On Certain Classes of Harmonic 𝑝-Valent Functions Defined by an Integral Operator

Abstract

We obtain coefficient characterization, extreme points, and distortion bounds of certain classes of harmonic 𝑝-valent functions defined by an integral operator.

1. Introduction

A continuous complex-valued function 𝑓=𝑢+𝑖𝑣 defined in a simply connected complex domain 𝐷 is said to be harmonic in 𝐷 if both 𝑢 and 𝑣 are real harmonic in 𝐷. In any simply connected domain, we can write 𝑓=ℎ+𝑔, (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^′(𝑧)|>|𝑔^′(𝑧)| in 𝐷 (see [1]).

Denote by 𝑆_𝐻 the class of functions 𝑓 of the form (1) that are harmonic univalent and sense preserving in the unit disc 𝑈={𝑧∶|𝑧|<1} for which 𝑓(0)=𝑓_𝑧(0)−1=0.

Recently, Jahangiri and Ahuja [2] defined the class ℋ_𝑝(𝑝∈ℕ={1,2,3,…}), consisting of all 𝑝-valent harmonic functions 𝑓=ℎ+𝑔 that are sense preserving in 𝑈 and ℎ, and 𝑔 are of the form ℎ(𝑧)=𝑧^𝑝+^∞_𝑘=𝑝+1𝑎_𝑘𝑧^𝑘,𝑔(𝑧)=^∞_𝑘=𝑝𝑏_𝑘𝑧^𝑘,𝑏_𝑝<1. (2)

For 𝑓=ℎ+𝑔 given by (2), we define the modified 𝑝-valent Salagean integral operator 𝐼^𝑛_𝑝,𝜆 of 𝑓 (see [3] and also [4] when 𝑝=1) as follows: 𝐼^𝑛_𝑝,𝜆𝑓(𝑧)=𝐼^𝑛_𝑝,𝜆ℎ(𝑧)+(−1)^𝑛𝐼^𝑛_𝑝,𝜆𝑔(𝑧), (3) where 𝐼^𝑛_𝑝,𝜆ℎ(𝑧)=𝑧^𝑝+^∞_𝑘=𝑝+1𝑝𝑝+𝜆𝑘−𝑝^𝑛𝑎_𝑘𝑧^𝑘𝑝∈ℕ;𝜆>0;𝑛∈ℕ₀{0,𝐼=ℕ∪}_{𝑛𝑝,𝜆}𝑔(𝑧)=^∞_𝑘=𝑝𝑝𝑝+𝜆𝑘−𝑝^𝑛𝑏_𝑘𝑧^𝑘𝑝∈ℕ;𝜆>0;𝑛∈ℕ₀. (4)

For 𝑝∈ℕ,𝜆>0,𝑛∈ℕ₀,0≤𝛼<1, and 𝑧∈𝑈, we let ℋ_𝑝,𝜆(𝑛;𝛼) denote the family of harmonic functions 𝑓 of the form (2) such that 𝐼Re^{𝑛𝑝,𝜆}𝑓(𝑧)𝐼^𝑛+1_𝑝,𝜆𝑓(𝑧)>𝛼, (5) where 𝐼^𝑛_𝑝,𝜆𝑓 is defined by (3).

We let the subclass ℋ⁻_𝑝,𝜆(𝑛;𝛼) consists of harmonic functions 𝑓_𝑛=ℎ+𝑔_𝑛 in ℋ_𝑝,𝜆(𝑛;𝛼) so that ℎ and 𝑔_𝑛 are of the form ℎ(𝑧)=𝑧^𝑝−^∞_𝑘=𝑝+1𝑎_𝑘𝑧^𝑘,𝑔_𝑛(𝑧)=(−1)^𝑛∞_𝑘=𝑝𝑏_𝑘𝑧^𝑘,𝑎_𝑘,𝑏_𝑘≥0. (6)

We note that ℋ⁻_𝑝,1(𝑛;𝛼)=ℋ⁻_𝑝(𝑛;𝛼), where the class ℋ⁻_𝑝(𝑛;𝛼) was defined and studied by Cotirla [5].

In this paper, we obtain coefficient characterization of the classes ℋ_𝑝,𝜆(𝑛;𝛼) and ℋ⁻_𝑝,𝜆(𝑛;𝛼). We also obtain extreme points and distortion bounds for functions in the class ℋ⁻_𝑝,𝜆(𝑛;𝛼).

