Some Inclusion Relationships of Certain Subclasses of 𝑝-Valent Functions Associated with a Family of Integral Operators

Abstract

By making use of the new integral operator ℜ^𝛼,𝛾_𝛽,𝑝, we introduce and investigate several new subclasses of 𝑝-valent starlike, 𝑝-valent convex, 𝑝-valent close-to-convex, and 𝑝-valent quasi-convex functions. In particular, we establish some inclusion relationships associated with the aforementioned integral operators. Some of the results established in this paper would provide extensions of those given in earlier works.

1. Introduction

Let 𝐴(𝑝) denote the class of functions of the form 𝑓(𝑧)=𝑧^𝑝+^∞_𝑘=1𝑎_𝑘+𝑝𝑧^𝑘+𝑝𝑝∈ℕ={1,2,3,…}, (1) which are analytic and 𝑝-valent in the unit disc 𝑈={𝑧∶𝑧∈ℂ, and |𝑧|<1} and let 𝐴(1)=𝐴.

A function 𝑓∈𝐴(𝑝) is said to be in the class 𝑆^∗_𝑝(𝜆) of 𝑝-valent starlike functions of order 𝜆 in 𝑈 if and only if Re𝑧𝑓^′(𝑧)𝑓(𝑧)>𝜆𝑧∈𝑈;0≤𝜆<𝑝. (2) The class 𝑆^∗_𝑝(𝜆) was introduced by Patil and Thakare [1].

Owa [2] introduced the class 𝐾_𝑝(𝜆) of 𝑝-valent convex of order 𝜆 in 𝑈 if and only if Re1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)>𝜆𝑧∈𝑈;0≤𝜆<𝑝. (3) It is easy to observe from (2) and (3) that 𝑓(𝑧)∈𝐾_𝑝(𝜆)⟺𝑧𝑓^′(𝑧)𝑝∈𝑆^∗_𝑝(𝜆). (4)

We denote by 𝑆^∗_𝑝=𝑆^∗_𝑝(0) and 𝐾_𝑝=𝐾_𝑝(0) where 𝑆^∗_𝑝 and 𝐾_𝑝 are the classes of 𝑝-valently starlike functions and 𝑝-valently convex functions, respectively, (see Goodman [3]).

For a function 𝑓∈𝐴(𝑝), we say that 𝑓∈𝐶_𝑝(𝜂,𝜆) if there exists a function 𝑔∈𝑆^∗_𝑝(𝜆) such that Re𝑧𝑓^′(𝑧)𝑔(𝑧)>𝜂𝑧∈𝑈;0≤𝜆,𝜂<𝑝. (5) Functions in the class 𝐶_𝑝(𝜂,𝜆) are called 𝑝-valent close-to-convex functions of order 𝜂 and type 𝜆. The class 𝐶_𝑝(𝜂,𝜆) was studied by Aouf [4] and the class 𝐶₁(𝜂,𝜆) was studied by Libera [5].

Noor [6, 7] introduced and studied the classes 𝐶^∗_𝑝(𝜂,𝜆) and 𝐶^∗₁(𝜂,𝜆) as follows.

A function 𝑓∈𝐴(𝑝) is said to be in the class 𝐶^∗_𝑝(𝜂,𝜆) of quasi-convex functions of order 𝜂 and type 𝜆 if there exists a function 𝑔∈𝐾_𝑝(𝜆) such that Re𝑧𝑓^′(𝑧)^′𝑔^′(𝑧)>𝜂𝑧∈𝑈;0≤𝜆,𝜂<𝑝. (6) It follows from (5) and (6) that 𝑓(𝑧)∈𝐶^∗_𝑝𝜂,𝜆⟺𝑧𝑓^′(𝑧)𝑝∈𝐶_𝑝𝜂,𝜆. (7)

For functions 𝑓∈𝐴(𝑝) given by (1) and 𝑔∈𝐴(𝑝) given by 𝑔(𝑧)=𝑧^𝑝+^∞_𝑘=1𝑏_𝑘+𝑝𝑧^𝑘+𝑝𝑝∈ℕ, (8) the Hadamard product (or convolution) of 𝑓 and 𝑔 is given by 𝑓∗𝑔(𝑧)=𝑧^𝑝+^∞_𝑘=1𝑎_𝑘+𝑝𝑏_𝑘+𝑝𝑧^𝑘+𝑝=𝑔∗𝑓(𝑧). (9)

For the function 𝑓∈𝐴(𝑝), we introduced the operator ℜ^𝛼,𝛾_𝛽,𝑝∶𝐴(𝑝)→𝐴(𝑝) as follows: ℜ^𝛼,𝛾_𝛽,𝑝𝑓(𝑧)=𝑝+𝛼+𝛽−𝛾𝑝+𝛽−1𝛼−𝛾+1𝑧^𝛽×^𝑧₀𝑡1−𝑧^𝛼−𝛾𝑡^𝛽−1𝑓(𝑡)=Γ𝑑𝑡𝑝+𝛼+𝛽−𝛾+1ΓΓ1𝑝+𝛽𝛼−𝛾+1𝑧^𝛽×^𝑧₀𝑡1−𝑧^𝛼−𝛾𝑡^𝛽−1𝑓(𝑡)𝑑𝑡=𝑧^𝑝+Γ𝑝+𝛼+𝛽−𝛾+1Γ×𝑝+𝛽_∞𝑘=1Γ𝛽+𝑝+𝑘Γ𝑎𝛼+𝛽+𝑝+𝑘−𝛾+1^𝑘+𝑝𝑧^𝑘+𝑝𝛽>−𝑝;𝛼≥𝛾−1;𝛾>0;𝑝∈ℕ;𝑧∈𝑈. (10)

