Properties of a Class of 𝑝-Harmonic Functions

Abstract

A 𝑝 times continuously differentiable complex-valued function 𝐹=𝑢+𝑖𝑣 in a domain 𝐷⊆ℂ is 𝑝-harmonic if 𝐹 satisfies the 𝑝-harmonic equation Δ⋯Δ𝐹=0, where 𝑝 is a positive integer. By using the generalized Salagean differential operator, we introduce a class of 𝑝-harmonic functions and investigate necessary and sufficient coefficient conditions, distortion bounds, extreme points, and convex combination of the class.

1. Introduction

A continuous complex-valued function 𝑓=𝑢+𝑖𝑣 in a domain 𝐷⊆ℂ is harmonic if both 𝑢 and 𝑣 are real harmonic in 𝐷; that is, Δ𝑢=0 and Δ𝑣=0. Here Δ represents the complex Laplacian operator 𝜕Δ=4²𝜕𝑧𝜕𝑧𝜕∶=²𝜕𝑥²+𝜕²𝜕𝑦². (1)

In any simply connected domain 𝐷 we can write 𝑓=ℎ+𝑔, 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 𝐽_𝑓=|𝑓_𝑧|²−|𝑓_𝑧|²>0 in 𝐷. See [1, 2].

Denote by 𝑆𝐻 the class of functions 𝑓=ℎ+𝑔 that are harmonic, univalent, and sense preserving in the unit disk 𝑈={𝑧∶|𝑧|<1} for which 𝑓(0)=𝑓_𝑧(0)−1=0. Then for 𝑓=ℎ+𝑔∈𝑆𝐻 we may express the analytic functions ℎ and 𝑔 as ℎ(𝑧)=𝑧+^∞_𝑗=2𝑎_𝑗𝑧^𝑗,𝑔(𝑧)=^∞_𝑗=1𝑏_𝑗𝑧^𝑗,𝑏₁<1. (2)

The properties of the class 𝑆𝐻 and its geometric subclasses have been investigated by many authors; see ([1–6]). Note that 𝑆𝐻 reduces to the class 𝑆 of normalized analytic univalent functions in 𝑈 if the coanalytic part of 𝑓 is identically zero.

A 𝑝 times continuously differentiable complex-valued function 𝐹=𝑢+𝑖𝑣 in a domain 𝐷⊆ℂ is 𝑝-harmonic if 𝐹 satisfies the 𝑝-harmonic equation Δ⋯Δ𝐹=0, where 𝑝 is a positive integer.

A function 𝐹 is 𝑝-harmonic in a simply connected domain 𝐷 if and only if 𝐹 has the following representation: 𝐹(𝑧)=^𝑝_𝑘=1|𝑧|^2(𝑘−1)𝑓_{𝑝−𝑘+1}(𝑧), (3) where Δ𝑓_{𝑝−𝑘+1}(𝑧)=0 in 𝐷 for each 𝑘∈{1,…,𝑝}. 𝑓_{𝑝−𝑘+1} has the form 𝑓_{𝑝−𝑘+1}=ℎ_{𝑝−𝑘+1}+𝑔_{𝑝−𝑘+1}, (4) where ℎ_𝑝(𝑧)=𝑧+^∞_𝑗=2𝑎_𝑗,𝑝𝑧^𝑗,ℎ_{𝑝−𝑘+1}(𝑧)=^∞_𝑗=1𝑎_{𝑗,𝑝−𝑘+1}𝑧^𝑗,(),𝑔𝑘≥2_{𝑝−𝑘+1}(𝑧)=^∞_𝑗=1𝑏_{𝑗,𝑝−𝑘+1}𝑧^𝑗,(𝑘≥1). (5)

Denote by 𝑆𝐻_𝑝 the class of functions 𝐹 of the form (3) that are harmonic, univalent, and sense-preserving in the unit disk. Apparently, if 𝑝=1 and 𝑝=2,  𝐹 is harmonic and biharmonic, respectively.

Biharmonic functions have been studied by several authors, such as, [7–9]. Also, biharmonic functions arise in many physical situations, particularly, in fluid dynamics and elasticity problems. They have many important applications in engineering, biology, and medicine, such as in [10, 11].

For a function 𝑓 in 𝑆, differential operator 𝐷^𝑛(𝑛∈ℕ₀) was introduced by Sălăgean [12]. Al-Oboudi [13] generalized 𝐷^𝑛 as follows: 𝐷⁰_𝜆𝑓(𝑧)(𝑧),𝐷=𝑓_1𝜆𝑓(𝑧)=(1−𝜆)𝑓(𝑧)+𝜆𝑧𝑓^′(𝑧)𝐷,𝜆≥0,_𝑛𝜆𝑓(𝑧)=𝐷¹_𝜆𝐷_𝜆^𝑛−1𝑓(𝑧). (6) When 𝜆=1, we get the Salagean differential operator.