2. Coefficient Characterization

Unless otherwise mentioned, we assume throughout this paper that 𝑝∈ℕ,𝑛∈ℕ₀,0≤𝛼<1,𝑎_𝑝=1, and 𝜆>0. We begin with a sufficient condition for functions in ℋ_𝑝,𝜆(𝑛;𝛼).

Theorem 1. Let 𝑓=ℎ+𝑔 so that ℎ and 𝑔 are given by (2). Furthermore, let ^∞_𝑘=𝑝Ψ_𝑝,𝜆(𝑛,𝑘,𝛼)𝑎_𝑘+Φ_𝑝,𝜆(𝑛,𝑘,𝛼)𝑏_𝑘≤2, (7) where Ψ_𝑝,𝜆()=𝑝𝑛,𝑘,𝛼𝑝+𝜆𝑘−𝑝^𝑛𝑝−𝛼𝑝+𝜆𝑘−𝑝^𝑛+1×(1−𝛼)⁻¹,Φ_𝑝,𝜆()=𝑝𝑛,𝑘,𝛼𝑝+𝜆𝑘−𝑝^𝑛𝑝+𝛼𝑝+𝜆𝑘−𝑝^𝑛+1×(1−𝛼)⁻¹. (8)(9) Then, 𝑓 is sense preserving in 𝑈 and 𝑓∈ℋ_𝑝,𝜆(𝑛;𝛼).