From (10), it is easy to verify that 𝑧ℜ^𝛼+1,𝛾_𝛽,𝑝𝑓(𝑧)^′=𝛼+𝛽+𝑝−𝛾+1ℜ^𝛼,𝛾_𝛽,𝑝𝑓(𝑧)−𝛼+𝛽−𝛾+1ℜ^𝛼+1,𝛾_𝛽,𝑝𝑓(𝑧). (11)

Remark 1. Consider (i) For 𝛾=1, ℜ^𝛼,1_𝛽,𝑝𝑓(𝑧)=𝑄^𝛼_𝛽,𝑝𝑓(𝑧)=𝑝+𝛼+𝛽−1𝛼𝑝+𝛽−1𝑧^𝛽𝑧₀𝑡1−𝑧^𝛼−1𝑡^𝛽−1𝑓(𝑡)=Γ𝑑𝑡𝑝+𝛼+𝛽ΓΓ(𝛼)1𝑝+𝛽𝑧^𝛽𝑧₀𝑡1−𝑧^𝛼−1𝑡^𝛽−1𝑓(𝑡)𝑑𝑡=𝑧^𝑝+Γ𝑝+𝛼+𝛽Γ×𝑝+𝛽_∞𝑘=1Γ𝛽+𝑝+𝑘Γ𝑎𝛼+𝛽+𝑝+𝑘^𝑘+𝑝𝑧^𝑘+𝑝𝛽>−𝑝;𝛼≥0;𝑝∈ℕ;𝑧∈𝑈, (12) where the operator 𝑄^𝛼_𝛽,𝑝 was introduced and studied by Liu and Owa [8], and 𝑄^𝛼_𝛽,1=𝑄^𝛼_𝛽, where the operator 𝑄^𝛼_𝛽 was introduced and studied by Jung et al. [9];
(ii) For 𝛼=𝛾=1 and 𝛽=𝑐, ℜ^1,1_𝑐,𝑝𝑓(𝑧)=𝐽_𝑐,𝑝(𝑧)=𝑝+𝑐𝑧^𝑐𝑧₀𝑡^𝑐−1𝑓(𝑡)𝑑𝑡=𝑧^𝑝+^∞_𝑘=1𝑐+𝑝𝑎𝑐+𝑘+𝑝^𝑘+𝑝𝑧^𝑘+𝑝𝑐>−𝑝;𝑝∈ℕ;𝑧∈𝑈, (13) where 𝐽_𝑐,𝑝(𝑧) is the familiar integral operator, which was defined by Cho and Kim [10]. The operator 𝐽_𝑐,1(𝑧)=𝐽_𝑐(𝑧) was introduced by Bernardi [11] and we note that 𝐽_1,1(𝑧)=𝐽(𝑧) was introduced and studied by Libera [12] and Livingston [13].
The main object of this paper is to investigate the various inclusion relationships for each of the following subclasses of the normalized analytic function class 𝐴(𝑝) which are defined here by means of the operator ℜ^𝛼,𝛾_𝛽,𝑝 given by (10).

Definition 2. In conjunction with (2) and (10), 𝑆^∗_{𝛼,𝛽,𝛾,𝑝}(𝜆)=𝑓∶𝑓∈𝐴𝑝,ℜ^𝛼,𝛾_𝛽,𝑝𝑓∈𝑆^∗_𝑝(𝜆),0≤𝜆<𝑝,𝑝∈ℕ. (14)

Definition 3. In conjunction with (3) and (10), 𝐾_{𝛼,𝛽,𝛾,𝑝}(𝜆)=𝑓∶𝑓∈𝐴𝑝,ℜ^𝛼,𝛾_𝛽,𝑝𝑓∈𝐾_𝑝(𝜆),0≤𝜆<𝑝,𝑝∈ℕ. (15)

Definition 4. In conjunction with (5) and (10), 𝐶_{𝛼,𝛽,𝛾,𝑝}𝜂,𝜆=𝑓∶𝑓∈𝐴𝑝,ℜ^𝛼,𝛾_𝛽,𝑝𝑓∈𝐶_𝑝,.𝜂,𝜆0≤𝜆,𝜂<𝑝,𝑝∈ℕ (16)

Definition 5. In conjunction with (6) and (10), 𝐶^∗_{𝛼,𝛽,𝛾,𝑝}𝜂,𝜆=𝑓∶𝑓∈𝐴𝑝,ℜ^𝛼,𝛾_𝛽,𝑝𝑓∈𝐶^∗_𝑝,.𝜂,𝜆0≤𝜆,𝜂<𝑝,𝑝∈ℕ (17)