For 𝑓(𝑧)=ℎ(𝑧)+𝑔(𝑧) given by (2), Li and Liu [14] defined the following generalized Salagean operator 𝐷^𝑛_𝜆 in 𝑆𝐻: 𝐷^𝑛_𝜆𝑓(𝑧)=𝐷^𝑛_𝜆ℎ(𝑧)+𝐷^𝑛_𝜆𝑔(𝑧),𝜆≥0, (7) where 𝐷^𝑛_𝜆ℎ(𝑧)=𝑧+^∞_𝑗=2[1+(𝑘−1)𝜆]^𝑛𝑎_𝑗𝑧^𝑗,𝐷^𝑛_𝜆𝑔(𝑧)=^∞_𝑗=1[1+(𝑘−1)𝜆]^𝑛𝑏_𝑗𝑧^𝑗. (8)

For a 𝑝-harmonic function 𝐹 given by (3), we define the following operator: 𝐷⁰_𝜆𝐹(𝑧)=𝐹(𝑧),𝐷¹_𝜆𝐹(𝑧)=(1−𝜆)𝐷⁰_𝜆𝐹(𝑧)+𝜆𝑧𝐷⁰_𝜆𝐹(𝑧)_𝑧+𝑧𝐷⁰_𝜆𝐹(𝑧)_𝑧𝐷,𝜆≥0,_𝑛𝜆𝐹(𝑧)=𝐷¹_𝜆𝐷_𝜆^𝑛−1𝐹(𝑧),(𝑛∈ℕ). (9)(10) If 𝐹 is given by (3), then from (10) we see that 𝐷^𝑛_𝜆𝐹(𝑧)=^𝑝_𝑘=1|𝑧|^{∞2(𝑘−1)}_𝑗=11+𝑗−1𝜆+2(𝑘−1)𝜆^𝑛×𝑎_{𝑗,𝑝−𝑘+1}𝑧^𝑗+^𝑝_𝑘=1|𝑧|^{∞2(𝑘−1)}_𝑗=11+𝑗−1𝜆+2(𝑘−1)𝜆^𝑛×𝑏_{𝑗,𝑝−𝑘+1}𝑧^𝑗,𝑎_1,𝑝=1,𝑏_1,𝑝<1. (11) When 𝑝=1, we get the generalized Salagean operator for harmonic univalent functions defined by Li and Liu [14].

Denote by 𝑆𝐻_𝑝(𝑛,𝜆,𝛼) the class of functions 𝐹 of the form (3) which satisfy the condition 𝐷Re^𝜆𝑛+1𝐹(𝑧)𝐷^𝑛_𝜆𝐹(𝑧)≥𝛼,0≤𝛼<1, (12) where 𝐷^𝑛_𝜆𝐹(𝑧) is defined by (11).

We let the subclass 𝑆𝐻_𝑝 of 𝑆𝐻_𝑝 consist of functions 𝐹 of the form (3) which include 𝑓_{𝑝−𝑘+1}=ℎ_{𝑝−𝑘+1}+𝑔_{𝑝−𝑘+1}, where ℎ_𝑝(𝑧)=𝑧−^∞_𝑗=2𝑎_𝑗,𝑝𝑧^𝑗,ℎ_{𝑝−𝑘+1}(𝑧)=−^∞_𝑗=1𝑎_{𝑗,𝑝−𝑘+1}𝑧^𝑗,(𝑘≥2),𝑔_{𝑝−𝑘+1}(𝑧)=−^∞_𝑗=1𝑏_{𝑗,𝑝−𝑘+1}𝑧^𝑗,(𝑘≥1). (13)(14) Define 𝑆𝐻_𝑝(𝑛,𝜆,𝛼)∶=𝑆𝐻_𝑝(𝑛,𝜆,𝛼)∩𝑆𝐻_𝑝.

The main object of the paper is to introduce a class of 𝑝-harmonic functions by using the generalized Salagean operator which was defined by Li and Liu [14]. We investigate necessary and sufficient coefficient conditions, extreme points, distortion bounds, and convex combination of the class.

