽₁−𝑧^𝛽₂+∑^∞_𝑘=2𝛽𝑎_𝑘𝑧₁^{𝑘+𝛽−1}−𝑧₂^{𝑘+𝛽−1}∑>1−^∞𝑘=1𝑘+𝛽−1𝑏_𝑘1−∑^∞_𝑘=2𝑘+𝛽−1𝑎_𝑘∑≥1−^∞𝑘=1(𝑘+𝛼)𝛽𝑘^𝑛/(1−𝛼)𝑏_𝑘1−∑^∞_𝑘=2(𝑘−𝛼)𝛽𝑘^𝑛/(1−𝛼)𝑎_𝑘≥0, (3.2) which proves univalence. Note that 𝑓 is sense-preserving in 𝕌. This is because ℎ^′(𝑧)^𝛽≥𝛽|𝑧|^𝛽−1−^∞_𝑘=2𝑘+𝛽−1𝑎_𝑘|𝑧|^{𝑘+𝛽−2}>𝛽1−^∞_𝑘=2(𝑘−𝛼)𝛽𝑘^𝑛𝑎1−𝛼^𝑘≥𝛽^∞_𝑘=1(𝑘+𝛼)𝛽𝑘^𝑛𝑏1−𝛼^𝑘≥^∞_𝑘=1𝛽𝑘+𝛽−1𝑏_𝑘|𝑧|^{𝑘+𝛽−2}≥𝑔^′(𝑧)^𝛽. (3.3)
By (2.6), 𝐷Re^𝑛+1𝑓(𝑧)^𝛽𝐷^𝑛𝑓(𝑧)^𝛽𝐷=Re^𝑛+1ℎ(𝑧)^𝛽+(−1)^𝑛+1𝐷^𝑛+1𝑔(𝑧)^𝛽𝐷^𝑛ℎ(𝑧)^𝛽+(−1)^𝑛𝐷^𝑛𝑔(𝑧)^𝛽>𝛼. (3.4) Using the fact that Re(𝑤)>𝛼 if and only if |1−𝛼+𝑤|≥|1+𝛼−𝑤|, it suffices to show that 𝐷1−𝛼+^𝑛+1𝑓(𝑧)^𝛽𝐷^𝑛𝑓(𝑧)^𝛽−𝐷1+𝛼−^𝑛+1𝑓(𝑧)^𝛽𝐷^𝑛𝑓(𝑧)^𝛽𝐷≥0,_𝑛+1𝑓(𝑧)^𝛽+(1−𝛼)𝐷^𝑛𝑓(𝑧)^𝛽−𝐷^𝑛+1𝑓(𝑧)^𝛽−(1+𝛼)𝐷^𝑛𝑓(𝑧)^𝛽≥0. (3.5)(3.6) Substituting for 𝐷^𝑛+1𝑓(𝑧)^𝛽, 𝐷^𝑛𝑓(𝑧)^𝛽 in (3.6), we have 𝐷^𝑛+1ℎ(𝑧)^𝛽+(−1)^𝑛+1𝐷^𝑛+1𝑔(𝑧)^𝛽+(1−𝛼)𝐷^𝑛ℎ(𝑧)^𝛽+(−1)^𝑛𝐷^𝑛𝑔(𝑧)^𝛽−𝐷^𝑛+1ℎ(𝑧)^𝛽+(−1)^𝑛+1𝐷^𝑛+1𝑔(𝑧)^𝛽−(1+𝛼)𝐷^𝑛ℎ(𝑧)^𝛽+(−1)^𝑛𝐷^𝑛𝑔(𝑧)^𝛽=𝑧^𝛽+^∞_𝑘=2𝛽𝑘^𝑛+1𝑎_𝑘𝑧^{𝛽+𝑘−1}+(−1)^𝑛+1∞_𝑘=1𝛽𝑘^𝑛+1𝑏_𝑘𝑧^{𝛽+𝑘−1}+(1−𝛼)𝑧^𝛽+^∞_𝑘=2𝛽𝑘^𝑛𝑎_𝑘𝑧^{𝛽+𝑘−1}+(−1)^𝑛∞_𝑘=1𝛽𝑘^𝑛𝑏_𝑘𝑧^{𝛽+𝑘−1}−𝑧^𝛽+^∞_𝑘=2𝛽𝑘^𝑛+1𝑎_𝑘𝑧^{𝛽+𝑘−1}+(−1)^𝑛+1∞_𝑘=1𝛽𝑘^𝑛+1𝑏_𝑘𝑧^{𝛽+𝑘−1}−(1+𝛼)𝑧^𝛽+^∞_𝑘=2𝛽𝑘^𝑛𝑎_𝑘𝑧^{𝛽+𝑘−1}+(−1)^𝑛∞_𝑘=1𝛽𝑘^𝑛𝑏_𝑘𝑧^{𝛽+𝑘−1}=(2−𝛼)𝑧^𝛽+^∞_𝑘=2𝛽(𝑘+1−𝛼)𝑘^𝑛𝑎_𝑘𝑧^{𝛽+𝑘−1}−(−1)^𝑛∞_𝑘=1𝛽(𝑘−1+𝛼)𝑘^𝑛𝑏_𝑘𝑧^{𝛽+𝑘−1}−(−𝛼)𝑧^𝛽+^∞_𝑘=2𝛽(𝑘−1−𝛼)𝑘^𝑛𝑎_𝑘𝑧^{𝛽+𝑘−1}−(−1)^𝑛∞_𝑘=1𝛽(𝑘+1+𝛼)𝑘^𝑛𝑏_𝑘𝑧^{𝛽+𝑘−1}≥2(1−𝛼)|𝑧|^𝛽−^∞_𝑘=22𝛽𝑘^𝑛(𝑘−𝛼)𝑎_𝑘𝑧^{𝛽+𝑘−1}−^∞_𝑘=12𝛽𝑘^𝑛(𝑘−𝛼)𝑏_𝑘𝑧^{𝛽+𝑘−1}=2(1−𝛼)1−^∞_𝑘=2𝛽𝑘^𝑛(𝑘−𝛼)𝑎1−𝛼^𝑘−^∞_𝑘=1𝛽𝑘^𝑛(𝑘+𝛼)𝑏1−𝛼^𝑘. (3.7) This last expression is nonnegative by (3.1), and so the proof is complete.