Remark 6. Consider (I) For 𝛾=1, in the above definitions, we have 𝑆^∗_{𝛼,𝛽,1,𝑝}(𝜆)=𝑆^∗_{𝛼,𝛽,𝑝}(𝜆)=𝑓∶𝑓∈𝐴𝑝,𝑄^𝛼_𝛽,𝑝𝑓∈𝑆^∗_𝑝(𝜆),,𝐾0≤𝜆<𝑝,𝑝∈ℕ_{𝛼,𝛽,1,𝑝}(𝜆)=𝐾_{𝛼,𝛽,𝑝}(𝜆)=𝑓∶𝑓∈𝐴𝑝,𝑄^𝛼_𝛽,𝑝𝑓∈𝐾_𝑝(𝜆),,𝐶0≤𝜆<𝑝,𝑝∈ℕ_{𝛼,𝛽,1,𝑝}𝜂,𝜆=𝐶_{𝛼,𝛽,𝑝}=𝑝𝜂,𝜆𝑓∶𝑓∈𝐴,𝑄^𝛼_𝛽,𝑝𝑓∈𝐶_𝑝,,𝐶𝜂,𝜆0≤𝜆,𝜂<𝑝,𝑝∈ℕ_{∗𝛼,𝛽,1,𝑝}𝜂,𝜆=𝐶^∗_{𝛼,𝛽,𝑝}=𝑝𝜂,𝜆𝑓∶𝑓∈𝐴,𝑄^𝛼_𝛽,𝑝𝑓∈𝐶^∗_𝑝,.𝜂,𝜆0≤𝜆,𝜂<𝑝,𝑝∈ℕ (18)
(II) For 𝛼=𝛾=1 and 𝛽=𝑐(𝑐>−𝑝), in the above definitions, we have 𝑆^∗_{1,𝑐,1,𝑝}(𝜆)=𝑆^∗_𝑐,𝑝(𝜆)=𝑓∶𝑓∈𝐴𝑝,𝐽_𝑐,𝑝𝑓(𝑧)∈𝑆^∗_𝑝(𝜆),,𝐾0≤𝜆<𝑝,𝑝∈ℕ_{1,𝑐,1,𝑝}(𝜆)=𝐾_𝑐,𝑝(𝜆)=𝑓∶𝑓∈𝐴𝑝,𝐽_𝑐,𝑝𝑓(𝑧)∈𝐾_𝑝(𝜆),,𝐶0≤𝜆<𝑝,𝑝∈ℕ_{1,𝑐,1,𝑝}𝜂,𝜆=𝐶_𝑐,𝑝=𝑝𝜂,𝜆𝑓∶𝑓∈𝐴,𝐽_𝑐,𝑝𝑓(𝑧)∈𝐶_𝑝,,𝐶𝜂,𝜆0≤𝜆,𝜂<𝑝,𝑝∈ℕ_{∗1,𝑐,1,𝑝}𝜂,𝜆=𝐶^∗_𝑐,𝑝=𝑝𝜂,𝜆𝑓∶𝑓∈𝐴,𝐽_𝑐,𝑝𝑓(𝑧)∈𝐶^∗_𝑝,.𝜂,𝜆0≤𝜆,𝜂<𝑝,𝑝∈ℕ (19)
(III) For 𝛾=𝑝=1, in the above definitions, we have 𝑆^∗_{𝛼,𝛽,1,1}(𝜆)=𝑆^∗_𝛼,𝛽(𝜆)=𝑓∶𝑓∈𝐴,𝑄^𝛼_𝛽𝑓∈𝑆^∗(𝜆),𝐾,0≤𝜆<1_{𝛼,𝛽,1,1}(𝜆)=𝐾_𝛼,𝛽(𝜆)=𝑓∶𝑓∈𝐴,𝑄^𝛼_𝛽(𝜆),𝐶𝑓∈𝐾,0≤𝜆<1_{𝛼,𝛽,1,1}𝜂,𝜆=𝐶_𝛼,𝛽=𝜂,𝜆𝑓∶𝑓∈𝐴,𝑄^𝛼_𝛽,𝐶𝑓∈𝐶𝜂,𝜆,0≤𝜆,𝜂<1_{∗𝛼,𝛽,1,1}𝜂,𝜆=𝐶^∗_𝛼,𝛽=𝜂,𝜆𝑓∶𝑓∈𝐴,𝑄^𝛼_𝛽𝑓∈𝐶^∗,,𝜂,𝜆0≤𝜆,𝜂<1 (20) where the classes 𝑆^∗_𝛼,𝛽(𝜆), 𝐾_𝛼,𝛽(𝜆), 𝐶_𝛼,𝛽(𝜂,𝜆), and 𝐶^∗_𝛼,𝛽(𝜂,𝜆) were introduced and studied by Gao et al. [14].

In order to establish our main results, we need the following lemma due to Miller and Mocanu [15].

Lemma 7 (see [15]). Let Θ be a complex-valued function such that Θ∶𝐷⟶ℂ,𝐷⊂ℂ×ℂℂisthecomplexplane, (21) and let 𝑢=𝑢₁+𝑖𝑢₂, 𝑣=𝑣₁+𝑖𝑣₂. Suppose that Θ(𝑢,𝑣) satisfies the following conditions:(i)Θ(𝑢,𝑣) is continuous in 𝐷;(ii)(1,0)∈𝐷 and Re{Θ(1,0)}>0;(iii)Re{Θ(𝑖𝑢₂,𝑣₁)}≤0 for all (𝑖𝑢₂,𝑣₁)∈𝐷 such that 𝑣₁≤−(1/2)(1+𝑢²₂).

Let 𝑞(𝑧)=1+𝑞₁𝑧+𝑞₂𝑧²+⋯ (22) be analytic in 𝑈 such that (𝑞(𝑧),𝑧𝑞^′(𝑧))∈𝐷 for all 𝑧∈𝑈. If ReΘ𝑞(𝑧),𝑧𝑞^′(𝑧)>0(𝑧∈𝑈), (23) then Re𝑞(𝑧)>0(𝑧∈𝑈). (24)

2. The Main Results

In this section, we give several inclusion relationships for analytic function classes, which are associated with the integral operator ℜ^𝛼,𝛾_𝛽,𝑝. Unless otherwise mentioned, we assume throughout this paper that 𝛽>−𝑝, 𝛼≥𝛾−1,  𝛾>0, 𝑝∈ℕ, and 𝑧∈𝑈.