Proof. According to (2) and (3), we only need to show that 𝐼Re^{𝑛𝑝,𝜆}𝑓(𝑧)−𝛼𝐼^𝑛+1_𝑝,𝜆𝑓(𝑧)𝐼^𝑛+1_𝑝,𝜆𝑓(𝑧)≥0(𝑧∈𝑈). (10) It follows that 𝐼Re^{𝑛𝑝,𝜆}𝑓(𝑧)−𝛼𝐼^𝑛+1_𝑝,𝜆𝑓(𝑧)𝐼^𝑛+1_𝑝,𝜆𝑓(𝑧)=Re(1−𝛼)𝑧^𝑝+^∞_𝑘=𝑝+1×𝑝𝑝+𝜆𝑘−𝑝^𝑛𝑝−𝛼𝑝+𝜆𝑘−𝑝^𝑛+1𝑎_𝑘𝑧^𝑘×𝑧^𝑝+^∞_𝑘=𝑝+1𝑝𝑝+𝜆𝑘−𝑝^𝑛+1𝑎_𝑘𝑧^𝑘+(−1)^∞𝑛+1_𝑘=𝑝𝑝𝑝+𝜆𝑘−𝑝^𝑛+1𝑏_𝑘𝑧^𝑘−1+(−1)^𝑛∞_𝑘=𝑝𝑝𝑝+𝜆𝑘−𝑝^𝑛𝑝+𝛼𝑝+𝜆𝑘−𝑝^𝑛+1𝑏_𝑘𝑧^𝑘×𝑧^𝑝+^∞_𝑘=𝑝+1𝑝𝑝+𝜆𝑘−𝑝^𝑛+1𝑎_𝑘𝑧^𝑘+(−1)^∞𝑛+1_𝑘=𝑝𝑝𝑝+𝜆𝑘−𝑝^𝑛+1𝑏_𝑘𝑧^𝑘−1)+=Re{((1−𝛼_{∞𝑘=𝑝+1}×𝑝𝑝+𝜆𝑘−𝑝^𝑛𝑝−𝛼𝑝+𝜆𝑘−𝑝^𝑛+1𝑎_𝑘𝑧^𝑘−𝑝×1+^∞_𝑘=𝑝+1𝑝𝑝+𝜆𝑘−𝑝^𝑛+1𝑎_𝑘𝑧^𝑘−𝑝+(−1)^∞𝑛+1_𝑘=𝑝𝑝𝑝+𝜆𝑘−𝑝^𝑛+1𝑏_𝑘𝑧^𝑘𝑧^−𝑝−1+(−1)^𝑛∞_𝑘=𝑝𝑝𝑝+𝜆𝑘−𝑝^𝑛𝑝+𝛼𝑝+𝜆𝑘−𝑝^𝑛+1×𝑏_𝑘𝑧^𝑘𝑧^−𝑝×1+^∞_𝑘=𝑝+1𝑝𝑝+𝜆𝑘−𝑝^𝑛+1×𝑎_𝑘𝑧^𝑘−𝑝+(−1)^∞𝑛+1_𝑘=𝑝𝑝𝑝+𝜆𝑘−𝑝^𝑛+1×𝑏_𝑘𝑧^𝑘𝑧^−𝑝−1(𝑧)=Re1−𝛼+𝐴(𝑧).1+𝐵 (11) For 𝑧=𝑟𝑒^𝑖𝜃, we have 𝐴𝑟𝑒^𝑖𝜃=^∞_𝑘=𝑝+1𝑝𝑝+𝜆𝑘−𝑝^𝑛𝑝−𝛼𝑝+𝜆𝑘−𝑝^𝑛+1𝑎_𝑘𝑟^𝑘−𝑝𝑒^{𝑖(𝑘−𝑝)𝜽}+(−1)^𝑛∞_𝑘=𝑝𝑝𝑝+𝜆𝑘−𝑝^𝑛𝑝+𝛼𝑝+𝜆𝑘−𝑝^𝑛+1×𝑏_𝑘𝑟^𝑘−𝑝𝑒^{−𝑖(𝑘+𝑝)𝜽},𝐵𝑟𝑒^𝑖𝜃=^∞_𝑘=𝑝+1𝑝𝑝+𝜆𝑘−𝑝^𝑛+1𝑎_𝑘𝑟^𝑘−𝑝𝑒^{𝑖(𝑘−𝑝)𝜽}+(−1)^∞𝑛+1_𝑘=𝑝𝑝𝑝+𝜆𝑘−𝑝^𝑛+1𝑏_𝑘𝑟^𝑘−𝑝𝑒^{−𝑖(𝑘+𝑝)𝜽}. (12) Setting that 1−𝛼+𝐴(𝑧)(𝑧)=()(𝑧)1+𝐵1−𝛼1+𝑤(𝑧),1−𝑤 (13) the proof will be complete if we can show that |𝑤(𝑧)|<1. Using the condition (7), we can write |𝑤(𝑧=𝐴(𝑧)−()𝐵(𝑧))|1−𝛼𝐴(𝑧)+()𝐵(𝑧)()=1−𝛼+21−𝛼_{∞𝑘=𝑝+1}𝑝𝑝+𝜆𝑘−𝑝^𝑛−𝑝𝑝+𝜆𝑘−𝑝^𝑛+1𝑎_𝑘𝑟^𝑘−𝑝𝑒^{𝑖(𝑘−𝑝)𝜽}×2(1−𝛼)+^∞_𝑘=𝑝+1𝑐_𝑘𝑎_𝑘𝑟^𝑘−𝑝𝑒^{𝑖(𝑘−𝑝)𝜽}+(−1)^𝑛∞_𝑘=𝑝𝑑_𝑘𝑏_𝑘𝑟^𝑘−𝑝𝑒^{−𝑖(𝑘+𝑝)𝜽−1}+(−1)^𝑛∞_𝑘=𝑝𝑝𝑝+𝜆𝑘−𝑝^𝑛𝑝+𝛼𝑝+𝜆𝑘−𝑝^𝑛+1×𝑏_𝑘𝑟^𝑘−𝑝𝑒^{−𝑖(𝑘+𝑝)𝜽}×2(1−𝛼)+^∞_𝑘=𝑝+1𝑐_𝑘𝑎_𝑘𝑟^𝑘−𝑝𝑒^{𝑖(𝑘−𝑝)𝜽}+(−1)^𝑛∞_𝑘=𝑝𝑑_𝑘𝑏_𝑘𝑟^𝑘−𝑝𝑒^{−𝑖(𝑘+𝑝)𝜽−1}≤^∞_𝑘=𝑝+1𝑝𝑝+𝜆𝑘−𝑝^𝑛−𝑝𝑝+𝜆𝑘−𝑝^𝑛+1𝑎_𝑘𝑟^𝑘−𝑝×2(1−𝛼)−^∞_𝑘=𝑝+1𝑐_𝑘𝑎_𝑘𝑟^𝑘−𝑝−^∞_𝑘=𝑝𝑑_𝑘𝑏_𝑘𝑟^{𝑘−𝑝−1}+^∞_𝑘=𝑝𝑝𝑝+𝜆𝑘−𝑝^𝑛+𝑝𝑝+𝜆𝑘−𝑝^𝑛+1𝑏_𝑘𝑟^𝑘−𝑝×2(1−𝛼)−^∞_𝑘=𝑝+1𝑐_𝑘𝑎_𝑘𝑟^𝑘−𝑝−^∞_𝑘=𝑝𝑑_𝑘𝑏_𝑘𝑟^{𝑘−𝑝−1}=^∞_𝑘=𝑝+1𝑝𝑝+𝜆𝑘−𝑝^𝑛−𝑝𝑝+𝜆𝑘−𝑝^𝑛+1𝑎_𝑘𝑟^𝑘−𝑝×4(1−𝛼)−^∞_𝑘=𝑝𝑐_𝑘𝑎_𝑘+𝑑_𝑘𝑏_𝑘𝑟^{𝑘−𝑝−1}+^∞_𝑘=𝑝𝑝𝑝+𝜆𝑘−𝑝^𝑛+𝑝𝑝+𝜆𝑘−𝑝^𝑛+1𝑏_𝑘𝑟^𝑘−𝑝×4(1−𝛼)−^∞_𝑘=𝑝𝑐_𝑘𝑎_𝑘+𝑑_𝑘𝑏_𝑘𝑟^{𝑘−𝑝−1}<^∞_𝑘=𝑝+1𝑝𝑝+𝜆𝑘−𝑝^𝑛−𝑝𝑝+𝜆𝑘−𝑝^𝑛+1𝑎_𝑘×4(1−𝛼)−^∞_𝑘=𝑝𝑐_𝑘𝑎_𝑘+𝑑_𝑘𝑏_𝑘⁻¹+^∞_𝑘=𝑝𝑝𝑝+𝜆𝑘−𝑝^𝑛+𝑝𝑝+𝜆𝑘−𝑝^𝑛+1𝑏_𝑘×4(1−𝛼)−^∞_𝑘=𝑝𝑐_𝑘𝑎_𝑘+𝑑_𝑘𝑏_𝑘⁻¹≤1, (14) where 𝑐_𝑘=𝑝𝑝+𝜆𝑘−𝑝^𝑛+()𝑝1−2𝛼𝑝+𝜆𝑘−𝑝^𝑛+1,𝑑_𝑘=𝑝𝑝+𝜆𝑘−𝑝^𝑛−()𝑝1−2𝛼𝑝+𝜆𝑘−𝑝^𝑛+1. (15)
The harmonic functions are as follows: 𝑓(𝑧)=𝑧^𝑝+^∞_𝑘=𝑝+11Ψ_𝑝,𝜆()𝑥𝑛,𝑘,𝛼^𝑘𝑧^𝑘+^∞_𝑘=𝑝1Φ_𝑝,𝜆(𝑛,𝑘,𝛼)𝑦_𝑘𝑧^𝑘, (16) where ∑^∞_𝑘=𝑝+1|𝑥_𝑘|+∑^∞_𝑘=𝑝|𝑦_𝑘|=1 show that the coefficient bound given by (7) is sharp. The functions of the form (8) are in the class ℋ_𝑝,𝜆(𝑛;𝛼) because ^∞_𝑘=𝑝Ψ_𝑝,𝜆(𝑛,𝑘,𝛼)𝑎_𝑘+Φ_𝑝,𝜆(𝑛,𝑘,𝛼)𝑏_𝑘=1+^∞_𝑘=𝑝+1𝑥_𝑘+^∞_𝑘=𝑝𝑦_𝑘=2. (17)
This completes the proof of Theorem 1.