The harmonic function𝑓(𝑧)^𝛽=𝑧^𝛽+^∞_𝑘=2𝛽1−𝛼(𝑘−𝛼)𝛽𝑘^𝑛𝑥_𝑘𝑧^{𝑘+𝛽−1}+^∞_𝑘=1𝛽1−𝛼(𝑘+𝛼)𝛽𝑘^𝑛𝑦_𝑘𝑧^{𝑘+𝛽−1}, (3.8) where 𝑛∈ℕ₀, 𝛽≥1, 0≤𝛼<1, and ∑^∞_𝑘=2|𝑥_𝑘|+∑^∞_𝑘=1|𝑦_𝑘|=1, shows that the coefficient bound given by (3.1) is sharp. The functions of the form (3.8) are in ℋ(𝑛,𝛽,𝛼) because ^∞_𝑘=1𝑘−𝛼𝑎1−𝛼^𝑘+𝑘+𝛼𝑏1−𝛼^𝑘𝛽𝑘^𝑛=𝛽+^∞_𝑘=2𝑥_𝑘+^∞_𝑘=1𝑦_𝑘=𝛽+1. (3.9)

In the following theorem, it is shown that the condition (3.1) is also necessary for functions 𝑓^𝛽_𝑛=ℎ^𝛽+𝑔^𝛽_𝑛 where ℎ^𝛽 and 𝑔^𝛽_𝑛 are of the form (2.7).