Theorem 8. Let 0≤𝜆<𝑝. Then 𝑆^∗_{𝛼,𝛽,𝛾,𝑝}(𝜆)⊂𝑆^∗_{𝛼+1,𝛽,𝛾,𝑝}(𝜆). (25)

Proof. Let 𝑓∈𝑆^∗_{𝛼,𝛽,𝛾,𝑝}(𝜆) and set 𝑧ℜ^𝛼+1,𝛾_𝛽,𝑝𝑓(𝑧)^′ℜ^𝛼+1,𝛾_𝛽,𝑝𝑓(𝑧)−𝜆=𝑝−𝜆𝑞(𝑧), (26) where 𝑞(𝑧) is given by (22). By using identity (11), we obtain ℜ^𝛼,𝛾_𝛽,𝑝𝑓(𝑧)ℜ^𝛼+1,𝛾_𝛽,𝑝𝑓(𝑧)=𝜆+𝑝−𝜆𝑞(𝑧)+𝛼+𝛽−𝛾+1.𝛼+𝛽+𝑝−𝛾+1 (27) Differentiating (27) logarithmically with respect to 𝑧, we obtain 𝑧ℜ^𝛼,𝛾_𝛽,𝑝𝑓(𝑧)^′ℜ^𝛼,𝛾_𝛽,𝑝𝑓(𝑧)=𝑧ℜ^𝛼+1,𝛾_𝛽,𝑝𝑓(𝑧)^′ℜ^𝛼+1,𝛾_𝛽,𝑝𝑓(𝑧)+𝑝−𝜆𝑧𝑞^′(𝑧)𝑞(𝑧)𝑞(𝑧)+𝑝−𝜆+𝜆+𝛼+𝛽−𝛾+1=𝜆+𝑝−𝜆𝑝−𝜆𝑧𝑞^′(𝑧)𝑞(𝑧).𝑝−𝜆+𝜆+𝛼+𝛽−𝛾+1 (28) We now choose 𝑢=𝑞(𝑧)=𝑢₁+𝑖𝑢₂ and 𝑣=𝑧𝑞^′(𝑧)=𝑣₁+𝑖𝑣₂, and define the function Θ by Θ()=𝑣𝑢,𝑣𝑝−𝜆𝑢+𝑝−𝜆.𝑝−𝜆𝑢+𝜆+𝛼+𝛽−𝛾+1 (29) Then, clearly Θ(𝑢,𝑣) satisfies the following conditions:(i)Θ(𝑢,𝑣) is continuous in 𝐷=(ℂ⧵{(𝜆+𝛼+𝛽−𝛾+1)/(𝜆−𝑝)})×ℂ;(ii)(1,0)∈𝐷 and Re{Θ(1,0)}=𝑝−𝜆>0;(iii)for all (𝑖𝑢₂,𝑣₁)∈𝐷 such that 𝑣₁≤−(1/2)(1+𝑢²₂) we have ReΘ𝑖𝑢₂,𝑣₁𝑣=Re𝑝−𝜆₁𝑝−𝜆𝑖𝑢₂=𝑣+𝜆+𝛼+𝛽−𝛾+1𝑝−𝜆𝜆+𝛼+𝛽−𝛾+1₁𝑝−𝜆²𝑢²₂+𝜆+𝛼+𝛽−𝛾+1²≤−𝑝−𝜆1+𝑢²₂𝜆+𝛼+𝛽−𝛾+12𝑝−𝜆²𝑢²₂+𝜆+𝛼+𝛽−𝛾+1²<0, (30)which shows that the function Θ satisfies the hypotheses of Lemma 7. Consequently, we easily obtain the inclusion relationship (25).

Theorem 9. Let 0≤𝜆<𝑝, 𝑝∈ℕ. Then 𝐾_{𝛼,𝛽,𝛾,𝑝}(𝜆)⊂𝐾_{𝛼+1,𝛽,𝛾,𝑝}(𝜆). (31)

Proof. Let 𝑓∈𝐾_{𝛼,𝛽,𝛾,𝑝}(𝜆). Then, from Definition 3, we have ℜ^𝛼,𝛾_𝛽,𝑝𝑓∈𝐾_𝑝(𝜆),0≤𝜆<𝑝;𝑝∈ℕ. (32) Furthermore, in view of the relationship (4), we find that 𝑧𝑝ℜ^𝛼,𝛾_𝛽,𝑝𝑓(𝑧)^′∈𝑆^∗_𝑝(𝜆), (33) that is, that ℜ^𝛼,𝛾_𝛽,𝑝𝑧𝑓^′(𝑧)𝑝∈𝑆^∗_𝑝(𝜆). (34) Thus, by using Definition 2 and Theorem 8, we have 𝑧𝑓^′(𝑧)𝑝∈𝑆^∗_𝑝𝛼,𝛽,𝛾,𝜆⊂𝑆^∗_𝑝𝛼+1,𝛽,𝛾,𝜆, (35) which implies that 𝐾_{𝛼,𝛽,𝛾,𝑝}(𝜆)⊂𝐾_{𝛼+1,𝛽,𝛾,𝑝}(𝜆). (36) The proof of Theorem 9 is thus completed.