2. Main Results

Theorem 1. Let 𝐹 be a 𝑝-harmonic function given by (3). Furthermore, let ^𝑝∞_𝑘=1𝑗=1()×()𝜆1+𝑗−1𝜆+2𝑘−1𝜆−𝛼1+𝑗−1𝜆+2𝑘−1^𝑛𝑎_{𝑗,𝑝−𝑘+1}+𝑏_{𝑗,𝑝−𝑘+1}≤2(1−𝛼), (15) where 𝜆≥1,𝑛∈ℕ,0≤𝛼<1,  and  𝑎_1,𝑝=1. Then 𝐹 is sense preserving, 𝑝-harmonic, and univalent in𝑈 and 𝐹∈𝑆𝐻_𝑝(𝑛,𝜆,𝛼).

Proof. Suppose 𝑧₁,𝑧₂∈𝑈 and 𝑧₁≠𝑧₂, so that |𝑧₁|≤|𝑧₂|<1: 𝐹𝑧₁−𝐹𝑧₂≥𝑓_𝑝𝑧₁−𝑓_𝑝𝑧₂−^𝑝_𝑘=2𝑧₁^2(𝑘−1)𝑓_{𝑝−𝑘+1}𝑧₁−𝑧₂^2(𝑘−1)𝑓_{𝑝−𝑘+1}𝑧₂≥𝑧₁−𝑧₂1−^∞_𝑗=2𝑧^𝑗₁−𝑧^𝑗₂𝑧₁−𝑧₂𝑎_𝑗,𝑝−^∞_𝑗=1𝑧^𝑗₁−𝑧^𝑗₂𝑧₁−𝑧₂𝑏_𝑗,𝑝−^𝑝∞_𝑘=2𝑗=1𝑧₂^2(𝑘−1)𝑧^𝑗₁−𝑧^𝑗₂𝑧₁−𝑧₂×𝑎_{𝑗,𝑝−𝑘+1}+𝑏_{𝑗,𝑝−𝑘+1}>𝑧₁−𝑧₂1−^∞_𝑗=2𝑗𝑎_𝑗,𝑝−^∞_𝑗=1𝑗𝑏_𝑗,𝑝−^𝑝∞_𝑘=2𝑗=1𝑗𝑎_{𝑗,𝑝−𝑘+1}+𝑏_{𝑗,𝑝−𝑘+1}≥𝑧₁−𝑧₂2−^𝑝∞_𝑘=1𝑗=1()×()𝜆1+𝑗−1𝜆+2𝑘−1𝜆−𝛼1+𝑗−1𝜆+2𝑘−1^𝑛×(1−𝛼)⁻¹𝑎_{𝑗,𝑝−𝑘+1}+𝑏_{𝑗,𝑝−𝑘+1}≥0, (16) which proves univalence.
In order to prove that 𝐹 is sense preserving, we need to show that |𝐹_𝑧(𝑧)|−|𝐹_𝑧(𝑧)|>0: 𝐹_𝑧(𝑧)−𝐹_𝑧(𝑧=)1+^∞_𝑗=2𝑗𝑎_𝑗,𝑝𝑧^𝑗−1+^𝑝_𝑘=2|𝑧|^{∞2(𝑘−1)}_𝑗=1𝑗+𝑘−1𝑎_{𝑗,𝑝−𝑘+1}𝑧^𝑗−1+^𝑝_𝑘=1|𝑧|^2(𝑘−1)𝑧^∞_𝑗=1(𝑘−1)𝑏_{𝑗,𝑝−𝑘+1}𝑧^𝑗−^𝑝_𝑘=1|𝑧|^2(𝑘−1)𝑧^∞_𝑗=1(𝑘−1)𝑎_{𝑗,𝑝−𝑘+1}𝑧^𝑗+^𝑝_𝑘=1|𝑧|^{∞2(𝑘−1)}_𝑗=1𝑗+𝑘−1𝑏_{𝑗,𝑝−𝑘+1}𝑧^𝑗−1>2−^𝑝∞_𝑘=1𝑗=1𝑗+2(𝑘−1)𝑎_{𝑗,𝑝−𝑘+1}+𝑏_{𝑗,𝑝−𝑘+1}≥2−^𝑝∞_𝑘=1𝑗=1()×()𝜆1+𝑗−1𝜆+2𝑘−1𝜆−𝛼1+𝑗−1𝜆+2𝑘−1^𝑛×(1−𝛼)⁻¹𝑎_{𝑗,𝑝−𝑘+1}+𝑏_{𝑗,𝑝−𝑘+1}≥0, (17) for all 𝑧∈𝑈.
Using the fact that Re𝑤≥𝛼 if and only if |1−𝛼+𝑤|≥|1+𝛼−𝑤|, it suffices to show that (1−𝛼)𝐷^𝑛_𝜆𝐹(𝑧)+𝐷_𝜆^𝑛+1𝐹(𝑧−)𝐷)(1+𝛼^𝑛_𝜆𝐹(𝑧)−𝐷_𝜆^𝑛+1𝐹(𝑧)≥0. (18) Substituting for 𝐷^𝑛_𝜆𝐹 in (18), we obtain (1−𝛼)𝐷^𝑛_𝜆𝐹(𝑧)+𝐷_𝜆^𝑛+1𝐹(𝑧−)𝐷)(1+𝛼^𝑛_𝜆𝐹(𝑧)−𝐷_𝜆^𝑛+1𝐹(𝑧(𝑧|)≥21−𝛼)|−2^∞_𝑗=21+𝑗−1𝜆−𝛼1+𝑗−1𝜆^𝑛𝑎_𝑗,𝑝|𝑧|^𝑗−2^∞_𝑗=11+𝑗−1𝜆−𝛼1+𝑗−1𝜆^𝑛𝑏_𝑗,𝑝|𝑧|^𝑗−2^𝑝_𝑘=2|𝑧|^{∞2(𝑘−1)}_𝑗=1()×()𝜆1+𝑗−1𝜆+2𝑘−1𝜆−𝛼1+𝑗−1𝜆+2𝑘−1^𝑛×𝑎_{𝑗,𝑝−𝑘+1}+𝑏_{𝑗,𝑝−𝑘+1}|𝑧|^𝑗>2(1−𝛼)|𝑧|2−^𝑝∞_𝑘=1𝑗=1()×()𝜆1+𝑗−1𝜆+2𝑘−1𝜆−𝛼1+𝑗−1𝜆+2𝑘−1^𝑛×(1−𝛼)⁻¹𝑎_{𝑗,𝑝−𝑘+1}+𝑏_{𝑗,𝑝−𝑘+1}. (19) This last expression is nonnegative by (15), and so the proof is complete.

