Some Properties of Subclasses of Multivalent Functions

Abstract

The authors introduce two new subclasses denoted by ℑ^∗(Ω,𝜆,𝑝,𝛼) and ℑ^∗𝜀(Ω,𝜆,𝑝,𝛼) of the class 𝐴(𝑝,𝑛) of 𝑝-valent analytic functions. They obtain coefficient inequality for the class ℑ^∗(Ω,𝜆,𝑝,𝛼). They investigate various properties of classes ℑ^∗(Ω,𝜆,𝑝,𝛼) and ℑ^∗𝜀(Ω,𝜆,𝑝,𝛼). Furthermore, they derive partial sums associated with the class ℑ^∗𝜀(Ω,𝜆,𝑝,𝛼).

1. Introduction and Definition

Let 𝐴(𝑝,𝑛) denote the class of functions of the form 𝑓(𝑧)=𝑧^𝑝+^∞_𝑘=𝑛𝑎_𝑝+𝑘𝑧^𝑝+𝑘(𝑛,𝑝∈ℕ={1,2,…}),(1.1) which are analytic and p-valent in the open unit disc 𝑈={𝑧∶𝑧∈ℂ,|𝑧|<1}. We write 𝐴(1,1)=𝐴.

A function 𝑓∈𝐴(𝑝,𝑛) is said to be in the class 𝑆(𝑝,𝑛,𝛼) of p-valently star-like functions of order 𝛼 if it satisfies the conditionRe𝑧𝑓^(𝑧)𝑓(𝑧)>𝛼(𝑧∈ℂ;0≤𝛼<𝑝).(1.2) Furthermore, a function 𝑓∈𝐴(𝑝,𝑛) is said to be in the class 𝐾(𝑝,𝑛,𝛼) of p-valently convex functions of order 𝛼 if it satisfies the conditionRe1+𝑧𝑓^(𝑧)𝑓^(𝑧)>𝛼(𝑧∈𝑈;0≤𝛼<𝑝).(1.3) The classes 𝑆(𝑝,𝑛,𝛼) and 𝐾(𝑝,𝑛,𝛼) were studied by Owa [1]. The class 𝑆^∗(𝑝,𝛼)∶=𝑆(𝑝,1,𝛼) was considered by Patil and Thakare [2].

We denote by 𝑇(𝑝,𝑛) the subclass of the class 𝐴(𝑝,𝑛) consisting of functions of the form 𝑓(𝑧)=𝑧^𝑝−^∞_𝑘=𝑛𝑎_𝑝+𝑘𝑧^𝑝+𝑘𝑎_𝑝+𝑘≥0;𝑛,𝑝∈ℕ(1.4) and define two further classes 𝑇^∗(𝑝,𝑛,𝛼) and 𝐶(𝑝,𝑛,𝛼) by𝑇^∗(𝑝,𝑛,𝛼)∶=𝑆(𝑝,𝑛,𝛼)∩𝑇(𝑝,𝑛),𝐶(𝑝,𝑛,𝛼)∶=𝐾(𝑝,𝑛,𝛼)∩𝑇(𝑝,𝑛).(1.5) For the classes 𝑇^∗(𝑝,𝛼)∶=𝑆^∗(𝑝,𝛼)∩𝑇(𝑝),𝐶(𝑝,𝛼)∶=𝐾(𝑝,𝛼)∩𝑇(𝑝).

The following lemmas were given by Owa [3].

Lemma 1.1. Let the function 𝑓 be defined by 𝑓(𝑧)=𝑧^𝑝−^∞_𝑘=1𝑎_𝑝+𝑘𝑧^𝑝+𝑘𝑎_𝑝+𝑘≥0;𝑝∈ℕ.(1.6) Then, 𝑓 is in the class 𝑇^∗(𝑝,𝛼) if and only if ^∞_𝑘=1(𝑝+𝑘−𝛼)𝑎_𝑝+𝑘≤𝑝−𝛼.(1.7) The result is sharp.

Lemma 1.2. Let the function 𝑓 be defined by (1.6). Then, 𝑓 is in the class 𝐶(𝑝,𝛼) if and only if ^∞_𝑘=1(𝑝+𝑘)(𝑝+𝑘−𝛼)𝑎_𝑝+𝑘≤𝑝(𝑝−𝛼).(1.8) The result is sharp.

For a function 𝑓 defined by (1.6) and in the class 𝑇^∗(𝑝,𝛼), Lemma 1.1 yields𝑎_𝑝+1≤𝑝−𝛼𝑝+1−𝛼.(1.9) On the other hand, for a function 𝑓 defined by (1.6) and in the class 𝐶(𝑝,𝛼), Lemma 1.2 yields𝑎_𝑝+1≤𝑝(𝑝−𝛼)(𝑝+1)(𝑝+1−𝛼).(1.10) In view of the coefficient inequalities (1.9) and (1.10), it would seem to be natural to introduce and study here two further classes 𝑇^∗𝜀(𝑝,𝛼) and 𝐶_𝜀(𝑝,𝛼) of analytic and p-valent functions, where 𝑇^∗𝜀(𝑝,𝛼) denotes the subclass of 𝑇^∗(𝑝,𝛼) consisting of functions of the form𝑓(𝑧)=𝑧^𝑝−(𝑝−𝛼)𝜀𝑝+1−𝛼𝑧^𝑝+1−^∞_𝑘=2𝑎_𝑝+𝑘𝑧^𝑝+𝑘𝑎_𝑝+𝑘≥0,𝑝∈ℕ,𝑘∈ℕ−{1};0≤𝛼<𝑝;0≤𝜀<1(1.11) and 𝐶_𝜀(𝑝,𝛼) denotes the subclass of 𝐶(𝑝,𝛼) consisting of functions of the form𝑓(𝑧)=𝑧^𝑝−𝑝(𝑝−𝛼)𝜀(𝑝+1)(𝑝+1−𝛼)𝑧^𝑝+1−^∞_𝑘=2𝑎_𝑝+𝑘𝑧^𝑝+𝑘𝑎_𝑝+𝑘≥0,𝑝∈ℕ,𝑘∈ℕ−{1};0≤𝛼<𝑝;0≤𝜀<1.(1.12) The classes 𝑇^∗𝜀(𝑝,𝛼) and 𝐶_𝜀(𝑝,𝛼) are studied by Aouf et al. [4].