Theorem 3.2. Let 𝑓^𝛽_𝑛=ℎ^𝛽+𝑔^𝛽_𝑛 be given by (2.7). Then 𝑓^𝛽_𝑛∈ℋ(𝑛,𝛽,𝛼), if and only if (𝑘−𝛼)𝑎^∞_𝑘=1𝑘+(𝑘+𝛼)𝑏_𝑘𝛽𝑘^𝑛≤1+𝛽(1−𝛼), (3.10) where 𝑎₁=1, 𝑛∈ℕ₀, 𝛽≥1, and 0≤𝛼<1.

Proof. Since ℋ(𝑛,𝛽,𝛼)⊂ℋ(𝑛,𝛽,𝛼), we only need to prove the “only if” part of the theorem. To this end, for functions 𝑓^𝛽_𝑛 of the form (2.7), we notice that the condition (2.6) is equivalent to ()𝑧Re1−𝛼^𝛽−∑^∞_𝑘=2(𝑘−𝛼)𝛽𝑘^𝑛𝑎_𝑘𝑧^{𝑘+𝛽−1}−(−1)^2𝑛∑^∞_𝑘=1(𝑘+𝛼)𝛽𝑘^𝑛𝑏_𝑘𝑧^{𝑘+𝛽−1}𝑧^𝛽−∑^∞_𝑘=2𝛽𝑘^𝑛𝑎_𝑘𝑧^{𝑘+𝛽−1}+(−1)^𝑛∑^∞_𝑘=1𝛽𝑘^𝑛𝑏_𝑘𝑧^{𝑘+𝛽−1}≥0. (3.11) The above required condition (3.11) must hold for all values of 𝑧 in 𝕌. Upon choosing the values of 𝑧 on the positive real axis where 0≤𝑧=𝑟<1, we must have 1−𝛼−∑^∞_𝑘=2(𝑘−𝛼)𝛽𝑘^𝑛𝑎_𝑘𝑟^𝑘−1−∑^∞_𝑘=1(𝑘+𝛼)𝛽𝑘^𝑛𝑏_𝑘𝑟^𝑘−11−∑^∞_𝑘=2𝛽𝑘^𝑛𝑎_𝑘𝑟^𝑘−1+∑^∞_𝑘=1𝛽𝑘^𝑛𝑏_𝑘𝑟^𝑘−1≥0. (3.12) If the condition (3.10) does not hold, then the numerator in (3.12) is negative for 𝑟 sufficiently close to 1. Hence there exist 𝑧₀=𝑟₀ in (0,1) for which the quotient in (3.12) is negative. This contradicts the required condition for 𝑓^𝛽_𝑛∈ℋ(𝑛,𝜆,𝛼) and so the proof is complete.

3.2. Distortion Bounds and Extreme Points

In this section, first we will obtain distortion bounds for functions in ℋ(𝑛,𝛽,𝛼).

Theorem 3.3. Let 𝑓^𝛽_𝑛∈ℋ(𝑛,𝛽,𝛼). Then for |𝑧|=𝑟<1, we have 𝑓_𝑛(𝑧)^𝛽≤1+𝑏₁𝑟^𝛽+1𝛽2^𝑛1−𝛼−2−𝛼1+𝛼𝛽𝑏2−𝛼¹𝑟^𝛽+1,𝑓_𝑛(𝑧)^𝛽≥1−𝑏₁𝑟^𝛽−1𝛽2^𝑛1−𝛼−2−𝛼1+𝛼𝛽𝑏2−𝛼¹𝑟^𝛽+1. (3.13)