Theorem 2. Let 𝐹 be given by (13) and (14). Then 𝐹∈𝑆𝐻_𝑝(𝑛,𝜆,𝛼) if and only if ^𝑝∞_𝑘=1𝑗=1()×()𝜆1+𝑗−1𝜆+2𝑘−1𝜆−𝛼1+𝑗−1𝜆+2𝑘−1^𝑛×𝑎_{𝑗,𝑝−𝑘+1}+𝑏_{𝑗,𝑝−𝑘+1}≤2(1−𝛼), (20) where 𝜆≥1,  𝑛∈ℕ,0≤𝛼<1,  and 𝑎_1,𝑝=1.

Proof. The “if" part follows from Theorem 1 upon noting that 𝑆𝐻_𝑝(𝑛,𝜆,𝛼)⊂𝑆𝐻_𝑝(𝑛,𝜆,𝛼). For the “only if" part, we show that 𝐹∉𝑆𝐻_𝑝(𝑛,𝜆,𝛼) if the condition (20) does not hold.
Note that a necessary and sufficient condition for 𝐹 given by (13) and (14) to be in 𝑆𝐻_𝑝(𝑛,𝜆,𝛼) is that the condition (12) should be satisfied.
This is equivalent to Re{𝐴(𝑧)/𝐵(𝑧)}≥0, where 𝐴(𝑧)=()𝑧−1−𝛼_∞𝑗=21+𝑗−1𝜆−𝛼1+𝑗−1𝜆^𝑛𝑎_𝑗,𝑝𝑧^𝑗−^∞_𝑗=11+𝑗−1𝜆−𝛼1+𝑗−1𝜆^𝑛𝑏_𝑗,𝑝𝑧^𝑗−^𝑝_𝑘=2|𝑧|^{∞2(𝑘−1)}_𝑗=1()×()𝜆1+𝑗−1𝜆+2𝑘−1𝜆−𝛼1+𝑗−1𝜆+2𝑘−1^𝑛×𝑎_{𝑗,𝑝−𝑘+1}𝑧^𝑗+𝑏_{𝑗,𝑝−𝑘+1}𝑧^𝑗,𝐵(𝑧)=𝑧−^∞_𝑗=21+𝑗−1𝜆^𝑛𝑎_𝑗,𝑝𝑧^𝑗−^∞_𝑗=11+𝑗−1𝜆^𝑛𝑏_𝑗,𝑝𝑧^𝑗−^𝑝_𝑘=2|𝑧|^{∞2(𝑘−1)}_𝑗=11+𝑗−1𝜆+2(𝑘−1)𝜆^𝑛×𝑎_{𝑗,𝑝−𝑘+1}𝑧^𝑗+𝑏_{𝑗,𝑝−𝑘+1}𝑧^𝑗. (21) The above condition must hold for all values of 𝑧, |𝑧|=𝑟<1. Upon choosing the values of 𝑧 on the positive real axis, where 0≤𝑧=𝑟<1 we must