The classes 𝑇^∗𝜀(𝛼)∶=𝑇^∗𝜀(1,𝛼),𝐶_𝜀(𝛼)∶=𝐶_𝜀(1,𝛼)(1.13)

were considered earlier by Silverman and Silvia [5].

Now, we give the following equalities for the functions 𝑓(𝑧) belonging to the class 𝐴(𝑝,𝑛):𝐷𝐷⁰𝑓(𝑧)=𝑓(𝑧),₁𝑓(𝑧)=𝐷𝑓(𝑧)=𝑧𝐷⁰𝑓(𝑧)^=𝑧𝑝𝑧^𝑝−1+^∞_𝑘=𝑛(𝑝+𝑘)𝑎_𝑝+𝑘𝑧^{𝑝+𝑘−1}=𝑝𝑧^𝑝+^∞_𝑘=𝑛𝐷₂(𝑝+𝑘)𝑎_𝑝+𝑘𝑧^𝑝+𝑘,𝑓(𝑧)=𝐷(𝐷𝑓(𝑧))=𝑧𝐷¹𝑓(𝑧)^=𝑧𝑝𝑧^𝑝+^∞_𝑘=𝑛(𝑝+𝑘)𝑎_𝑝+𝑘𝑧^𝑝+𝑘^=𝑝²𝑧^𝑝+^∞_𝑘=𝑛𝐷_Ω⋮(𝑝+𝑘)²𝑎_𝑝+𝑘𝑧^𝑝+𝑘,𝑓(𝑧)=𝐷𝐷^Ω−1𝑓(𝑧)=𝑝^Ω𝑧^𝑝+^∞_𝑘=𝑛(𝑘+𝑝)^Ω𝑎_𝑝+𝑘𝑧^𝑝+𝑘.(1.14) We define ℘∶𝐴(𝑝,𝑛)→𝐴(𝑝,𝑛) such that℘(Ω,𝜆,𝑝)=1𝑝^Ω−𝜆𝐷^Ω𝑓(𝑧)+𝜆𝑝𝑧𝐷^Ω𝑓(𝑧)^0≤𝜆≤𝑝1^Ω,Ω∈ℕ∪{0}.(1.15)

A function 𝑓(𝑧)∈𝐴(𝑝,𝑛) is said to be in the class ℑ(Ω,𝜆,𝑝,𝛼) if it satisfies the inequalityRe𝑧(℘(Ω,𝜆,𝑝))′℘(Ω,𝜆,𝑝)=Re𝑧1/𝑝^Ω+(1/𝑝−1)𝜆𝐷^Ω𝑓(𝑧)^+(𝜆/𝑝)𝑧𝐷^Ω𝑓(𝑧)^1/𝑝^Ω−𝜆𝐷^Ω𝑓(𝑧)+(𝜆/𝑝)𝑧𝐷^Ω𝑓(𝑧)^>𝛼,(1.16) for some 𝛼(0≤𝛼<𝑝),0≤𝜆≤1/𝑝^Ω,  Ω∈ℕ∪{0} and for all 𝑧∈𝑈.

If Ω=0 and 𝜆=0, we obtain the condition (1.2). Furthermore, we obtain the condition (1.3) for Ω=0 and 𝜆=1.

We denote by 𝑇(𝑝,𝑛) the subclass of the class 𝐴(𝑝,𝑛) consisting of functions of the form 𝑓(𝑧)=𝑧^𝑝−^∞_𝑘=𝑛𝑎_𝑘+𝑝𝑧^𝑘+𝑝𝑎_𝑘+𝑝≥0;𝑛,𝑝∈ℕ,(1.17) and define the class ℑ^∗(Ω,𝜆,𝑝,𝛼) byℑ^∗(Ω,𝜆,𝑝,𝛼)=ℑ(Ω,𝜆,𝑝,𝛼)∩𝑇(𝑝,𝑛).(1.18)

Furthermore, we denote by ℑ^∗𝜀(Ω,𝜆,𝑝,𝛼) the subclass of ℑ^∗(Ω,𝜆,𝑝,𝛼) consisting of functions of the form 𝑓(𝑧)=𝑧^𝑝−(𝑝−𝛼)𝜀((𝑝+𝑛)/𝑝)^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼)𝑧^𝑝+𝑛−^∞_𝑘_=𝑛+1𝑎_𝑝+𝑘𝑧^𝑝+𝑘(0≤𝜀<1).(1.19) The main object of the present paper is to investigate interesting properties and characteristics of the classes ℑ^∗(Ω,𝜆,𝑝,𝛼) and ℑ^∗𝜀(Ω,𝜆,𝑝,𝛼). Also, the partial sums is defined for 𝑓 function defined by (1.19).