In the following theorem, it is shown that the condition (7) is also necessary for functions 𝑓_𝑛=ℎ+𝑔_𝑛, where ℎ and 𝑔_𝑛 are of the form (6).

Theorem 2. Let 𝑓_𝑛=ℎ+𝑔_𝑛, where ℎ and 𝑔_𝑛 are given by (6). Then, 𝑓_𝑛∈ℋ⁻_𝑝,𝜆(𝑛;𝛼) if and only if ^∞_𝑘=𝑝Ψ_𝑝,𝜆(𝑛,𝑘,𝛼)𝑎_𝑘+Φ_𝑝,𝜆(𝑛,𝑘,𝛼)𝑏_𝑘≤2, (18) where Ψ_𝑝,𝜆(𝑛,𝑘,𝛼) and Φ_𝑝,𝜆(𝑛,𝑘,𝛼) are given by (8) and (9), respectively.

Proof. Since ℋ⁻_𝑝,𝜆(𝑛;𝛼)⊂ℋ_𝑝,𝜆(𝑛;𝛼), we only need to prove the “only if” part of the theorem. To this end, for functions 𝑓_𝑛=ℎ+𝑔_𝑛, where ℎ and 𝑔_𝑛 are given by (6), we notice that the condition Re{𝐼^𝑛_𝑝,𝜆𝑓(𝑧)/𝐼^𝑛+1_𝑝,𝜆𝑓(𝑧)}>𝛼is equivalent toRe(1−𝛼)𝑧^𝑝−^∞_𝑘=𝑝+1×𝑝𝑝+𝜆𝑘−𝑝^𝑛𝑝−𝛼𝑝+𝜆𝑘−𝑝^𝑛+1𝑎_𝑘𝑧^𝑘×𝑧^𝑝−^∞_𝑘=𝑝+1𝑝𝑝+𝜆𝑘−𝑝^𝑛+1𝑎_𝑘𝑧^𝑘+(−1)^∞2𝑛_𝑘=𝑝𝑝𝑝+𝜆𝑘−𝑝^𝑛+1𝑏_𝑘𝑧^𝑘−1+(−1)^∞2𝑛−1_𝑘=𝑝×𝑝𝑝+𝜆𝑘−𝑝^𝑛𝑝+𝛼𝑝+𝜆𝑘−𝑝^𝑛+1𝑏_𝑘𝑧^𝑘×𝑧^𝑝−^∞_𝑘=𝑝+1𝑝𝑝+𝜆𝑘−𝑝^𝑛+1𝑎_𝑘𝑧^𝑘+(−1)^∞2𝑛_𝑘=𝑝𝑝𝑝+𝜆𝑘−𝑝^𝑛+1𝑏_𝑘𝑧^𝑘−1≥0. (19)
The previous required condition (19) 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=𝑝+1𝑝𝑝+𝜆𝑘−𝑝^𝑛𝑝−𝛼𝑝+𝜆𝑘−𝑝^𝑛+1𝑎_𝑘𝑟^𝑘−𝑝×1−^∞_𝑘=𝑝+1𝑝𝑝+𝜆𝑘−𝑝^𝑛+1𝑎_𝑘𝑟^𝑘−𝑝+^∞_𝑘=𝑝𝑝𝑝+𝜆𝑘−𝑝^𝑛+1𝑏_𝑘𝑟^{𝑘−𝑝−1}+−^∞_𝑘=𝑝𝑝𝑝+𝜆𝑘−𝑝^𝑛𝑝+𝛼𝑝+𝜆𝑘−𝑝^𝑛+1𝑏_𝑘𝑟^𝑘−𝑝×1−^∞_𝑘=𝑝+1𝑝𝑝+𝜆𝑘−𝑝^𝑛+1𝑎_𝑘𝑟^𝑘−𝑝+^∞_𝑘=𝑝𝑝𝑝+𝜆𝑘−𝑝^𝑛+1𝑏_𝑘𝑟^{𝑘−𝑝−1}≥0. (20)
If the condition (18) does not hold, then the numerator in (20) is negative for 𝑟 sufficiently close to 1. Hence there exists 𝑧₀=𝑟₀ in (0,1) for which the quotient in (20) is negative. This contradicts the required condition for 𝑓_𝑛∈ℋ⁻_𝑝,𝜆(𝑛;𝛼), and so the proof of Theorem 2 is completed.

3. Extreme Points and Distortion Theorem

Our next theorem is on the extreme points of convex hulls of the class ℋ⁻_𝑝,𝜆(𝑛;𝛼) denoted by 𝑐𝑙𝑐𝑜ℋ⁻_𝑝,𝜆(𝑛;𝛼).