Proof. We only prove the right-hand inequality. The proof for the left-hand inequality is similar and will be omitted. Let 𝑓^𝛽_𝑛∈ℋ(𝑛,𝛽,𝛼). Taking the absolute value of 𝑓^𝛽_𝑛, we obtain 𝑓_𝑛(𝑧)^𝛽=𝑧^𝛽+^∞_𝑘=2𝑎_𝑘𝑧^{𝑘+𝛽−1}+(−1)^𝑛∞_𝑘=1𝑏_𝑘𝑧^{𝑘+𝛽−1}≤1+𝑏₁𝑟^𝛽+^∞_𝑘=2𝑎_𝑘+𝑏_𝑘𝑟^{𝑘+𝛽−1}≤1+𝑏₁𝑟^𝛽+𝑟^𝛽+1∞_𝑘=2𝑎_𝑘+𝑏_k≤1+𝑏₁𝑟^𝛽+1−𝛼(2−𝛼)𝛽2^{𝑛∞𝑘=2}(2−𝛼)𝛽2^𝑛𝑎1−𝛼^𝑘+(2−𝛼)𝛽2^𝑛𝑏1−𝛼^𝑘𝑟^𝛽+1≤1+𝑏₁𝑟^𝛽+1−𝛼(2−𝛼)𝛽2^{𝑛∞𝑘=2}(𝑘−𝛼)𝛽𝑘^𝑛𝑎1−𝛼^𝑘+(𝑘+𝛼)𝛽𝑘^𝑛𝑏1−𝛼^𝑘𝑟^𝛽+1≤1+𝑏₁𝑟^𝛽+1−𝛼(2−𝛼)𝛽2^𝑛1−1+𝛼𝛽𝑏1−𝛼¹𝑟^𝛽+1, (3.14) for |𝑏₁|<1. This shows that the bounds given in Theorem 3.3 are sharp.

The following covering result follows from the left-hand inequality in Theorem 3.3.

Corollary 3.4. If function 𝑓^𝛽_𝑛=ℎ^𝛽+𝑔^𝛽, where ℎ^𝛽 and 𝑔^𝛽 are given by (2.7), is in ℋ(𝑛,𝛽,𝛼), then 𝑤∶|𝑤|<𝛽2^𝑛+1−1−𝛽2^𝑛−1𝛼𝛽2^𝑛()−22−𝛼^𝑛+1+12^𝑛()𝑏2−𝛼¹⊂𝑓_𝑛(𝕌). (3.15)

Next we determine the extreme points of closed convex hulls of ℋ(𝑛,𝛽,𝛼) denoted by clcoℋ(𝑛,𝛽,𝛼).

Theorem 3.5. Let 𝑓^𝛽_𝑛=ℎ^𝛽+𝑔^𝛽, where ℎ^𝛽 and 𝑔^𝛽 are given by (2.7). Then 𝑓^𝛽_𝑛∈ℋ(𝑛,𝛽,𝛼) if and only if 𝑓_𝑛(𝑧)^𝛽=^∞_𝑘=1𝑋_𝑘ℎ_𝑘(𝑧)+𝑌_𝑘𝑔_𝑛𝑘(𝑧), (3.16) where ℎ₁(𝑧)^𝛽=𝑧^𝛽, ℎ_𝑘(𝑧)^𝛽=𝑧^𝛽−(1−𝛼)/((𝑘−𝛼)𝑘^𝑛)𝑧^{𝑘+𝛽−1}(𝑘=2,3,…), 𝑔_𝑛𝑘(𝑧)^𝛽=𝑧^𝛽+(−1)^𝑛(1−𝛼)/((𝑘+𝛼)𝑘^𝑛)𝑧^{𝑘+𝛽−1}(𝑘=1,2,3,..), and  ∑^∞_𝑘=1(𝑋_𝑘+𝑌_𝑘)=1,  𝑋_𝑘≥0, 𝑌_𝑘≥0. In particular, the extreme points of ℋ(𝑛,𝛽,𝛼) are {ℎ_𝑘} and {𝑔_𝑛𝑘}.