2. A Coefficient Inequality for the Class ℑ^∗(Ω,𝜆,𝑝,𝛼) and Some Theorems for the Class ℑ^∗𝜀(Ω,𝜆,𝑝,𝛼)

First, we give a coefficient inequality for the class ℑ^∗(Ω,𝜆,𝑝,𝛼).

Theorem 2.1. Let the function 𝑓 be defined by (1.17). Then, 𝑓 is in the class ℑ^∗(Ω,𝜆,𝑝,𝛼) if and only if ^∞_𝑘=n𝑘+𝑝𝑝^Ω1+𝜆𝑘𝑝^Ω−1(𝑘+𝑝−𝛼)𝑎_𝑝+𝑘≤𝑝−𝛼.(2.1)

Proof. Suppose that 𝑓(𝑧)∈ℑ^∗(Ω,𝜆,𝑝,𝛼). Then, we find from (1.16) that Re𝑧(℘(Ω,𝜆,𝑝))^℘(Ω,𝜆,𝑝)=Re𝑧1/𝑝^Ω+(1/𝑝−1)𝜆𝐷^Ω𝑓(𝑧)^+(𝜆/𝑝)𝑧𝐷^Ω𝑓(𝑧)^=Re𝑝𝑧_𝑝1/𝑝^Ω−𝜆𝐷^Ω𝑓(𝑧)+(𝜆/𝑝)𝑧𝐷^Ω𝑓(𝑧)^−∑^∞𝑘=𝑛(𝑘+𝑝)^Ω+11/𝑝^Ω+𝜆𝑘/𝑝𝑎_𝑝+𝑘𝑧^𝑝+𝑘𝑧^𝑝−∑^∞𝑘=𝑛(𝑘+𝑝)^Ω1/𝑝^Ω+𝜆𝑘/𝑝𝑎_𝑝+𝑘𝑧^𝑝+𝑘>𝛼.(2.2) If we choose 𝑧 to be real and let 𝑧→1⁻, we get 𝑝−∑^∞𝑘=𝑛(𝑘+𝑝)^Ω+11/𝑝^Ω+𝜆𝑘/𝑝𝑎_𝑝+𝑘1−∑^∞𝑘=𝑛(𝑘+𝑝)^Ω1/𝑝^Ω+𝜆𝑘/𝑝𝑎_𝑝+𝑘≥𝛼(2.3) or, equivalently, 𝑝−^∞_𝑘=𝑛(𝑘+𝑝)^Ω+11/𝑝^Ω+𝜆𝑘/𝑝𝑎_𝑝+𝑘≥𝛼1−^∞_𝑘=𝑛(𝑘+𝑝)^Ω1/𝑝^Ω+𝜆𝑘/𝑝𝑎_𝑝+𝑘.(2.4) Thus, we have ^∞_𝑘=𝑛(𝑘+𝑝)^Ω+11/𝑝^Ω+𝜆𝑘/𝑝−(𝑘+𝑝)^Ω1/𝑝^Ω+𝜆𝑘/𝑝𝑎_𝑝+𝑘≤𝑝−𝛼(2.5) or ^∞_𝑘=𝑛𝑘+𝑝𝑝^Ω1+𝜆𝑘𝑝^Ω−1(𝑘+𝑝−𝛼)𝑎_𝑝+𝑘≤𝑝−𝛼.(2.6) Conversely, suppose that the inequality (2.1) holds true and let 𝑧∈𝜕𝑈={𝑧∶𝑧∈ℂ,|𝑧|=1}.(2.7) Then, we find from the definition (1.4) that ||||𝑧(℘(Ω,𝜆,𝑝))^℘(Ω,𝜆,𝑝)−𝑝||||=|||||𝑧1/𝑝^Ω+(1/𝑝−1)𝜆𝐷^Ω𝑓(𝑧)^+(𝜆/𝑝)𝑧𝐷^Ω𝑓(𝑧)^1/𝑝^Ω−𝜆𝐷^Ω𝑓(𝑧)+(𝜆/𝑝)𝑧𝐷^Ω𝑓(𝑧)^=||||𝑝𝑧_𝑝−𝑝|||||−∑^∞𝑘=𝑛((𝑘+𝑝)/𝑝)^Ω(𝑘+𝑝)1+𝜆𝑘𝑝^Ω−1𝑎_𝑝+𝑘𝑧^𝑝+𝑘𝑧^𝑝−∑^∞𝑘=𝑛((𝑘+𝑝)/𝑝)^Ω1+𝜆𝑘𝑝^Ω−1𝑎_𝑝+𝑘𝑧^𝑝+𝑘≤∑−𝑝_∞𝑘=𝑛||||𝑘((𝑘+𝑝)/𝑝)^Ω1+𝜆𝑘𝑝^Ω−1𝑎_𝑝+𝑘1−∑^∞𝑘=𝑛((𝑘+𝑝)/𝑝)^Ω1+𝜆𝑘𝑝^Ω−1𝑎_𝑝+𝑘.(2.8) By means of inequality (2.1), we can write ^∞_𝑘=𝑛𝑘+𝑝𝑝^Ω1+𝜆𝑘𝑝^Ω−1(𝑘+𝑝−𝛼)𝑎_𝑝+𝑘≤𝑝−𝛼⟹^∞_𝑘=𝑛𝑘𝑘+𝑝𝑝^Ω1+𝜆𝑘𝑝^Ω−1𝑎_𝑝+𝑘≤𝑝−𝛼−^∞_𝑘=𝑛𝑘+𝑝𝑝^Ω1+𝜆𝑘𝑝^Ω−1(𝑝−𝛼)𝑎_𝑝+𝑘(2.9) or ^∞_𝑘=𝑛𝑘𝑘+𝑝𝑝^Ω1+𝜆𝑘𝑝^Ω−1𝑎_𝑝+𝑘≤𝑝−𝛼−(𝑝−𝛼)^∞_𝑘=𝑛𝑘+𝑝𝑝^Ω1+𝜆𝑘𝑝^Ω−1𝑎_𝑝+𝑘.(2.10) Thus, we obtain |||||𝑧1/𝑝_Ω+(1/𝑝−1)𝜆𝐷^Ω𝑓(𝑧)^+(𝜆/𝑝)𝑧𝐷^Ω𝑓(𝑧)^1/𝑝^Ω−𝜆𝐷^Ω𝑓(𝑧)+(𝜆/𝑝)𝑧𝐷^Ω𝑓(𝑧)^=||||∑−𝑝_∞𝑘=𝑛|||||𝑘((𝑘+𝑝)/𝑝)^Ω1+𝜆𝑘𝑝^Ω−1𝑎_𝑝+𝑘𝑧^𝑘1−∑^∞𝑘=𝑛((𝑘+𝑝)/𝑝)^Ω1+𝜆𝑘𝑝^Ω−1𝑎_𝑝+𝑘𝑧^𝑘≤𝑝−𝛼−(𝑝−𝛼)∑_∞𝑘=𝑛||||((𝑘+𝑝)/𝑝)^Ω1+𝜆𝑘𝑝^Ω−1𝑎_𝑝+𝑘1−∑^∞𝑘=𝑛((𝑘+𝑝)/𝑝)^Ω1+𝜆𝑘𝑝^Ω−1𝑎_𝑝+𝑘=𝑝−𝛼.(2.11) This evidently completes the proof of Theorem 2.1.