Theorem 3. Let 𝑓_𝑛=ℎ+𝑔_𝑛, where ℎ and 𝑔_𝑛 are given by (6). Then, 𝑓_𝑛∈ℋ⁻_𝑝,𝜆(𝑛;𝛼) if and only if 𝑓_𝑛(𝑧)=^∞_𝑘=𝑝𝑥_𝑘ℎ_𝑘(𝑧)+𝑦_𝑘𝑔_𝑛𝑘(𝑧), (21) where ℎ₁(𝑧)=𝑧^𝑝,ℎ_𝑘(𝑧)=𝑧^𝑝−1Ψ_𝑝,𝜆()𝑧𝑛,𝑘,𝛼^𝑘,𝑔𝑘=𝑝+1,𝑝+2,𝑝+3,…_𝑛𝑘(𝑧)=𝑧^𝑝+(−1)^𝑛−11Φ_𝑝,𝜆(𝑛,𝑘,𝛼)𝑧^𝑘,𝑥𝑘=𝑝,𝑝+1,𝑝+2,…_𝑘,𝑦_𝑘≥0,𝑥_𝑝=1−^∞_𝑘=𝑝+1𝑥_𝑘−^∞_𝑘=𝑝𝑦_𝑘. (22)
In particular, the extreme points of the class ℋ⁻_𝑝,𝜆(𝑛;𝛼) are {ℎ_𝑘} and {𝑔_𝑛𝑘}.

Proof. Suppose that 𝑓_𝑛(𝑧)=^∞_𝑘=𝑝𝑥_𝑘ℎ_𝑘(𝑧)+𝑦_𝑘𝑔_𝑛𝑘(𝑧=_∞𝑘=𝑝)𝑥_𝑘+𝑦_𝑘𝑧^𝑝−^∞_𝑘=𝑝+11Ψ_𝑝,𝜆()𝑥𝑛,𝑘,𝛼^𝑘𝑧^𝑘+(−1)^∞𝑛−1_𝑘=11Φ_𝑝,𝜆()𝑦𝑛,𝑘,𝛼^𝑘𝑧^𝑘. (23) Then, ^∞_𝑘=𝑝+1Ψ_𝑝,𝜆(1𝑛,𝑘,𝛼)Ψ_𝑝,𝜆()𝑥𝑛,𝑘,𝛼^𝑘+^∞_𝑘=1Φ_𝑝,𝜆()1𝑛,𝑘,𝛼Φ_𝑝,𝜆()𝑦𝑛,𝑘,𝛼^𝑘=^∞_𝑘=𝑝+1𝑥_𝑘+^∞_𝑘=𝑝𝑦_𝑘=1−𝑥_𝑝≤1, (24) and so 𝑓_𝑛∈𝑐𝑙𝑐𝑜ℋ⁻_𝑝,𝜆(𝑛;𝛼).
Conversely, if 𝑓_𝑛∈𝑐𝑙𝑐𝑜ℋ⁻_𝑝,𝜆(𝑛;𝛼), then 𝑎_𝑘≤1Ψ_𝑝,𝜆(𝑛,𝑘,𝛼),𝑏_𝑘≤1Φ_𝑝,𝜆().𝑛,𝑘,𝛼 (25) Set that 𝑥_𝑘=Ψ_𝑝,𝜆(𝑛,𝑘,𝛼)𝑎_𝑘,𝑦𝑘=𝑝+1,𝑝+2,𝑝+3,…_𝑘=Φ_𝑝,𝜆(𝑛,𝑘,𝛼)𝑏_𝑘𝑘=𝑝,𝑝+1,𝑝+2,…. (26) Then note that by Theorem 2, 0≤𝑥_𝑘≤1,(𝑘=𝑝+1,𝑝+2,𝑝+3,…), and 0≤𝑦_𝑘≤1,(𝑘=𝑝,𝑝+1,𝑝+2,…). We define that 𝑥_𝑝=1−∑^∞_𝑘=𝑝+1𝑥_𝑘−∑^∞_𝑘=𝑝𝑦_𝑘 and note that by Theorem 2, 𝑥_𝑝≥0. Consequently, we obtain 𝑓_𝑛(𝑧)=∑^∞_𝑘=𝑝{𝑥_𝑘ℎ_𝑘(𝑧)+𝑦_𝑘𝑔_𝑘(𝑧)} as required.

The following theorem gives the distortion bounds for functions in the class ℋ⁻_𝑝,𝜆(𝑛;𝛼) which yields a covering result for this class.

