A Class of Nonlinear Integral Operators Preserving Double Subordinations

Abstract

The purpose of the present paper is to investigate some subordination- and superordination-preserving properties of certain integral operators defined on the space of meromorphic functions in the punctured open unit disk. The sandwich-type theorem for these integral operators is also considered.

1. Introduction

Let ℋ=ℋ(𝕌) denote the class of analytic functions in the open unit disk 𝕌={𝑧∈ℂ∶|𝑧|<1}. For 𝑎∈ℂ, let

ℋ[𝑎,𝑛]={𝑓∈ℋ∶𝑓(𝑧)=𝑎+𝑎_𝑛𝑧^𝑛+𝑎_𝑛+1𝑧^𝑛+1+⋯}. (1.1)

Let 𝑓 and 𝐹 be members of ℋ. The function 𝑓 is said to be subordinate to 𝐹, or 𝐹 is said to be superordinate to 𝑓, if there exists a function 𝑤 analytic in 𝕌, with 𝑤(0)=0 and |𝑤(𝑧)|<1, and such that 𝑓(𝑧)=𝐹(𝑤(𝑧)). In such a case, we write 𝑓≺𝐹 or 𝑓(𝑧)≺𝐹(𝑧). If the function 𝐹 is univalent in 𝕌, then we have 𝑓≺𝐹 if and only if 𝑓(0)=𝐹(0) and 𝑓(𝕌)⊂𝐹(𝕌) (cf. [1]).

Definition 1.1 (see [1]). Let 𝜙∶ℂ²→ℂ and let ℎ be univalent in 𝕌. If 𝑝 is analytic in 𝕌 and satisfies the differential subordination
𝜙(𝑝(𝑧),𝑧𝑝^′(𝑧))≺ℎ(𝑧), (1.2) then 𝑝 is called a solution of the differential subordination. The univalent function 𝑞 is called a dominant of the solutions of the differential subordination, or more simply a dominant if 𝑝≺𝑞 for all 𝑝 satisfying (1.2). A dominant ^0𝑥0𝑞^02𝑑𝑐 that satisfies ^0𝑥0𝑞^02𝑑𝑐≺𝑞 for all dominants 𝑞 of (1.2) is said to be the best dominant.

Definition 1.2 (see [2]). Let 𝜑∶ℂ²→ℂ and let ℎ be analytic in 𝕌. If 𝑝 and 𝜑(𝑝(𝑧),𝑧𝑝^′(𝑧)) are univalent in 𝕌 and satisfy the differential superordination
ℎ(𝑧)≺𝜑(𝑝(𝑧),𝑧𝑝^′(𝑧)), (1.3) then 𝑝 is called a solution of the differential superordination. An analytic function 𝑞 is called a subordinant of the solutions of the differential superordination, or more simply a subordinant if 𝑞≺𝑝 for all 𝑝 satisfying (1.3). A univalent subordinant ^0𝑥0𝑞^02𝑑𝑐 that satisfies 𝑞≺𝑞^{0𝑥002𝑑𝑐} for all subordinants 𝑞 of (1.3) is said to be the best subordinant.

Definition 1.3 (see [2]). One denotes by 𝒬 the class of functions 𝑓 that are analytic and injective on 𝕌⧵𝐸(𝑓), where
𝐸(𝑓)={𝜁∈𝜕𝕌∶lim_𝑧→𝜁𝑓(𝑧)=∞}, (1.4) and are such that 𝑓^′(𝜁)≠0 for 𝜁∈𝜕𝕌⧵𝐸(𝑓).

Let Σ denote the class of functions of the form

1𝑓(𝑧)=𝑧+^∞_𝑛=0𝑎_𝑛𝑧^𝑛 (1.5) which are analytic in the punctured open unit disk 𝔻=𝕌⧵{0}. Let Σ^∗ and Σ_𝑘 be the subclasses of Σ consisting of all functions which are, respectively, meromorphic starlike and meromorphic convex in 𝔻 (see, for details, [3–5]).

For a function 𝑓∈Σ, we introduce the following integral operators 𝐼_𝛽,𝛾 defined by

𝐼_𝛽,𝛾(𝑓)(𝑧)∶=(𝛾−𝛽+1𝑧^𝛾+1𝑧₀𝑡^𝛾𝑓^𝛽(𝑡)𝑑𝑡)^1/𝛽(𝑓∈Σ;𝛽,𝛾∈ℂ;𝛽∈ℂ⧵{0};Re{𝛾−𝛽}>0). (1.6) The integral operator 𝐼_𝛽,𝛾(𝑓) defined by (1.6) has been extensively studied by many authors [6–10] with suitable restrictions on the parameters 𝛽 and 𝛾, and for 𝑓 belonging to some favored classes of meromorphic functions. In particular, Bajpai [6] showed that the integral operator 𝐼_1,1(𝑓) belongs to the classes Σ^∗ and Σ_𝑘, whenever 𝑓 belongs to the classes Σ^∗ and Σ_𝑘, respectively. Moreover, the operator 𝐼_𝛽,𝛾 for the case 𝛽=1 is related to the generalized Libera transform introduced by Stević (see, e.g., [11–13]).