Now, we give a characterization theorem for the class ℑ^∗𝜀(Ω,𝜆,𝑝,𝛼).

Theorem 2.2. Let the function 𝑓 be defined by (1.19). Then, 𝑓 is in the class ℑ^∗𝜀(Ω,𝜆,𝑝,𝛼) if and only if ^∞_𝑘=𝑛+1𝑘+𝑝𝑝^Ω1+𝜆𝑘𝑝^Ω−1(𝑘+𝑝−𝛼)𝑎_𝑝+𝑘≤(𝑝−𝛼)(1−𝜀)0≤𝛼<𝑝,0≤𝜀<1,0≤𝜆≤1𝑝^Ω.(2.12) The result is sharp for the function 𝑓 given by 𝑓(𝑧)=𝑧^𝑝−(𝑝−𝛼)𝜀((𝑝+𝑛)/𝑝)^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼)𝑧^𝑝+𝑛−(𝑝−𝛼)(1−𝜀)((𝑝+𝑘)/𝑝)^Ω1+𝜆𝑘𝑝^Ω−1(𝑘+𝑝−𝛼)𝑧^𝑝+𝑘(𝑘=𝑛+1,𝑛+2,…;𝑛∈ℕ).(2.13)

Proof. Using inequality (2.1), we have 𝑛+𝑝𝑝^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼)𝑎_𝑝+𝑛+^∞_𝑘_=𝑛+1𝑘+𝑝𝑝^Ω1+𝜆𝑘𝑝^Ω−1(𝑘+𝑝−𝛼)𝑎_𝑝+𝑘≤𝑝−𝛼(2.14) or ^∞_𝑘=𝑛+1𝑘+𝑝𝑝^Ω1+𝜆𝑘𝑝^Ω−1(𝑘+𝑝−𝛼)𝑎_𝑝+𝑘≤𝑝−𝛼−𝑛+𝑝𝑝^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼)𝑎_𝑝+𝑛.(2.15) Thus, by setting 𝑎_𝑝+𝑛=(𝑝−𝛼)𝜀((𝑝+𝑛)/𝑝)^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼),(2.16) we obtain ^∞_𝑘=𝑛+1𝑘+𝑝𝑝^Ω1+𝜆𝑘𝑝^Ω−1(𝑘+𝑝−𝛼)𝑎_𝑝+𝑘≤𝑝−𝛼−𝑛+𝑝𝑝^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼)((𝑝+𝑛)/𝑝)^Ω1(𝑝+−𝜆𝑛𝛼𝑝)𝜀_Ω−1(𝑛+𝑝−𝛼)(2.17) or ^∞_𝑘=𝑛+1𝑘+𝑝𝑝^Ω1+𝜆𝑘𝑝^Ω−1(𝑘+𝑝−𝛼)𝑎_𝑝+𝑘≤(p−𝛼)(1−𝜀).(2.18) A closure theorem for the class ℑ^∗𝜀(Ω,𝜆,𝑝,𝛼) is given by the following.

Theorem 2.3. Let 𝑓_𝑠(𝑧)=𝑧^𝑝−(𝑝−𝛼)𝜀((𝑝+𝑛)/𝑝)^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼)𝑧^𝑝+𝑛−^∞_𝑘_=𝑛+1𝑎_{𝑝+𝑘,𝑠}𝑧^𝑝+𝑘𝑎_{𝑝+𝑘,𝑠}≥0;𝑝,𝑛∈ℕ;0≤𝜀<1;𝑠=1,2,3,…,𝑚.(2.19) If 𝑓_𝑠∈ℑ^∗𝜀(Ω,𝜆,𝑝,𝛼), then the function 𝑔 given by 𝑔(𝑧)=𝑧^𝑝−(𝑝−𝛼)𝜀((𝑝+𝑛)/𝑝)^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼)𝑧^𝑝+𝑛−^∞_k=_𝑛+1𝑏_𝑝+𝑘𝑧^𝑝+𝑘,(2.20) with 𝑏_𝑝+𝑘∶=1𝑚^𝑚_𝑠=1𝑎_{𝑝+𝑘,𝑠}≥0,(2.21) is also in the class ℑ^∗𝜀(Ω,𝜆,𝑝,𝛼).