Theorem 10. Let 0≤𝜆, 𝜂<𝑝. Then 𝐶_{𝛼,𝛽,𝛾,𝑝}𝜂,𝜆⊂𝐶_{𝛼+1,𝛽,𝛾,𝑝}𝜂,𝜆. (37)

Proof. Let 𝑓∈𝐶_{𝛼,𝛽,𝛾,𝑝}(𝜂,𝜆). Then there exists a function Ψ∈𝑆^∗_𝑝(𝜆) such that 𝑧ℜRe^{𝛼,𝛾𝛽,𝑝}𝑓(𝑧)^′Ψ(𝑧)>𝜂0≤𝜆,𝜂<𝑝,𝑧∈𝑈. (38) We put ℜ^𝛼,𝛾_𝛽,𝑝𝑔(𝑧)=Ψ(𝑧), (39) so that we have 𝑔∈𝑆^∗_{𝛼,𝛽,𝛾,𝑝}(𝜆)𝑧ℜ,Re^{𝛼,𝛾𝛽,𝑝}𝑓(𝑧)^′ℜ^𝛼,𝛾_𝛽,𝑝𝑔(𝑧)>𝜂(𝑧∈𝑈). (40) We next put 𝑧ℜ^𝛼+1,𝛾_𝛽,𝑝𝑓(𝑧)^′ℜ^𝛼+1,𝛾_𝛽,𝑝𝑔(𝑧)=𝜂+𝑝−𝜂𝑞(𝑧), (41) where 𝑞(𝑧) is given by (22). Thus, by using identity (11), we obtain 𝑧ℜ^𝛼,𝛾_𝛽,𝑝𝑓(𝑧)^′ℜ^𝛼,𝛾_𝛽,𝑝𝑔(𝑧)=ℜ^𝛼,𝛾_𝛽,𝑝𝑧𝑓^′(𝑧)ℜ^𝛼,𝛾_𝛽,𝑝𝑔(𝑧)=𝑧ℜ^𝛼+1,𝛾_𝛽,𝑝𝑧𝑓^′(𝑧)^′+𝛼+𝛽−𝛾+1ℜ^𝛼+1,𝛾_𝛽,𝑝𝑧𝑓^′(𝑧)×𝑧ℜ^𝛼+1,𝛾_𝛽,𝑝𝑔(𝑧)^′+𝛼+𝛽−𝛾+1ℜ^𝛼+1,𝛾_𝛽,𝑝𝑔(𝑧)⁻¹=𝑧ℜ^𝛼+1,𝛾_𝛽,𝑝𝑧𝑓^′(𝑧)^′ℜ^𝛼+1,𝛾_𝛽,𝑝𝑔(𝑧)+ℜ𝛼+𝛽−𝛾+1^{𝛼+1,𝛾𝛽,𝑝}𝑧𝑓^′(𝑧)ℜ^𝛼+1,𝛾_𝛽,𝑝𝑔(𝑧)×𝑧ℜ^𝛼+1,𝛾_𝛽,𝑝𝑔(𝑧)^′ℜ^𝛼+1,𝛾_𝛽,𝑝𝑔(𝑧)+𝛼+𝛽−𝛾+1⁻¹. (42) Since 𝑔∈𝑆^∗_{𝛼,𝛽,𝛾,𝑝}(𝜆), then from Theorem 8 we have 𝑔∈𝑆^∗_{𝛼+1,𝛽,𝛾,𝑝}(𝜆), so that we can put 𝑧ℜ^𝛼+1,𝛾_𝛽,𝑝𝑔(𝑧)^′ℜ^𝛼+1,𝛾_𝛽,𝑝𝑔(𝑧)=𝜆+𝑝−𝜆𝐺(𝑧), (43) where 𝐺(𝑧)=𝑔₁𝑥,𝑦+𝑖𝑔₂,(𝐺(𝑧𝑥,𝑦Re))=𝑔₁𝑥,𝑦>0(𝑧∈𝑈). (44) Then 𝑧ℜ^𝛼,𝛾_𝛽,𝑝𝑓(𝑧)^′ℜ^𝛼,𝛾_𝛽,𝑝𝑔(𝑧)=𝑧ℜ^𝛼+1,𝛾_𝛽,𝑝𝑧𝑓^′(𝑧)^′ℜ^𝛼+1,𝛾_𝛽,𝑝𝑔(𝑧)+𝑞(𝑧×𝐺(𝑧)𝛼+𝛽−𝛾+1𝜂+𝑝−𝜂)𝑝−𝜆+𝜆+𝛼+𝛽−𝛾+1⁻¹. (45) We thus find from (41) that 𝑧ℜ^𝛼+1,𝛾_𝛽,𝑝𝑓(𝑧)^′=ℜ^𝛼+1,𝛾_𝛽,𝑝𝑔(𝑧)𝜂+𝑝−𝜂𝑞(𝑧). (46) Differentiating both sides of (46) with respect to 𝑧, we obtain 𝑧𝑧ℜ^𝛼+1,𝛾_𝛽,𝑝𝑓(𝑧)^′′ℜ^𝛼+1,𝛾_𝛽,𝑝𝑔(𝑧)=𝑝−𝜂𝑧𝑞^′(𝑧)+𝑧ℜ^𝛼+1,𝛾_𝛽,𝑝𝑔(𝑧)^′ℜ^𝛼+1,𝛾_𝛽,𝑝𝑔(𝑧)𝑞(𝑧=𝜂+𝑝−𝜂)𝑝−𝜂𝑧𝑞^′(𝑧)+𝜆+𝑝−𝜆𝐺(𝑧)𝜂+𝑝−𝜂𝑞(𝑧). (47) By substituting (47) into (45), we have 𝑧ℜ^𝛼,𝛾_𝛽,𝑝𝑓(𝑧)^′ℜ^𝛼,𝛾_𝛽,𝑝𝑔(𝑧)𝑞(𝑧)+−𝜂=𝑝−𝜂𝑝−𝜂𝑧𝑞^′(𝑧)𝐺(𝑧).𝑝−𝜆+𝜆+𝛼+𝛽−𝛾+1 (48)
The remainder of our proof of Theorem 10 is much akin to that of Theorem 8. We, therefore, choose to omit the details involved.