Making use of the principle of subordination between analytic functions, Miller et al. [14] obtained some subordination-preserving properties for certain integral operators (see also [15]). Moreover, Miller and Mocanu [2] considered differential superordinations as the dual concept of differential subordinations (see also [16]). In the present paper, we obtain the subordination- and superordination-preserving properties of the integral operator 𝐼_𝛽,𝛾 defined by (1.6) with the sandwich-type theorem.

Throughout this paper, we denote the class Σ_𝛽,𝛾 by

Σ_𝛽,𝛾∶={𝑓∈Σ∶𝑧𝑓(𝑧)≠0and𝑧𝐼_𝛽,𝛾(𝑓)(𝑧)≠0(𝑧∈𝕌;𝛽≠1)}, (1.7) where 𝐼_𝛽,𝛾 is the integral operator defined by (1.6). For various interesting developments involving functions in the class Σ_𝛽,𝛾, the reader may be referred, for example, to the work of Dwivedi et al. [9].

2. A Set of Lemmas

The following lemmas will be required in our present investigation.

Lemma 2.1 (see [17]). Suppose that the function 𝐻∶ℂ²→ℂ satisfies the following condition:
Re{𝐻(𝑖𝑠,𝑡)}≤0 (2.1) for all real 𝑠 and 𝑡≥−𝑛(1+𝑠²)/2, where 𝑛 is a positive integer. If the function 𝑝(𝑧)=1+𝑝_𝑛𝑧^𝑛+⋯ is analytic in 𝕌 and
Re{𝐻(𝑝(𝑧),𝑧𝑝^′(𝑧))}>0(𝑧∈𝕌), (2.2) then Re{𝑝(𝑧)}>0(𝑧∈𝕌).

Lemma 2.2 (see [5]). Let 𝛽,𝛾∈ℂ with 𝛽≠0 and let ℎ∈ℋ(𝕌) with ℎ(0)=𝑐. If Re{𝛽ℎ(𝑧)+𝛾}>0(𝑧∈𝕌), then the solution of the differential equation
𝑞(𝑧)+𝑧𝑞^′(𝑧)𝛽𝑞(𝑧)+𝛾=ℎ(𝑧)(𝑧∈𝕌;𝑞(0)=𝑐) (2.3) is analytic in 𝕌 and satisfies Re{𝛽𝑞(𝑧)+𝛾}>0(𝑧∈𝕌).

Lemma 2.3 (see [1]). Let 𝑝∈𝒬 with 𝑝(0)=𝑎 and let 𝑞(𝑧)=𝑎+𝑎_𝑛𝑧^𝑛+⋯ be analytic in 𝕌 with 𝑞(𝑧)≢𝑎 and 𝑛≥1. If 𝑞 is not subordinate to 𝑝, then there exist points 𝑧₀=𝑟₀𝑒^𝑖𝜃∈𝕌 and 𝜁₀∈𝜕𝕌⧵𝐸(𝑓), for which
𝑞(𝕌_𝑟0)⊂𝑝(𝕌),𝑞(𝑧₀)=𝑝(𝜁₀),𝑧₀𝑞^′(𝑧₀)=𝑚𝜁₀𝑝^′(𝜁₀)(𝑚≥𝑛). (2.4)

A function 𝐿(𝑧,𝑡) defined on 𝕌×[0,∞) is the subordination chain (or Löwner chain) if 𝐿(⋅,𝑡) is analytic and univalent in 𝕌 for all 𝑡∈[0,∞), 𝐿(𝑧,⋅) is continuously differentiable on [0,∞) for all 𝑧∈𝕌, and 𝐿(𝑧,𝑠)≺𝐿(𝑧,𝑡) for 0≤𝑠<𝑡.

Lemma 2.4 (see [2]). Let 𝑞∈ℋ[𝑎,1], set 𝜑∶ℂ²→ℂ, and let 𝜑(𝑞(𝑧),𝑧𝑞^′(𝑧))≡ℎ(𝑧)(𝑧∈𝕌). If 𝐿(𝑧,𝑡)=𝜑(𝑞(𝑧),𝑡𝑧𝑞^′(𝑧)) is a subordination chain and 𝑝∈ℋ[𝑎,1]∩𝒬, then
ℎ(𝑧)≺𝜑(𝑝(𝑧),𝑧𝑝^′(𝑧)) (2.5) implies that
𝑞(𝑧)≺𝑝(𝑧). (2.6) Furthermore, if 𝜑(𝑞(𝑧),𝑧𝑝^′(𝑧))=ℎ(𝑧) has a univalent solution 𝑞∈𝒬, then 𝑞 is the best subordinant.