Proof. For functions 𝑓^𝛽_𝑛=ℎ^𝛽+𝑔^𝛽, where ℎ^𝛽 and 𝑔^𝛽 are given by (3.16), we have 𝑓_𝑛(𝑧)^𝛽=^∞_𝑘=1𝑋_𝑘ℎ_𝑘(𝑧)+𝑌_𝑘𝑔_𝑛𝑘(𝑧)=^∞_𝑘=1𝑋_𝑘+𝑌_𝑘𝑧^𝛽−^∞_𝑘=21−𝛼(𝑘−𝛼)𝑘^𝑛𝑋_𝑘𝑧^{𝑘+𝛽−1}+(−1)^𝑛∞_𝑘=11−𝛼(𝑘+𝛼)𝑘^𝑛𝑌_𝑘𝑧^{𝑘+𝛽−1}. (3.17) Then ^∞_𝑘=2(𝑘−𝛼)𝛽𝑘^𝑛𝑎1−𝛼^𝑘+^∞_𝑘=1(𝑘+𝛼)𝛽𝑘^𝑛𝑏1−𝛼^𝑘=^∞_𝑘=2𝑋_𝑘+^∞_𝑘=1𝑌_𝑘=1−𝑋₁≤1, (3.18) and so 𝑓^𝛽_𝑛∈clcoℋ(𝑛,𝛽,𝛼).
Conversely, suppose that 𝑓^𝛽_𝑛∈clcoℋ(𝑛,𝛽,𝛼). Setting 𝑋_𝑘=(𝑘−𝛼)𝛽𝑘^𝑛𝑎1−𝛼^𝑘0≤𝑋_𝑘(),𝑌≤1𝑘=2,3,…_𝑘=(𝑘+𝛼)𝛽𝑘^𝑛𝑏1−𝛼^𝑘0≤𝑌_𝑘≤1(𝑘=1,2,3,…), (3.19) and 𝑋₁=1−∑^∞_𝑘=2X_𝑘−∑^∞_𝑘=1𝑌_𝑘; therefore, 𝑓^𝛽_𝑛 can be written as 𝑓_𝑛(𝑧)^𝛽=𝑧^𝛽−^∞_𝑘=2𝛽𝑎_𝑘𝑧^{𝑘+𝛽−1}+(−1)^𝑛∞_𝑘=1𝛽𝑏_𝑘𝑧^{𝑘+𝛽−1}=𝑧^𝛽−^∞_𝑘=2(1−𝛼)𝑋_𝑘(𝑘−𝛼)𝑘^𝑛𝑧^{𝑘+𝛽−1}+(−1)^𝑛∞_𝑘=1(1−𝛼)𝑌_𝑘(𝑘+𝛼)𝑘^𝑛𝑧^{𝑘+𝛽−1}=𝑧^𝛽+^∞_𝑘=2ℎ_𝑘(𝑧)^𝛽−𝑧^𝛽𝑋_𝑘+^∞_𝑘=1𝑔_𝑛𝑘(𝑧)^𝛽−𝑧^𝛽𝑌_𝑘=^∞_𝑘=2ℎ_𝑘(𝑧)^𝛽𝑋_𝑘+^∞_𝑘=1𝑔_𝑛𝑘(𝑧)^𝛽𝑌_𝑘+𝑧^𝛽1−^∞_𝑘=2𝑋_𝑘−^∞_𝑘=1𝑌_𝑘=^∞_𝑘=1ℎ_𝑘(𝑧)^𝛽𝑋_𝑘+𝑔_𝑛𝑘(𝑧)^𝛽𝑌_𝑘,asrequired. (3.20)

3.3. Convolution and Convex Combination

In this section, we show that the class ℋ(𝑛,𝛽,𝛼) is invariant under convolution and convex combination of its member.