Proof. Since 𝑓_𝑠∈ℑ^∗𝜀(Ω,𝜆,𝑝,𝛼) for 𝑠=1,2,…,𝑚, it follows from Theorem 2.2 that ^∞_𝑘=𝑛+1𝑘+𝑝𝑝^Ω1+𝜆𝑘𝑝^Ω−1(𝑘+𝑝−𝛼)𝑎_{𝑝+𝑘,𝑠}≤(𝑝−𝛼)(1−𝜀)(𝑠=1,2,…,𝑚).(2.22) By applying (2.22) and the definition (2.21), we write ^∞_𝑘=𝑛+1𝑘+𝑝𝑝^Ω1+𝜆𝑘𝑝^Ω−1(𝑘+𝑝−𝛼)𝑏_𝑝+𝑘=^∞_𝑘_=𝑛+1𝑘+𝑝𝑝^Ω1+𝜆𝑘𝑝^Ω−1(𝑘+𝑝−𝛼)𝑚1^𝑚_𝑠=1𝑎_{𝑝+𝑘,𝑠}=1𝑚^𝑚_𝑠=1⎧⎪⎨⎪⎩^∞_𝑘_=𝑛+1𝑘+𝑝𝑝^Ω1+𝜆𝑘𝑝^Ω−1(𝑘+𝑝−𝛼)𝑎_{𝑝+𝑘,𝑠}≤1⎫⎪⎬⎪⎭𝑚^𝑚_𝑠=1(𝑝−𝛼)(1−𝜀)=(𝑝−𝛼)(1−𝜀).(2.23)

Theorem 2.4. Let 𝑓_𝑝+𝑛(𝑧)=𝑧^𝑝−(𝑝−𝛼)𝜀((𝑝+𝑛)/𝑝)^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼)𝑧^𝑝+𝑛𝑓_𝑝+𝑘,(𝑧)=𝑧^𝑝−(𝑝−𝛼)𝜀((𝑝+𝑛)/𝑝)^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼)𝑧^𝑝+𝑛−(𝑝−𝛼)(1−𝜀)((𝑝+𝑘)/𝑝)^Ω1+𝜆𝑘𝑝^Ω−1(𝑘+𝑝−𝛼)𝑧^𝑝+𝑘,(2.24) where 𝑘∈{𝑛+1,𝑛+2,…}.
Then, 𝑓 is in the class ℑ^∗𝜀(Ω,𝜆,𝑝,𝛼) if and only if it can be expressed in the form 𝑓(𝑧)=^∞_𝑘=𝑛𝜂_𝑝+𝑘𝑓_𝑝+𝑘(𝑧)𝜂_𝑝+𝑘≥0;^∞_𝑘=𝑛𝜂_𝑝+𝑘=1.(2.25)

Proof. Suppose that 𝑓 is given by (2.25), so that we find from (2.24) that 𝑓(𝑧)=𝑧^𝑝−(𝑝−𝛼)𝜀((𝑝+𝑛)/𝑝)^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼)𝑧^𝑝+𝑛−^∞_𝑘_=𝑛+1(𝑝−𝛼)(1−𝜀)((𝑝+𝑘)/𝑝)^Ω1+𝜆𝑘𝑝^Ω−1(𝑘+𝑝−𝛼)𝜂^𝑝+𝑘𝑧^𝑝+𝑘,(2.26) where the coefficients 𝜂_𝑝+𝑘 are given with ∑^∞𝑘=𝑛𝜂_𝑝+𝑘=1,  𝜂_𝑝+𝑘≥0. Then, since, ^∞_𝑘=𝑛+1𝑘+𝑝𝑝^Ω1+𝜆𝑘𝑝^Ω−1(𝑘+𝑝−𝛼)((𝑘+𝑝)/𝑝)^Ω(𝑝−𝛼)(1−𝜀)1+𝜆𝑘𝑝^Ω−1(𝑘+𝑝−𝛼)𝜂^𝑝+𝑘=(𝑝−𝛼)(1−𝜀)^∞_𝑘_=𝑛+1𝜂_𝑝+𝑘=≤((𝑝𝑝−−𝛼𝛼))((11−−𝜀𝜀)),1−𝜂_𝑝+𝑛(2.27) we conclude from Theorem 2.2 that 𝑓∈ℑ^∗𝜀(Ω,𝜆,𝑝,𝛼).
Conversely, let us assume that the function 𝑓 defined by (1.19) is in the class ℑ^∗𝜀(Ω,𝜆,𝑝,𝛼). Then, 𝑎_𝑝+𝑘≤(𝑝−𝛼)(1−𝜀)((𝑘+𝑝)/𝑝)^Ω1+𝜆𝑘𝑝^Ω−1(𝑘+𝑝−𝛼),(2.28) which follows readily from (2.12) for 𝑘∈{𝑛+1,𝑛+2,…}.
For 𝑘∈{𝑛+1,𝑛+2,…}, setting 𝜂_𝑝+𝑘=((𝑘+𝑝)/𝑝)^Ω1+𝜆𝑘𝑝^Ω−1(𝑘+𝑝−𝛼)(𝑝−𝛼)(1−𝜀)𝑎^𝑝+𝑘(2.29) and 𝜂_𝑝+𝑛=1−∑^{∞𝑘=𝑛+1}𝜂_𝑝+𝑘, we thus arrive at (2.25). This completes the proof of Theorem 2.4.