Lemma 2.5 (see [18]). The function 𝐿(𝑧,𝑡)=𝑎₁(𝑡)𝑧+⋯ with 𝑎₁(𝑡)≠0 and lim_𝑡→∞|𝑎₁(𝑡)|=∞ is a subordination chain if and only if
Re{𝑧𝜕𝐿(𝑧,𝑡)/𝜕𝑧𝜕𝐿(𝑧,𝑡)/𝜕𝑡}>0(𝑧∈𝕌;0≤𝑡<∞). (2.7)

3. Main Results

Subordination theorem involving the integral operator 𝐼_𝛽,𝛾 defined by (1.6) is contained in Theorem 3.1 below.

Theorem 3.1. Let 𝑓,𝑔∈Σ_𝛽,𝛾. Suppose that
Re{1+𝑧𝜙^′′(𝑧)𝜙^′(𝑧)}>−𝛿(𝑧∈𝕌;𝜙(𝑧)∶=𝑧(𝑧𝑔(𝑧))^𝛽), (3.1) where
𝛿=1+|𝛾−𝛽|²−|1−(𝛾−𝛽)²|4Re{𝛾−𝛽}(Re{𝛾−𝛽}>0). (3.2) Then, the subordination
𝑧(𝑧𝑓(𝑧))^𝛽≺𝑧(𝑧𝑔(𝑧))^𝛽 (3.3) implies that
𝑧(𝑧𝐼_𝛽,𝛾(𝑓)(𝑧))^𝛽≺𝑧(𝑧𝐼_𝛽,𝛾(𝑔)(𝑧))^𝛽, (3.4) where 𝐼_𝛽,𝛾 is the integral operator defined by (1.6). Moreover, the function 𝑧(𝑧𝐼_𝛽,𝛾(𝑔)(𝑧))^𝛽 is the best dominant.