have(1−𝛼)−^∞_𝑗=21+𝑗−1𝜆−𝛼1+𝑗−1𝜆^𝑛𝑎_𝑗,𝑝𝑟^𝑗−1−^∞_𝑗=11+𝑗−1𝜆−𝛼1+𝑗−1𝜆^𝑛𝑏_𝑗,𝑝𝑟^𝑗−1×1−^∞_𝑗=21+𝑗−1𝜆^𝑛𝑎_𝑗,𝑝−^∞_𝑗=11+𝑗−1𝜆^𝑛𝑏_𝑗,𝑝−^𝑝∞_𝑘=2𝑗=11+𝑗−1𝜆+2(𝑘−1)𝜆^𝑛×𝑎_{𝑗,𝑝−𝑘+1}+𝑏_{𝑗,𝑝−𝑘+1}⁻¹−^𝑝∞_𝑘=2𝑗=1()×()𝜆1+𝑗−1𝜆+2𝑘−1𝜆−𝛼1+𝑗−1𝜆+2𝑘−1^𝑛×𝑎_{𝑗,𝑝−𝑘+1}+𝑏_{𝑗,𝑝−𝑘+1}𝑟^{𝑗+2𝑘−3}×1−^∞_𝑗=21+𝑗−1𝜆^𝑛𝑎_𝑗,𝑝−^∞_𝑗=11+𝑗−1𝜆^𝑛𝑏_𝑗,𝑝−^𝑝∞_𝑘=2𝑗=11+𝑗−1𝜆+2(𝑘−1)𝜆^𝑛×𝑎_{𝑗,𝑝−𝑘+1}+𝑏_{𝑗,𝑝−𝑘+1}⁻¹≥0. (22)
If the condition (20) does not hold, then the numerator in (22) is negative for 𝑟 is sufficiently close to 1. Hence there exist 𝑧₀=𝑟₀ in (0,1) for which the quotient in (22) is negative. This contradicts the required condition for 𝐹∈𝑆𝐻_𝑝(𝑛,𝜆,𝛼) and so the proof is complete.

Theorem 3. Let 𝐹 be given by (13) and (14). Then 𝐹∈𝑆𝐻_𝑝(𝑛,𝜆,𝛼) if and only if 𝐹(𝑧)=^𝑝∞_𝑘=1𝑗=1𝑋_{𝑗,𝑝−𝑘+1}ℎ_{𝑗,𝑝−𝑘+1}(𝑧)+𝑌_{𝑗,𝑝−𝑘+1}𝑔_{𝑗,𝑝−𝑘+1}(𝑧), (23) where ℎ_1,𝑝(𝑧)ℎ=𝑧,_𝑗,𝑝(𝑧)2()=𝑧−1−𝛼1+𝑗−1𝜆−𝛼1+𝑗−1𝜆^𝑛𝑧^𝑗,,𝑔𝑗≥2_𝑗,𝑝(𝑧)2()=𝑧−1−𝛼1+𝑗−1𝜆−𝛼1+𝑗−1𝜆^𝑛𝑧^𝑗,,ℎ𝑗≥1_{𝑗,𝑝−𝑘+1}(𝑧)=𝑧−|𝑧|^2(𝑘−1)(2(×()×()𝜆1−𝛼))1+𝑗−1𝜆+2𝑘−1𝜆−𝛼1+𝑗−1𝜆+2𝑘−1^𝑛−1𝑧^𝑗,,𝑔𝑗≥1,2≤𝑘≤𝑝_{𝑗,𝑝−𝑘+1}(𝑧)=𝑧−|𝑧|^2(𝑘−1)(2(×()×()𝜆1−𝛼))1+𝑗−1𝜆+2𝑘−1𝜆−𝛼1+𝑗−1𝜆+2𝑘−1^𝑛−1𝑧^𝑗,𝑗≥1,2≤𝑘≤𝑝, (24) and ∑^𝑝_𝑘=1∑^∞_𝑗=1(𝑋_{𝑗,𝑝−𝑘+1}+𝑌_{𝑗,𝑝−𝑘+1})=1,   𝑋_{𝑗,𝑝−𝑘+1}≥0,  𝑌_{𝑗,𝑝−𝑘+1}≥0.
In particular, the extreme points of 𝑆𝐻_𝑝(𝑛,𝜆,𝛼) are {ℎ_{𝑗,𝑝−𝑘+1}(𝑧)} and {𝑔_{𝑗,𝑝−𝑘+1}(𝑧)}, where 𝑗≥1 and 1≤𝑘≤𝑝.