Theorem 2.5. Let 𝑓 be given by (1.19) and define the partial sums 𝑠_𝑚(𝑧) by 𝑠_𝑚(𝑧)=⎧⎪⎪⎨⎪⎪⎩𝑧^𝑝−𝑝^Ω(𝑝−𝛼)𝜀(𝑝+𝑛)^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼)𝑧^𝑝+𝑛𝑧,𝑚=𝑛,_𝑝−𝑝^Ω(𝑝−𝛼)𝜀(𝑝+𝑛)^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼)𝑧^𝑝+𝑛−^𝑚_𝑘_=𝑛+1𝑎_𝑝+𝑘𝑧^𝑝+𝑘,𝑚=𝑛+1,𝑛+2,….(2.30) Suppose also that ^∞_𝑘=𝑛+1𝑑_𝑝+𝑘𝑎_𝑝+𝑘≤1−𝑝^Ω(𝑝−𝛼)𝜀where𝑑_𝑝+𝑘(𝑝+𝑛)^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼),=(𝑝+𝑘)^Ω1+𝜆𝑘𝑝^Ω−1(𝑘+𝑝−𝛼)𝑝^Ω(𝑝−𝛼)(1−𝜀);0≤𝛼<𝑝,0≤𝜀<1,0≤𝜆≤𝑝1^Ω.(2.31) Then, for 𝑚≥𝑛+1, one has Re𝑓(𝑧)𝑠_𝑚(𝑧)>1−𝑝^Ω(𝑝−𝛼)(1−𝜀)Re𝑠_𝑚(𝑝+𝑚+1)^Ω1+𝜆(𝑚+1)𝑝^Ω−1(𝑝+𝑚+1−𝛼),(2.32)(𝑧)𝑓(𝑧)>(𝑝+𝑚+1)^Ω1+𝜆(𝑚+1)𝑝^Ω−1(𝑚+𝑝+1−𝛼)𝑝^Ω(𝑝−𝛼)(1−𝜀)+(𝑝+𝑚+1)^Ω1+𝜆(𝑚+1)𝑝^Ω−1(𝑚+𝑝+1−𝛼).(2.33) Each of the bounds in (2.32) and (2.33) is the best possible.