For harmonic functions 𝑓_𝑛(𝑧)^𝛽=𝑧^𝛽−∑^∞_𝑘=2|𝑎_𝑘|𝑧^{𝑘+𝛽−1}+(−1)^𝑛∑^∞_𝑘=1|𝑏_𝑘|𝑧^{𝑘+𝛽−1} and 𝐹_𝑛(𝑧)^𝛽=𝑧^𝛽−∑^∞_𝑘=2|𝐴_𝑘|𝑧^{𝑘+𝛽−1}+(−1)^𝑛∑^∞_𝑘=1|𝐵_𝑘|𝑧^{𝑘+𝛽−1}.

The convolution of 𝑓^𝛽_𝑛 and 𝐹^𝛽_𝑛 is given by𝑓^𝛽_𝑛∗𝐹^𝛽_𝑛(𝑧)=𝑓_𝑛(𝑧)^𝛽∗𝐹_𝑛(𝑧)^𝛽=𝑧^𝛽−^∞_𝑘=2𝑎_𝑘𝐴_𝑘𝑧^{𝑘+𝛽−1}+(−1)^𝑛∞_𝑘=1𝑏_𝑘𝐵_𝑘𝑧^{𝑘+𝛽−1}. (3.21)

Theorem 3.6. For 0≤𝜆≤𝛼<1, let 𝑓^𝛽_𝑛∈ℋ(𝑛,𝛽,𝛼) and 𝐹^𝛽_𝑛∈ℋ(𝑛,𝛽,𝛽). Then 𝑓^𝛽_𝑛∗𝐹^𝛽_𝑛∈ℋ(𝑛,𝛽,𝛼)⊂ℋ(𝑛,𝛽,𝜆).

Proof. Let the functions 𝑓_𝑛(𝑧)^𝛽=𝑧^𝛽−∑^∞_𝑘=2|𝑎_𝑘|𝑧^{𝑘+𝛽−1}+(−1)^𝑛∑^∞_𝑘=1|𝑏_𝑘|𝑧^{𝑘+𝛽−1} be in the class ℋ(𝑛,𝛽,𝛼) and let the functions 𝐹_𝑛(𝑧)^𝛽=𝑧^𝛽−∑^∞_𝑘=2|𝐴_𝑘|𝑧^{𝑘+𝛽−1}+(−1)^𝑛∑^∞_𝑘=1|𝐵_𝑘|𝑧^{𝑘+𝛽−1} be in the class ℋ(𝑛,𝛽,𝜆). Then the convolution 𝑓^𝛽_𝑛∗𝐹^𝛽_𝑛 is given by (3.21). We wish to show that the coefficients of 𝑓^𝛽_𝑛∗𝐹^𝛽_𝑛 satisfy the required condition given in Theorem 3.2. For 𝐹^𝛽_𝑛∈ℋ(𝑛,𝛽,𝜆), we note that |𝐴_𝑘|≤1 and |𝐵_𝑘|≤1. Now, for the convolution function 𝑓^𝛽_𝑛∗𝐹^𝛽_𝑛, we obtain _∞𝑘=2𝑘−𝛽𝛽𝑘^𝑛𝑎1−𝛽^𝑘𝐴_𝑘+^∞_𝑘=1𝑘+𝛽𝛽𝑘^𝑛𝑏1−𝛽^𝑘𝐵_𝑘≤^∞_𝑘=2𝑘−𝛽𝛽𝑘^𝑛𝑎1−𝛽^𝑘+^∞_𝑘=1𝑘+𝛽𝛽𝑘^𝑛𝑏1−𝛽^𝑘≤^∞_𝑘=2(𝑘−𝛼)𝛽𝑘^𝑛𝑎1−𝛼^𝑘+^∞_𝑘=1(𝑘+𝛼)𝛽𝑘^𝑛𝑏1−𝛼^𝑘≤1, (3.22) since 0≤𝜆≤𝛼<1 and 𝑓^𝛽_𝑛∈ℋ(𝑛,𝛽,𝛼). Therefore, 𝑓^𝛽_𝑛∗𝐹^𝛽_𝑛∈ℋ(𝑛,𝛽,𝛼)⊂ℋ(𝑛,𝛽,𝜆).