Theorem 11. Let 0≤𝜆, 𝜂<𝑝. Then 𝐶^∗_{𝛼,𝛽,𝛾,𝑝}𝜂,𝜆⊂𝐶^∗_{𝛼+1,𝛽,𝛾,𝑝}𝜂,𝜆. (49)

Proof. Just as we derived Theorem 9 as a consequence of Theorem 8 by using the equivalence (4), we can also prove Theorem 11 by using Theorem 10 in conjunction with the equivalence (7).

Our main results in Theorems 8–11, can thus be applied with a view to deducing the following corollaries.

Taking 𝛾=1 in Theorems 8–11 above, we obtain the following corollary.

Corollary 12. Let 0≤𝜆, 𝜂<𝑝. Then 𝑆^∗_{𝛼,𝛽,𝑝}(𝜆)⊂𝑆^∗_{𝛼+1,𝛽,𝑝}(𝜆),𝐾_{𝛼,𝛽,𝑝}(𝜆)⊂𝐾_{𝛼+1,𝛽,𝑝}(𝜆),𝐶_{𝛼,𝛽,𝑝}𝜂,𝜆⊂𝐶_{𝛼+1,𝛽,𝑝},𝐶𝜂,𝜆_{∗𝛼,𝛽,𝑝}𝜂,𝜆⊂𝐶^∗_{𝛼+1,𝛽,𝑝}𝜂,𝜆. (50)

Remark 13. Taking 𝑝=1 in Corollary 12, we obtain the results obtained by Gao et al. [14, Theorems 1-4].
Taking 𝛼=𝛾=1 and 𝛽=𝑐(𝑐>−𝑝) in Theorems 8–11, we obtain the following corollary.

Corollary 14. Let 0≤𝜆, 𝜂<𝑝. Then 𝑆^∗_1,𝑐,𝑝(𝜆)⊂𝑆^∗_2,𝑐,𝑝(𝜆),𝐾_1,𝑐,𝑝(𝜆)⊂𝐾_2,𝑐,𝑝(𝜆),𝐶_1,𝑐,𝑝𝜂,𝜆⊂𝐶_2,𝑐,𝑝,𝐶𝜂,𝜆_{∗1,𝑐,𝑝}𝜂,𝜆⊂𝐶^∗_2,𝑐,𝑝𝜂,𝜆. (51)

Remark 15. Taking 𝑝=1 in Corollary 14, we obtain the results obtained by Gao et al. [14, Corollary  1–4].

3. A Set of Integral-Preserving Properties

In this section, we present several integral-preserving properties of the analytic function classes introduced here. In order to obtain the integral-preserving properties involving the integral operator 𝐽_𝑐,𝑝 defined by (13).

Theorem 16. Let 𝑐 be any real number and 𝑐>−𝑝. If 𝑓(𝑧)∈𝑆^∗_{𝛼,𝛽,𝛾,𝑝}(𝜆), then 𝐽_𝑐,𝑝(𝑧)∈𝑆^∗_{𝛼,𝛽,𝛾,𝑝}(𝜆), where 𝐽_𝑐,𝑝𝑓(𝑧) is defined by (13).

Proof. From (13), we have 𝑧ℜ^𝛼,𝛾_𝛽,𝑝𝐽_𝑐,𝑝(𝑧)^′=𝑐+𝑝ℜ^𝛼,𝛾_𝛽,𝑝−𝑐ℜ^𝛼,𝛾_𝛽,𝑝𝐽_𝑐,𝑝(𝑧). (52)
Let 𝑓∈𝑆^∗_{𝛼,𝛽,𝛾,𝑝}(𝜆) and set 𝑧ℜ^𝛼,𝛾_𝛽,𝑝𝐽_𝑐,𝑝𝑓(𝑧)^′ℜ^𝛼,𝛾_𝛽,𝑝𝐽_𝑐,𝑝𝑓(𝑧)−𝜆=𝑝−𝜆𝑞(𝑧), (53) where 𝑞(𝑧) is given by (22). By using identity (52), we obtain ℜ^𝛼,𝛾_𝛽,𝑝𝑓(𝑧)ℜ^𝛼,𝛾_𝛽,𝑝𝐽_𝑐,𝑝𝑓(𝑧)=𝜆+𝑝−𝜆𝑞(𝑧)+𝑐.𝑐+𝑝 (54) Differentiating (54) logarithmically with respect to 𝑧, we obtain 𝑧ℜ^𝛼,𝛾_𝛽,𝑝𝑓(𝑧)^′ℜ^𝛼,𝛾_𝛽,𝑝𝑓(𝑧)=𝑧ℜ^𝛼,𝛾_𝛽,𝑝𝐽_𝑐,𝑝𝑓(𝑧)^′ℜ^𝛼,𝛾_𝛽,𝑝𝐽_𝑐,𝑝𝑓(𝑧)+𝑝−𝜆𝑧𝑞^′(𝑧)𝑞(𝑧)𝑞(𝑧)+𝑝−𝜆+𝜆+𝑐=𝜆+𝑝−𝜆𝑝−𝜆𝑧𝑞^′(𝑧)𝑞(𝑧).𝑝−𝜆+𝜆+𝑐 (55) We now choose 𝑢=𝑞(𝑧)=𝑢₁+𝑖𝑢₂, and 𝑣=𝑧𝑞^′(𝑧)=𝑣₁+𝑖𝑣₂, and define the function Θ by Θ()=𝑣𝑢,𝑣𝑝−𝜆𝑢+𝑝−𝜆.𝑝−𝜆𝑢+𝜆+𝑐 (56) It is easy to see that the function Θ(𝑢,𝑣) satisfies the conditions of Lemma 7, and the remaining part of the proof of Theorem 16 is similar to that of Theorem 8.