Proof. For functions 𝐹 of the form (13) and (14) we have 𝐹(𝑧)=^𝑝∞_𝑘=1𝑗=1𝑋_{𝑗,𝑝−𝑘+1}ℎ_{𝑗,𝑝−𝑘+1}(𝑧)+𝑌_{𝑗,𝑝−𝑘+1}𝑔_{𝑗,𝑝−𝑘+1}(𝑧)=𝑧−^∞_𝑗=2(2(1−𝛼))1+𝑗−1𝜆−𝛼1+𝑗−1𝜆^𝑛−1×𝑋_𝑗,𝑝𝑧^𝑗−^∞_𝑗=1(2(1−𝛼))1+𝑗−1𝜆−𝛼1+𝑗−1𝜆^𝑛−1×𝑌_𝑗,𝑝𝑧^𝑗−^𝑝_𝑘=2|𝑧|^2(𝑘−1)×^∞_𝑗=1(2(()×()𝜆1−𝛼))1+𝑗−1𝜆+2𝑘−1𝜆−𝛼1+𝑗−1𝜆+2𝑘−1^𝑛−1×𝑋_{𝑗,𝑝−𝑘+1}𝑧^𝑗+𝑌_{𝑗,𝑝−𝑘+1}𝑧^𝑗. (25) Then 1+𝑗−1𝜆−𝛼1+𝑗−1𝜆^∞_𝑗=2^𝑛2()×2()1−𝛼1−𝛼1+𝑗−1𝜆−𝛼1+𝑗−1𝜆^𝑛𝑋_𝑗,𝑝+^∞_𝑗=11+𝑗−1𝜆−𝛼1+𝑗−1𝜆^𝑛2()×2()1−𝛼1−𝛼1+𝑗−1𝜆−𝛼1+𝑗−1𝜆^𝑛𝑌_𝑗,𝑝+^𝑝∞_𝑘=2𝑗=1()×()𝜆1+𝑗−1𝜆+2𝑘−1𝜆−𝛼1+𝑗−1𝜆+2𝑘−1^𝑛(2(1−𝛼))⁻¹×(2(()×()𝜆1−𝛼))1+𝑗−1𝜆+2𝑘−1𝜆−𝛼1+𝑗−1𝜆+2𝑘−1^𝑛−1×𝑋_{𝑗,𝑝−𝑘+1}+𝑌_{𝑗,𝑝−𝑘+1}=^∞_𝑗=2𝑋_𝑗,𝑝+^∞_𝑗=1𝑌_𝑗,𝑝+^𝑝∞_𝑘=2𝑗=1𝑋_{𝑗,𝑝−𝑘+1}+𝑌_{𝑗,𝑝−𝑘+1}=1−𝑋_1,𝑝≤1, (26) and so 𝐹∈𝑆𝐻_𝑝(𝑛,𝜆,𝛼). Conversely, if 𝐹∈𝑆𝐻_𝑝(𝑛,𝜆,𝛼), then 𝑎_𝑗,𝑝≤2(1−𝛼)1+𝑗−1𝜆−𝛼1+𝑗−1𝜆^𝑛,,𝑎𝑗≥2_{𝑗,𝑝−𝑘+1}≤(2(()×()𝜆1−𝛼))1+𝑗−1𝜆+2𝑘−1𝜆−𝛼1+𝑗−1𝜆+2𝑘−1^𝑛−1,,𝑏𝑗≥1,2≤𝑘≤𝑝_𝑗,𝑝≤2(1−𝛼)1+𝑗−1𝜆−𝛼1+𝑗−1𝜆^𝑛,,𝑏𝑗≥2_{𝑗,𝑝−𝑘+1}≤(2(()×()𝜆1−𝛼))1+𝑗−1𝜆+2𝑘−1𝜆−𝛼1+𝑗−1𝜆+2𝑘−1^𝑛−1,𝑗≥1,2≤𝑘≤𝑝. (27) Set 𝑋_𝑗,𝑝=×𝜆1+𝑗−1𝜆−𝛼1+𝑗−1^𝑛(2(1−𝛼))⁻¹𝑎_𝑗,𝑝,,𝑌𝑗≥2_𝑗,𝑝=1+𝑗−1𝜆−𝛼1+𝑗−1𝜆^𝑛×(2(1−𝛼))⁻¹𝑏_𝑗,𝑝,,𝑋𝑗≥1_{𝑗,𝑝−𝑘+1}=()×()𝜆1+𝑗−1𝜆+2𝑘−1𝜆−𝛼1+𝑗−1𝜆+2𝑘−1^𝑛×(2(1−𝛼))⁻¹𝑎_{𝑗,𝑝−𝑘+1},,𝑌𝑗≥1,2≤𝑘≤𝑝_{𝑗,𝑝−𝑘+1}=()×()𝜆1+𝑗−1𝜆+2𝑘−1𝜆−𝛼1+𝑗−1𝜆+2𝑘−1^𝑛×(2(1−𝛼))⁻¹𝑏_{𝑗,𝑝−𝑘+1},,𝑋𝑗≥1,2≤𝑘≤𝑝_1,𝑝=1−^∞_𝑗=2𝑋_𝑗,𝑝−^∞_𝑗=1𝑌_𝑗,𝑝−^𝑝∞_𝑘=2𝑗=1𝑋_{𝑗,𝑝−𝑘+1}+𝑌_{𝑗,𝑝−𝑘+1}, (28) where 𝑋_1,𝑝≥0. Then, as required, we obtain 𝐹(𝑧)=^𝑝∞_𝑘=1𝑗=1𝑋_{𝑗,𝑝−𝑘+1}ℎ_{𝑗,𝑝−𝑘+1}(𝑧)+𝑌_{𝑗,𝑝−𝑘+1}𝑔_{𝑗,𝑝−𝑘+1}(𝑧). (29)