Proof. From (2.31) and Theorem 2.2, we have that 𝑓∈ℑ^∗𝜀(Ω,𝜆,𝑝,𝛼). By definition 𝑑_𝑝+𝑘, we can write 𝑑_𝑝+𝑘=((𝑝+𝑘)/𝑝)^Ω1+𝜆𝑘𝑝^Ω−1(𝑘+𝑝−𝛼)=1+𝑘(𝑝−𝛼)(1−𝜀)=1+𝑘𝑝^Ω1+𝜆𝑘𝑝^Ω−1(𝑝𝑝−−𝛼)𝛼(1+−𝑘𝜀)𝑝^Ω1+𝜆𝑘𝑝^Ω−11−1𝜀+(𝑝−𝛼)𝑘(1−𝜀)>1.(2.34) Under the hypothesis of this theorem, we can see from (2.31) that 𝑑_𝑝+𝑘+1>𝑑_𝑝+𝑘>1, for 𝑘=𝑛+1,𝑛+2…. Therefore, we have ^𝑚_𝑘=𝑛+1𝑎_𝑝+𝑘+𝑑_{∞𝑝+𝑚+1𝑘=}_𝑚+1𝑎_𝑝+𝑘≤^∞_𝑘_=𝑛+1𝑑_𝑝+𝑘𝑎_𝑝+𝑘≤1−𝑝^Ω(𝑝−𝛼)𝜀(𝑝+𝑛)^Ω1+𝜆𝑛𝑝^Ω−1(𝑝+𝑛−𝛼).(2.35) Using (1.19) and (2.30), we can write 𝑓(𝑧)𝑠_𝑚(𝑧)=𝑧^𝑝−𝑝^Ω(𝑝−𝛼)𝜀/(𝑝+𝑛)^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼)𝑧^𝑝+𝑛−∑^{∞𝑘=𝑛+1}𝑎_𝑝+𝑘𝑧^𝑝+𝑘𝑧^𝑝−𝑝^Ω(𝑝−𝛼)𝜀/(𝑝+𝑛)^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼)𝑧^𝑝+𝑛−∑^{𝑚𝑘=𝑛+1}𝑎_𝑝+𝑘𝑧^𝑝+𝑘=1−∑^{∞𝑘=𝑚+1}𝑎_𝑝+𝑘𝑧^𝑝+𝑘𝑧^𝑝−𝑝^Ω(𝑝−𝛼)𝜀/(𝑝+𝑛)^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼)𝑧^𝑝+𝑛−∑^{𝑚𝑘=𝑛+1}𝑎_𝑝+𝑘𝑧^𝑝+𝑘.(2.36) Set Ψ₁(𝑧)=𝑑_𝑝+𝑚+1=1−𝑠𝑓_𝑚((𝑧𝑧))−1−𝑑1_𝑝+𝑚+1𝑑_𝑝+𝑚+1∑^{∞𝑘=𝑚+1}𝑎_𝑝+𝑘𝑧^𝑘1−𝑝^Ω(𝑝−𝛼)𝜀/(𝑝+𝑛)^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼)𝑧^𝑛−∑^{𝑚𝑘=𝑛+1}𝑎_𝑝+𝑘𝑧^𝑘.(2.37) By applying (2.35) and (2.37), we find that ||||𝜓₁(𝑧)−1𝜓₁(=𝑧)||||||||||+1||||−𝑑_𝑝+𝑚+1∑^{∞𝑘=𝑚+1}𝑎_𝑝+𝑘𝑧^𝑘2−2𝑝^Ω(𝑝−𝛼)𝜀(𝑝+𝑛)^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼)𝑧^𝑛−2∑^{𝑚𝑘=𝑛+1}𝑎_𝑝+𝑘𝑧^𝑘−𝑑_𝑝+𝑚+1∑^{∞𝑘=𝑚+1}𝑎_𝑝+𝑘𝑧^𝑘≤𝑑_𝑝+𝑚+1||||||||||∑^{∞𝑘=𝑚+1}𝑎_𝑝+𝑘𝑧^𝑝+𝑘21−𝑝^Ω(𝑝−𝛼)𝜀(𝑝+𝑛)^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼)−∑^{𝑚𝑘=𝑛+1}𝑎_𝑝+𝑘−𝑑^𝑝+𝑚+1∑^{∞𝑘=𝑚+1}𝑎_𝑝+𝑘≤1(𝑧∈𝑈;𝑚≥𝑛+1),(2.38) which shows that ReΨ₁(𝑧)>0. Thus, we obtain ReΨ₁(𝑧)=Re𝑑_𝑝+𝑚+1𝑠𝑓_𝑚((𝑧𝑧))−1−𝑑1_𝑝+𝑚+1>0⟹Re𝑠𝑓_𝑚((𝑧𝑧))>1−𝑑1_𝑝+𝑚+1(2.39) or Re𝑓(𝑧)𝑠_𝑚(𝑧)>1−𝑝^Ω(𝑝−𝛼)(1−𝜀)(𝑝+𝑚+1)^Ω1+𝜆(𝑚+1)𝑝^Ω−1(𝑝+𝑚+1−𝛼).(2.40) Let 𝑓(𝑧)=𝑧^𝑝−𝑝^Ω(𝑝−𝛼)𝜀(𝑝+𝑛)^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼)𝑧^𝑝+𝑛−1−𝑝^Ω(𝑝−𝛼)𝜀/(𝑝+𝑛)^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼)𝑑_𝑝+𝑚+1𝑧^𝑝+𝑚+1.(2.41) Then, 𝑓(𝑧) satisfies the condition (2.31) and 𝑓∈ℑ^∗𝜀(Ω,𝜆,𝑝,𝛼). Thus, we can write 𝑓(𝑧)𝑠_𝑚(𝑧)=1−𝑝^Ω(𝑝−𝛼)𝜀/(𝑝+𝑛)^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼)𝑧^𝑛1−𝑝^Ω(𝑝−𝛼)𝜀/(𝑝+𝑛)^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼)𝑧^𝑛−1−𝑝^Ω(𝑝−𝛼)𝜀/(𝑝+𝑛)^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼)/𝑑_𝑝+𝑚+1𝑧^𝑚+11−𝑝^Ω(𝑝−𝛼)𝜀/(𝑝+𝑛)^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼)𝑧^𝑛=1−1−𝑝^Ω(𝑝−𝛼)𝜀/(𝑝+𝑛)^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼)/𝑑_𝑝+𝑚+1𝑧^𝑚+11−𝑝^Ω(𝑝−𝛼)𝜀/(𝑝+𝑛)^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼)𝑧^𝑛(2.42) and taking as 𝑧→1⁻⟶1−1𝑑_𝑝+𝑚+1=1−𝑝^Ω(𝑝−𝛼)(1−𝜀)(𝑝+𝑚+1)^Ω1+𝜆(𝑚+1)𝑝^Ω−1(𝑝+𝑚+1−𝛼),(2.43) which shows that the bound in (2.32) is the best possible.
By using definitions of 𝑓(𝑧) and 𝑠_𝑚(𝑧), we can write 𝑠_𝑚(𝑧)𝑓(𝑧)=𝑧^𝑝−(𝑝−𝛼)𝜀/((𝑝+𝑛)/𝑝)^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼)𝑧^𝑝+𝑛−∑^{𝑚𝑘=𝑛+1}𝑎_𝑝+𝑘𝑧^𝑝+𝑘𝑧^𝑝−(𝑝−𝛼)𝜀/((𝑝+𝑛)/𝑝)^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼)𝑧^𝑝+𝑛−∑^{∞𝑘=𝑛+1}𝑎_𝑝+𝑘𝑧^𝑝+𝑘=𝑧^𝑝−(𝑝−𝛼)𝜀/((𝑝+𝑛)/𝑝)^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼)𝑧^𝑝+𝑛𝑧^𝑝−(𝑝−𝛼)𝜀/((𝑝+𝑛)/𝑝)^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼)𝑧^𝑝+𝑛−∑^{∞𝑘=𝑛+1}𝑎_𝑝+𝑘𝑧^𝑝+𝑘−∑^{∞𝑘=𝑛+1}𝑎_𝑝+𝑘𝑧^𝑝+𝑘+∑^{∞𝑘=𝑚+1}𝑎_𝑝+𝑘𝑧^𝑝+𝑘𝑧^𝑝−(𝑝−𝛼)𝜀/((𝑝+𝑛)/𝑝)^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼)𝑧^𝑝+𝑛−∑^{∞𝑘=𝑛+1}𝑎_𝑝+𝑘𝑧^𝑝+𝑘=1+∑^{∞𝑘=𝑚+1}𝑎_𝑝+𝑘𝑧^𝑘1−(𝑝−𝛼)𝜀/((𝑝+𝑛)/𝑝)^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼)𝑧^𝑛−∑^{∞𝑘=𝑛+1}𝑎_𝑝+𝑘𝑧^𝑘.(2.44) If we put 𝜓₂(𝑧)=1+𝑑_𝑝+𝑚+1𝑠^𝑚(𝑧)=1+𝑓(𝑧)−1+𝑑𝑑1+^𝑝+𝑚+1𝑑_𝑝+𝑚+1_𝑝+𝑚+1∑^{∞𝑘=𝑚+1}𝑎_𝑝+𝑘𝑧^𝑘1−(𝑝−𝛼)𝜀/((𝑝+𝑛)/𝑝)^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼)𝑧^𝑛−∑^{∞𝑘=𝑛+1}𝑎_𝑝+𝑘𝑧^𝑘(2.45) and make use of (2.35), we can deduce that ||||𝜓₂(𝑧)−1=𝜓₂|||||||||||||||||(𝑧)+1||||1+𝑑_𝑝+𝑚+1∑^{∞𝑘=𝑚+1}𝑎_𝑝+𝑘𝑧^𝑘2−⎛⎜⎜⎜⎜⎝2(𝑝−𝛼)𝜀𝑝+𝑝𝑛^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼)⎞⎟⎟⎟⎟⎠𝑧^𝑛−2∑^{𝑚𝑘=𝑛+1}𝑎_𝑝+𝑘𝑧^𝑘+𝑑_𝑝+𝑚+1−1∑^{∞𝑘=𝑚+1}𝑎_𝑝+𝑘𝑧^𝑘≤1+𝑑_𝑝+𝑚+1|||||||||||||||||∑^{∞𝑘=𝑚+1}𝑎_𝑝+𝑘2−⎛⎜⎜⎜⎜⎝2(𝑝−𝛼)𝜀𝑝+𝑝𝑛^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼)⎞⎟⎟⎟⎟⎠−2∑^{𝑚𝑘=𝑛+1}𝑎_𝑝+𝑘−𝑑_𝑝+𝑚+1−1∑^{∞𝑘=𝑚+1}𝑎_𝑝+𝑘=1+𝑑^𝑝+𝑚+1∑^{∞𝑘=𝑚+1}𝑎_𝑝+𝑘2⎧⎪⎪⎨⎪⎪⎩1−⎛⎜⎜⎜⎜⎝(𝑝−𝛼)𝜀𝑝+𝑝𝑛^Ω1+𝜆𝑛𝑝^Ω−1(𝑛+𝑝−𝛼)⎞⎟⎟⎟⎟⎠−∑^{𝑚𝑘=𝑛+1}𝑎_𝑝+𝑘⎫⎪⎪⎬⎪⎪⎭−𝑑_𝑝+𝑚+1−1∑^{∞𝑘=𝑚+1}𝑎_𝑝+𝑘≤1+𝑑^𝑝+𝑚+1∑^{∞𝑘=𝑚+1}𝑎_𝑝+𝑘2𝑑_𝑝+𝑚+1∑^{∞𝑘=𝑚+1}𝑎_𝑝+𝑘−𝑑_𝑝+𝑚+1−1∑^∞_k=𝑚+1𝑎_𝑝+𝑘=1,(2.46)|(𝜓₂(𝑧)−1)/(𝜓₂(𝑧)+1)|≤1 requires that Re{𝜓₂(𝑧)}>0. Thus, we obtain ReΨ₂(𝑧)=1+𝑑_𝑝+𝑚+1Re𝑠^𝑚(𝑧)𝑓(𝑧)−1+𝑑𝑑^𝑝+𝑚+1_𝑝+𝑚+1>0(2.47) or Re𝑠^𝑚(𝑧)𝑓(𝑧)>1+𝑑𝑑^𝑝+𝑚+1_𝑝+𝑚+1.(2.48) It follows from the last inequality that assertion (2.33) of the Theorem 2.5 holds.
The bound in (2.33) is sharp with the extremal function given by (2.41). The proof of the theorem is thus completed.