Taking 𝛾=1 in Theorem 16, we obtain the following corollary.

Corollary 17. Let 𝑐 be any real number and 𝑐>−𝑝. If 𝑓(𝑧)∈𝑆^∗_{𝛼,𝛽,𝑝}(𝜆), then 𝐽_𝑐,𝑝(𝑧)∈𝑆^∗_{𝛼,𝛽,𝑝}(𝜆), where 𝐽_𝑐,𝑝𝑓(𝑧) is defined by (13).

Theorem 18. Let 𝑐 be any real number and 𝑐>−𝑝. If 𝑓(𝑧)∈𝐾_{𝛼,𝛽,𝛾,𝑝}(𝜆), then 𝐽_𝑐,𝑝(𝑧)∈𝐾_{𝛼,𝛽,𝛾,𝑝}(𝜆), where 𝐽_𝑐,𝑝𝑓(𝑧) is defined by (13).

Proof. By applying Theorem 16 in conjunction with the equivalence (4), it follows that 𝑓(𝑧)∈𝐾_{𝛼,𝛽,𝛾,𝑝}(𝜆)⟺𝑧𝑝𝑓^′(𝑧)∈𝑆^∗_{𝛼,𝛽,𝛾,𝑝}(𝜆)⟹𝐽_𝑐,𝑝𝑧𝑝𝑓^′(𝑧)∈𝑆^∗_{𝛼,𝛽,𝛾,𝑝}(𝜆)⟺𝑧𝑝𝐽_𝑐,𝑝𝑓(𝑧)^′∈𝑆^∗_{𝛼,𝛽,𝛾,𝑝}(𝜆)⟹𝐽_𝑐,𝑝𝑓(𝑧)∈𝐾_{𝛼,𝛽,𝛾,𝑝}(𝜆), (57) which proves Theorem 18.

Taking 𝛾=1 in Theorem 18, we obtain the following corollary.

Corollary 19. Let 𝑐 be any real number and 𝑐>−𝑝. If 𝑓(𝑧)∈𝐾_{𝛼,𝛽,𝑝}(𝜆), then 𝐽_𝑐,𝑝(𝑧)∈𝐾_{𝛼,𝛽,𝑝}(𝜆), where 𝐽_𝑐,𝑝𝑓(𝑧) is defined by (13).

Theorem 20. Let 𝑐 be any real number and 𝑐>−𝑝. If 𝑓(𝑧)∈𝐶_{𝛼,𝛽,𝛾,𝑝}(𝜂,𝜆), then 𝐽_𝑐,𝑝(𝑧)∈𝐶_{𝛼,𝛽,𝛾,𝑝}(𝜂,𝜆), where 𝐽_𝑐,𝑝𝑓(𝑧) is defined by (13).