Theorem 4. Let 𝐹∈𝑆𝐻_𝑝(𝑛,𝜆,𝛼). Then for |𝑧|=𝑟<1 we have |𝐹(𝑧≤𝑏)|1+_1,𝑝+^𝑝_𝑘=2𝑎_{1,𝑝−𝑘+1}+𝑏_{1,𝑝−𝑘+1}𝑟+2(1−𝛼)(1+𝜆−𝛼)[1+𝜆]^𝑛−^𝑝_𝑘=1[1+2(𝑘−1)𝜆−𝛼][1+2(𝑘−1)𝜆]^𝑛(1+𝜆−𝛼)[1+𝜆]^𝑛×𝑎_{1,𝑝−𝑘+1}+𝑏_{1,𝑝−𝑘+1}𝑟²,|𝐹(𝑧≥𝑏)|1−_1,𝑝−^𝑝_𝑘=2𝑎_{1,𝑝−𝑘+1}+𝑏_{1,𝑝−𝑘+1}𝑟−2(1−𝛼)(1+𝜆−𝛼)[1+𝜆]^𝑛−^𝑝_𝑘=1[1+2(𝑘−1)𝜆−𝛼][1+2(𝑘−1)𝜆]^𝑛(1+𝜆−𝛼)[1+𝜆]^𝑛×𝑎_{1,𝑝−𝑘+1}+𝑏_{1,𝑝−𝑘+1}𝑟². (30)

Proof. We only prove the right-hand inequality. The proof for the left-hand inequality is similar and will be omitted. Let 𝐹∈𝑆𝐻_𝑝(𝑛,𝜆,𝛼). Taking the absolute value of 𝐹 we have |𝐹(𝑧)|≤^𝑝_𝑘=1𝑎_{1,𝑝−𝑘+1}+𝑏_{1,𝑝−𝑘+1}𝑟+^𝑝∞_𝑘=1𝑗=2𝑎_{𝑗,𝑝−𝑘+1}+𝑏_{𝑗,𝑝−𝑘+1}𝑟²≤^𝑝_𝑘=1𝑎_{1,𝑝−𝑘+1}+𝑏_{1,𝑝−𝑘+1}𝑟+2(1−𝛼)𝑟²(1+𝜆−𝛼)[1+𝜆]^𝑛×^𝑝∞_𝑘=1𝑗=2()×()𝜆1+𝑗−1𝜆+2𝑘−1𝜆−𝛼1+𝑗−1𝜆+2𝑘−1^𝑛×(2(1−𝛼))⁻¹𝑎_{𝑗,𝑝−𝑘+1}+𝑏_{𝑗,𝑝−𝑘+1}≤1+𝑏_1,𝑝+^𝑝_𝑘=2𝑎_{1,𝑝−𝑘+1}+𝑏_{1,𝑝−𝑘+1}𝑟+2(1−𝛼)𝑟²(1+𝜆−𝛼)[1+𝜆]^𝑛×1−^𝑝_𝑘=1[1+2(𝑘−1)𝜆−𝛼][1+2(𝑘−1)𝜆]^𝑛2()×𝑎1−𝛼_{1,𝑝−𝑘+1}+𝑏_{1,𝑝−𝑘+1}. (31)

The following covering result follows from the left-hand inequality in Theorem 4.