Theorem 4. Let 𝑓_𝑛(𝑧)∈ℋ⁻_𝑝,𝜆(𝑛;𝛼). Then, for |𝑧|=𝑟<1, we have 1−𝑏_𝑝𝑟^𝑝−Γ_𝑝,𝜆(𝑛,𝛼)−Δ_𝑝,𝜆(𝑛,𝛼)𝑟^𝑝+1≤𝑓_𝑛(𝑧≤)1+𝑏_𝑝𝑟^𝑝+Γ_𝑝,𝜆(𝑛,𝛼)−Δ_𝑝,𝜆(𝑛,𝛼)𝑏_𝑝𝑟^𝑝+1, (27) where Γ_𝑝,𝜆(𝑛,𝛼)=1−𝛼(𝑝/(𝑝+𝜆))^𝑛−𝛼(𝑝/(𝑝+𝜆))^𝑛+1,Δ_𝑝,𝜆(𝑛,𝛼)=1+𝛼(𝑝/(𝑝+𝜆))^𝑛−𝛼𝑝/(𝑝+𝜆)^𝑛+1. (28)
The result is sharp.

Proof. We only prove the right-hand inequality. The proof for the left-hand inequality is similar and will be omitted. Let 𝑓_𝑛(𝑧)∈ℋ⁻_𝑝,𝜆(𝑛;𝛼). Taking the absolute value of 𝑓_𝑛, we have 𝑓_𝑛(𝑧)≤1+𝑏_𝑝𝑟^𝑝+^∞_𝑘=𝑝+1𝑎_𝑘+𝑏_𝑘𝑟^𝑘≤1+𝑏_𝑝𝑟^𝑝+^∞_𝑘=𝑝+1𝑎_𝑘+𝑏_𝑘𝑟^𝑝+1=1+𝑏_𝑝𝑟^𝑝+Γ_𝑝,𝜆()×𝑛,𝛼_{∞𝑘=𝑝+1}1Γ_𝑝,𝜆()𝑎𝑛,𝛼^𝑘+𝑏_𝑘𝑟^𝑝+1≤1+𝑏_𝑝𝑟^𝑝+Γ_𝑝,𝜆(𝑛,𝛼)𝑟^𝑝+1×^∞_𝑘=𝑝+1Ψ_𝑝,𝜆(𝑛,𝑘,𝛼)𝑎_𝑘+Φ_𝑝,𝜆(𝑛,𝑘,𝛼)𝑏_𝑘≤1+𝑏_𝑝𝑟^𝑝+Γ_𝑝,𝜆(𝑛,𝑘,𝛼)−Δ_𝑝,𝜆(𝑛,𝑘,𝛼)𝑏_𝑝𝑟^𝑝+1. (29)
The bounds given in Theorem 4 for functions 𝑓_𝑛=ℎ+𝑔_𝑛, whereℎ and 𝑔_𝑛of form (6), also hold for functions of form (2) if the coefficient condition (7) is satisfied. The upper bound given for 𝑓∈ℋ⁻_𝑝,𝜆(𝑛;𝛼) is sharp, and the equality occurs for the functions 𝑓(𝑧)=𝑧^𝑝+𝑏_𝑝𝑧^𝑝−1−𝛼𝑝/(𝑝+𝜆)^𝑛−𝛼𝑝/𝑝+𝜆^𝑛+1−1−𝛼𝑝/(𝑝+𝜆)^𝑛+𝛼𝑝/𝑝+𝜆^𝑛+1𝑏_𝑝𝑧^𝑝+1,𝑓(𝑧)=𝑧^𝑝+𝑏_𝑝𝑧^𝑝−1−𝛼𝑝/(𝑝+𝜆)^𝑛−𝛼𝑝/𝑝+𝜆^𝑛+1−1−𝛼𝑝/(𝑝+𝜆)^𝑛+𝛼𝑝/𝑝+𝜆^𝑛+1𝑏_𝑝𝑧^𝑝+1 (30)(31) showing that the bounds given in Theorem 4 are sharp.

Remark 5. (i) Putting 𝜆=1 in the previous results, we obtain the results of Cotirla [5].
(ii) Putting 𝜆=1 in the previous results, we obtain the results of Cotirla [6], when 𝛽=0.

Acknowledgment

The author is grateful to the referees for their valuable suggestions.