We now examine the convex combination of ℋ(𝑛,𝛽,𝛼).

Let the functions 𝑓_𝑛𝑗(𝑧)^𝛽 be defined, for 𝑗=1,2,…, by𝑓_𝑛j(𝑧)^𝛽=𝑧^𝛽−^∞_𝑘=2𝑎_𝑘,𝑗𝑧^{𝑘+𝛽−1}+(−1)^𝑛∞_𝑘=1𝑏_𝑘,𝑗𝑧^{𝑘+𝛽−1}. (3.23)

Theorem 3.7. Let the functions 𝑓_𝑛𝑗(𝑧)^𝛽 defined by (3.23) be in the class ℋ(𝑛,𝛽,𝛼) for every 𝑗=1,2,…,𝑚. Then the functions 𝑡_𝑗(𝑧)^𝛽 defined by 𝑡_𝑗(𝑧)^𝛽=^𝑚_𝑗=1𝑐_𝑗𝑓_𝑛𝑗(𝑧)0≤𝑐_𝑗≤1 (3.24) are also in the class ℋ(𝑛,𝛽,𝛼) where ∑^𝑚_𝑗=1𝑐_𝑗=1.

Proof. According to the definition of 𝑡^𝛽, we can write 𝑡(𝑧)^𝛽=𝑧^𝛽−^∞_𝑘=2^𝑚_𝑗=1𝑐_𝑗𝑎_𝑘,𝑗𝑧^{𝑘+𝛽−1}+(−1)^𝑛∞_𝑘=1^𝑚_𝑗=1𝑐_𝑗𝑏_𝑛,𝑗𝑧^{𝑘+𝛽−1}. (3.25) Further, since 𝑓_𝑛𝑗(𝑧)^𝛽 are in ℋ(𝑛,𝛽,𝛼) for every (𝑗=1,2,…), then by (3.1) we have (𝑘−𝛼)^∞_𝑘=1^𝑚_𝑗=1𝑐_𝑗𝑎_𝑘,𝑗+(𝑘+𝛼)^𝑚_𝑗=1𝑐_𝑗𝑏_𝑘,𝑗𝛽𝑘^𝑛=^𝑚_𝑗=1𝑐_𝑗^∞_𝑘=1(𝑘−𝛼)𝑎_𝑛,𝑗+(𝑘+𝛼)𝑏_𝑛,𝑗𝛽𝑘^𝑛≤^𝑚_𝑗=1𝑐_𝑗2(1−𝛼)≤2(1−𝛼). (3.26) Hence the theorem follows.

Corollary 3.8. The class ℋ(𝑛,𝛽,𝛼) is close under convex linear combination.

Proof. Let the functions 𝑓_𝑛𝑗(𝑧)^𝛽(𝑗=1,2) defined by (3.23) be in the class 𝑀_ℋ(𝑛,𝜆,𝛼). Then the function Ψ(𝑧)^𝛽 defined by Ψ(𝑧)^𝛽=𝜇𝑓_𝑛1(𝑧)^𝛽+1−𝜇𝑓_𝑛2(𝑧)^𝛽0≤𝜇≤1 (3.27) is in the class 𝑀_ℋ(𝑛,𝜆,𝛼). Also, by taking 𝑚=2, 𝑡₁=𝜇, and 𝑡₂=(1−𝜇) in Theorem 3.7, we have the above corollary.

Acknowledgments

The first author was an ex-postgraduate student under the supervision of Professor Maslina Darus. Both the first and second authors are supported by UKM-ST-06-FRGS0107-2009. The third author made her contribution to the work while she was a Visiting Researcher at the African Institute for Mathematical Sciences, South Africa where she enjoyed numerous stimulating discussions with Professor Alan Beardon. The authors also would like to thank the anonymous referees for the informative and creative comments given to the paper.