Corollary 5. Let 𝐹 of the form (13) and (14) be so that 𝐹∈𝑆𝐻_𝑝(𝑛,𝜆,𝛼). Then |𝑤|<(]𝑤∶1+𝜆−𝛼)[1+𝜆^𝑛−(1−𝛼)(1+𝜆−𝛼)[1+𝜆]^𝑛+(1−𝛼)−(1+𝜆−𝛼)[1+𝜆]^𝑛(1+𝜆−𝛼)[1+𝜆]^𝑛𝑏_1,𝑝+^𝑝_𝑘=2[1+2(𝑘−1)𝜆−𝛼][1+2(𝑘−1)𝜆]^𝑛−(1+𝜆−𝛼)[1+𝜆]^𝑛(1+𝜆−𝛼)[1+𝜆]^𝑛−1×𝑎_{1,𝑝−𝑘+1}+𝑏_{1,𝑝−𝑘+1}⊂𝐹(𝑈). (32)

Theorem 6. The class 𝑆𝐻_𝑝(𝑛,𝜆,𝛼) is closed under convex combinations.

Proof. Let 𝐹_𝑖∈𝑆𝐻_𝑝(𝑛,𝜆,𝛼) for 𝑖=1,2,…, where 𝐹_𝑖 is given by 𝐹_𝑖(𝑧)=𝑧−^∞_𝑗=2𝑎_{𝑖𝑗,𝑝}𝑧^𝑗−^∞_𝑗=1𝑏_{𝑖𝑗,𝑝}𝑧^𝑗−^𝑝_𝑘=2|𝑧|^{∞2(𝑘−1)}_𝑗=1𝑎_{𝑖𝑗,𝑝−𝑘+1}𝑧^𝑗+𝑏_{𝑖𝑗,𝑝−𝑘+1}𝑧^𝑗. (33) Then by (20), ^𝑝∞_𝑘=1𝑗=1()×()𝜆1+𝑗−1𝜆+2𝑘−1𝜆−𝛼1+𝑗−1𝜆+2𝑘−1^𝑛(2(1−𝛼))⁻¹×𝑎_{𝑖𝑗,𝑝−𝑘+1}+𝑏_{𝑖𝑗,𝑝−𝑘+1}≤1. (34) For ∑^∞_𝑖=1𝑡_𝑖=1, 0≤𝑡_𝑖≤1, the convex combination of 𝐹_𝑖 may be written as ^∞_𝑖=1𝑡_𝑖𝐹_𝑖(𝑧)=𝑧−^∞_𝑗=2^∞_𝑖=1𝑡_𝑖𝑎_{𝑖𝑗,𝑝}𝑧^𝑗+𝑏_{𝑖𝑗,𝑝}𝑧^𝑗−^𝑝_𝑘=2|𝑧|^{∞2(𝑘−1)}_𝑗=1^∞_𝑖=1𝑡_𝑖𝑎_{𝑖𝑗,𝑝−𝑘+1}𝑧^𝑗+𝑏_{𝑖𝑗,𝑝−𝑘+1}𝑧^𝑗. (35) Then by (34), ^𝑝∞_𝑘=1𝑗=1()×()𝜆1+𝑗−1𝜆+2𝑘−1𝜆−𝛼1+𝑗−1𝜆+2𝑘−1^𝑛(2(1−𝛼))⁻¹×^∞_𝑖=1𝑡_𝑖𝑎_{𝑖𝑗,𝑝−𝑘+1}+𝑏_{𝑖𝑗,𝑝−𝑘+1}=^∞_𝑖=1𝑡_𝑖^𝑝∞_𝑘=1𝑗=1()×()𝜆1+𝑗−1𝜆+2𝑘−1𝜆−𝛼1+𝑗−1𝜆+2𝑘−1^𝑛×(2(1−𝛼))⁻¹𝑎_{𝑖𝑗,𝑝−𝑘+1}+𝑏_{𝑖𝑗,𝑝−𝑘+1}≤^∞_𝑖=1𝑡_𝑖=1. (36) This is the condition required by (20) and so ∑^∞_𝑖=1𝑡_𝑖𝐹_𝑖(𝑧)∈𝑆𝐻_𝑝(𝑛,𝜆,𝛼).