Proof. Let 𝑓∈𝐶_{𝛼,𝛽,𝛾,𝑝}(𝜂,𝜆). Then there exists a function Ψ∈𝑆^∗_𝑝(𝜆) such that 𝑧ℜRe^{𝛼,𝛾𝛽,𝑝}𝑓(𝑧)^′Ψ(𝑧)>𝜂0≤𝜆,𝜂<𝑝,𝑧∈𝑈. (58) We put ℜ^𝛼,𝛾_𝛽,𝑝𝑔(𝑧)=Ψ(𝑧), (59) so that we have 𝑔∈𝑆^∗_{𝛼,𝛽,𝛾,𝑝}(𝜆)𝑧ℜ,Re^{𝛼,𝛾𝛽,𝑝}𝑓(𝑧)^′ℜ^𝛼,𝛾_𝛽,𝑝𝑔(𝑧)>𝜂(𝑧∈𝑈). (60) We next put 𝑧ℜ^𝛼,𝛾_𝛽,𝑝𝐽_𝑐,𝑝𝑓(𝑧)^′ℜ^𝛼,𝛾_𝛽,𝑝𝐽_𝑐,𝑝𝑔(𝑧)=𝜂+𝑝−𝜂𝑞(𝑧), (61) where 𝑞(𝑧) is given by (22). Thus, by using identity (52), we obtain 𝑧ℜ^𝛼,𝛾_𝛽,𝑝𝐽_𝑐,𝑝𝑓(𝑧)^′ℜ^𝛼,𝛾_𝛽,𝑝𝐽_𝑐,𝑝𝑔(𝑧)=ℜ^𝛼,𝛾_𝛽,𝑝𝑧𝐽_𝑐,𝑝𝑓(𝑧)^′ℜ^𝛼,𝛾_𝛽,𝑝𝐽_𝑐,𝑝𝑔(𝑧)=𝑧ℜ^𝛼,𝛾_𝛽,𝑝𝑧𝐽_𝑐,𝑝𝑓(𝑧)^′′+𝑐ℜ^𝛼,𝛾_𝛽,𝑝𝑧𝐽_𝑐,𝑝𝑓(𝑧)^′×𝑧ℜ^𝛼,𝛾_𝛽,𝑝𝐽_𝑐,𝑝𝑔(𝑧)^′+𝑐ℜ^𝛼,𝛾_𝛽,𝑝𝐽_𝑐,𝑝𝑔(𝑧)⁻¹=𝑧ℜ^𝛼,𝛾_𝛽,𝑝𝑧𝐽_𝑐,𝑝𝑓(𝑧)^′′ℜ^𝛼,𝛾_𝛽,𝑝𝐽_𝑐,𝑝𝑔(𝑧)ℜ+𝑐^{𝛼,𝛾𝛽,𝑝}𝑧𝐽_𝑐,𝑝𝑓(𝑧)^′ℜ^𝛼,𝛾_𝛽,𝑝𝐽_𝑐,𝑝𝑔(𝑧)×𝑧ℜ^𝛼,𝛾_𝛽,𝑝𝐽_𝑐,𝑝𝑔(𝑧)^′ℜ^𝛼,𝛾_𝛽,𝑝𝐽_𝑐,𝑝𝑔(𝑧)+𝑐⁻¹. (62) Since 𝑔∈𝑆^∗_{𝛼,𝛽,𝛾,𝑝}(𝜆), then from Theorem 16 we have 𝐽_𝑐,𝑝𝑔(𝑧)∈𝑆^∗_{𝛼,𝛽,𝛾,𝑝}(𝜆), so that we can put 𝑧ℜ^𝛼,𝛾_𝛽,𝑝𝐽_𝑐,𝑝𝑔(𝑧)^′ℜ^𝛼,𝛾_𝛽,𝑝𝐽_𝑐,𝑝𝑔(𝑧)=𝜆+𝑝−𝜆𝐺(𝑧), (63) where 𝐺(𝑧)=𝑔₁𝑥,𝑦+𝑖𝑔₂𝑥,𝑦,R𝑒(𝐺(𝑧))=𝑔₁().𝑥,𝑦>0𝑧∈𝑈 (64) Then 𝑧ℜ^𝛼,𝛾_𝛽,𝑝𝐽_𝑐,𝑝𝑓(𝑧)^′ℜ^𝛼,𝛾_𝛽,𝑝𝐽_𝑐,𝑝𝑔(𝑧)=𝑧ℜ^𝛼,𝛾_𝛽,𝑝𝑧𝐽_𝑐,𝑝𝑓(𝑧)^′′ℜ^𝛼,𝛾_𝛽,𝑝𝐽_𝑐,𝑝𝑔(𝑧)𝑞(𝑧)×𝐺(𝑧)+𝑐𝜂+𝑝−𝜂𝑝−𝜆+𝜆+𝑐⁻¹. (65) We thus find from (61) that 𝑧ℜ^𝛼,𝛾_𝛽,𝑝𝐽_𝑐,𝑝𝑓(𝑧)^′=ℜ^𝛼,𝛾_𝛽,𝑝𝐽_𝑐,𝑝𝑔(𝑧)𝜂+𝑝−𝜂𝑞(𝑧). (66) Differentiating both sides of (66) with respect to 𝑧, we obtain 𝑧𝑧ℜ^𝛼,𝛾_𝛽,𝑝𝐽_𝑐,𝑝𝑓(𝑧)^′′ℜ^𝛼,𝛾_𝛽,𝑝𝐽_𝑐,𝑝𝑔(𝑧)=𝑝−𝜂𝑧𝑞^′(𝑧)+𝑧ℜ^𝛼,𝛾_𝛽,𝑝𝐽_𝑐,𝑝𝑔(𝑧)^′ℜ^𝛼,𝛾_𝛽,𝑝𝐽_𝑐,𝑝𝑔(𝑧)𝑞(𝑧=𝜂+𝑝−𝜂)𝑝−𝜂𝑧𝑞^′(𝑧)+𝜆+𝑝−𝜆𝐺(𝑧)𝜂+𝑝−𝜂𝑞(𝑧). (67) By substituting (67) into (65), we have 𝑧ℜ^𝛼,𝛾_𝛽,𝑝𝑓(𝑧)^′ℜ^𝛼,𝛾_𝛽,𝑝𝑔(𝑧)−𝜂=𝑝−𝜂𝑞(𝑧)+𝑝−𝜂𝑧𝑞^′(𝑧)𝐺(𝑧).𝑝−𝜆+𝜆+𝑐 (68)
The remainder of our proof of Theorem 20 is much akin to that of Theorem 8. We, therefore, choose to omit the details involved.

Taking 𝛾=1 in Theorem 20, we obtain the following corollary.

Corollary 21. Let 𝑐 be any real number and 𝑐>−𝑝. If 𝑓(𝑧)∈𝐶_{𝛼,𝛽,𝑝}(𝜂,𝜆), then 𝐽_𝑐,𝑝(𝑧)∈𝐶_{𝛼,𝛽,𝑝}(𝜂,𝜆), where 𝐽_𝑐,𝑝𝑓(𝑧) is defined by (13).

Theorem 22. Let 𝑐 be any real number and 𝑐>−𝑝. If 𝑓(𝑧)∈𝐶^∗_{𝛼,𝛽,𝛾,𝑝}(𝜂,𝜆), then 𝐽_𝑐,𝑝(𝑧)∈𝐶^∗_{𝛼,𝛽,𝛾,𝑝}(𝜂,𝜆), where 𝐽_𝑐,𝑝𝑓(𝑧) is defined by (13).

Proof. Just as we derived Theorem 18 as a consequence of Theorem 16 by using the equivalence (4), we can also prove Theorem 22 by using Theorem 20 in conjunction with the equivalence (7).

Taking 𝛾=1 in Theorem 22, we obtain the following corollary.

Corollary 23. Let 𝑐 be any real number and 𝑐>−𝑝. If 𝑓(𝑧)∈𝐶^∗_{𝛼,𝛽,𝛾,𝑝}(𝜂,𝜆), then 𝐽_𝑐,𝑝(𝑧)∈𝐶^∗_{𝛼,𝛽,𝛾,𝑝}(𝜂,𝜆), where 𝐽_𝑐,𝑝𝑓(𝑧) is defined by (13).