If Ω=0, 𝜆=0, and 𝑛=1 are taken in Theorem 2.5, the following result is obtained given by Liu [6].

Corollary 2.6. Let 𝑓 be given by (1.19) and define the partial sums 𝑠_𝑚(𝑧) by 𝑠_𝑚(𝑧)=⎧⎪⎪⎨⎪⎪⎩𝑧^𝑝−(𝑝−𝛼)𝜀𝑝+1−𝛼𝑧^𝑝+1𝑧,𝑚=1,_𝑝−(𝑝−𝛼)𝜀(𝑛+𝑝−𝛼)𝑧^𝑝+1−^𝑚_𝑘=2𝑎_𝑝+𝑘𝑧^𝑝+𝑘,𝑚=2,3,….(2.49) Suppose also that ^∞_𝑘=2𝑑_𝑝+𝑘𝑎_𝑝+𝑘≤1−(𝑝−𝛼)𝜀𝑝+1−𝛼where𝑑^𝑝+𝑘=𝑘+𝑝−𝛼(𝑝−𝛼)(1−𝜀);0≤𝛼<𝑝,0≤𝜀<1.(2.50) Then, for 𝑚≥2, one has Re𝑓(𝑧)Re𝑠_𝑚𝑠_𝑚(𝑧)>(𝑝𝑝−+𝛼𝑚)𝜀++1𝑚−+𝛼1(𝑧)𝑓(𝑧)>(𝑝−𝑝𝛼)+(2𝑚−+𝜀1)−+𝛼𝑚+1.(2.51) Each of the bounds in (2.51) is the best possible.