Proof. Let us define the functions 𝐹 and 𝐺 by
𝐹(𝑧)∶=𝑧(𝑧𝐼_𝛽,𝛾(𝑓)(𝑧))^𝛽,𝐺(𝑧)∶=𝑧(𝑧𝐼_𝛽,𝛾(𝑔)(𝑧))^𝛽, (3.5) respectively. Without loss of generality, we can assume that 𝐺 is analytic and univalent on 𝕌 and that
𝐺^′(𝜁)≠0(|𝜁|=1). (3.6)
We first show that if the function 𝑞 is defined by
𝑞(𝑧)∶=1+𝑧𝐺^′′(𝑧)𝐺^′(𝑧)(𝑧∈𝕌), (3.7) then
Re{𝑞(𝑧)}>0(𝑧∈𝕌). (3.8) In terms of the function 𝜙 involved in (3.1), the definition (1.6) readily yields
𝛽𝑧(𝐼_𝛽,𝛾(𝑔)(𝑧))^′𝐼_𝛽,𝛾(𝑔)(𝑧)=−(𝛾+1)+(𝛾−𝛽+1)𝜙(𝑧)𝐺(𝑧). (3.9) We also have
𝛽𝑧(𝐼_𝛽,𝛾(𝑔)(𝑧))^′𝐼_𝛽,𝛾(𝑔)(𝑧)=−(1+𝛽)+𝑧𝐺^′(𝑧)𝐺(𝑧). (3.10) By a simple calculation with (3.9) and (3.10), we obtain the relationship
1+𝑧𝜙^′′(𝑧)𝜙^′(𝑧)=𝑞(𝑧)+𝑧𝑞^′(𝑧)𝑞(𝑧)+𝛾−𝛽≡ℎ(𝑧). (3.11) From (3.1), we note that
Re{ℎ(𝑧)+𝛾−𝛽}>0(𝑧∈𝕌) (3.12) and by using Lemma 2.2, we conclude that the differential equation (3.11) has a solution 𝑞∈ℋ(𝕌) with
𝑞(0)=ℎ(0)=1. (3.13) Let us put
𝑣𝐻(𝑢,𝑣)=𝑢+𝑢+𝛾−𝛽+𝛿, (3.14) where 𝛿 is given by (3.2). From (3.1), (3.11), and (3.14), we obtain
Re{𝐻(𝑞(𝑧),𝑧𝑞^′(𝑧))}>0(𝑧∈𝕌). (3.15) Now, we proceed to show that Re{𝐻(𝑖𝑠,𝑡)}≤0 for all real 𝑠 and 𝑡≤−(1+𝑠²)/2. From (3.14), we have𝑡Re{𝐻(𝑖𝑠,𝑡)}=Re{𝑖𝑠+=𝑖𝑠+𝛾−𝛽+𝛿}𝑡Re{𝛾−𝛽}|𝛾−𝛽+𝑖𝑠|²𝐸+𝛿≤−^𝛿(𝑠)2|𝛾−𝛽+𝑖𝑠|², (3.16) where𝐸_𝛿(𝑠)∶=(Re{𝛾−𝛽}−2𝛿)𝑠²−4𝛿Im{𝛾−𝛽}𝑠−2𝛿|𝛾−𝛽|²+Re{𝛾−𝛽}. (3.17) For 𝛿 given by (3.2), we note that the coefficient of 𝑠² in the quadratic expression 𝐸_𝛿(𝑠) given by (3.17) is positive or equal to zero. Moreover, for the assumed value of 𝛿 given by (3.2), the quadratic expression 𝐸_𝛿(𝑠) by 𝑠 in (3.17) is a perfect square. Hence, from (3.16), we see that Re{𝐻(𝑖𝑠,𝑡)}≤0 for all real 𝑠 and 𝑡≤−(1+𝑠²)/2. Thus, by using Lemma 2.1, we conclude that
Re{𝑞(𝑧)}>0(𝑧∈𝕌), (3.18) that is, 𝐺 is convex in 𝕌.
Next, we prove that the subordination condition (3.3) implies that
𝐹(𝑧)≺𝐺(𝑧) (3.19) for the functions 𝐹 and 𝐺 defined by (3.5). For this purpose, we consider the function 𝐿(𝑧,𝑡) given by
𝐿(𝑧,𝑡)∶=𝛾−𝛽𝛾−𝛽+1𝐺(𝑧)+1+𝑡𝛾−𝛽+1𝑧𝐺^′(𝑧)(𝑧∈𝕌;0≤𝑡<∞). (3.20) Since 𝐺 is convex in 𝕌 and Re{𝛾−𝛽}>0, we obtain that
𝜕𝐿(𝑧,𝑡)|𝜕𝑧^𝑧0=𝐺^′𝑡(0)(1+𝛾−𝛽+1)≠0(𝑧∈𝕌;0≤𝑡<∞),Re{𝑧𝜕𝐿(𝑧,𝑡)/𝜕𝑧𝜕𝐿(𝑧,𝑡)/𝜕𝑡}=Re{𝛾−𝛽+(1+𝑡)(1+𝑧𝐺^′′(𝑧)𝐺^′(𝑧))}>0(𝑧∈𝕌). (3.21) Therefore, by virtue of Lemma 2.5, 𝐿(𝑧,𝑡) is a subordination chain. We observe from the definition of a subordination chain that
𝜙(𝑧)=𝛾−𝛽1𝛾−𝛽+1𝐺(𝑧)+𝛾−𝛽+1𝑧𝐺^′(𝑧)=𝐿(𝑧,0),𝐿(𝑧,0)≺𝐿(𝑧,𝑡)(0≤𝑡<∞). (3.22) This implies that
𝐿(𝜁,𝑡)∉𝐿(𝕌,0)=𝜙(𝕌)(𝜁∈𝜕𝕌;0≤𝑡<∞). (3.23)
Now suppose that 𝐹 is not subordinate to 𝐺, then by Lemma 2.3, there exist points 𝑧₀∈𝕌 and 𝜁₀∈𝜕𝕌 such that
𝐹(𝑧₀)=𝐺(𝜁₀),𝑧₀𝐹(𝑧₀)=(1+𝑡)𝜁₀𝐺^′(𝜁₀)(0≤𝑡<∞). (3.24) Hence, we have𝐿(𝜁₀,𝑡)=𝛾−𝛽𝛾−𝛽+1𝐺(𝜁₀)+1+𝑡𝜁𝛾−𝛽+1⁰𝐺^′(𝜁₀)=𝛾−𝛽𝛾−𝛽+1𝐹(𝜁₀1)+𝑧𝛾−𝛽+1⁰𝐹^′(𝑧₀)=𝑧₀(𝑧₀𝑓(𝑧₀))^𝛽∈𝜙(𝕌), (3.25) by virtue of the subordination condition (3.3). This contradicts the above observation that 𝐿(𝜁₀,𝑡)∉𝜙(𝕌). Therefore, the subordination condition (3.3) must imply the subordination given by (3.19). Considering 𝐹(𝑧)=𝐺(𝑧), we see that the function 𝐺(𝑧) is the best dominant. This evidently completes the proof of Theorem 3.1.

Remark 3.2. We note that 𝛿 given by (3.2) in Theorem 3.1 satisfies the inequality 0<𝛿≤1/2.

We next prove a dual problem of Theorem 3.1 in the sense that the subordinations are replaced by superordinations.

Theorem 3.3. Let 𝑓,𝑔∈Σ_𝛽,𝛾. Suppose that
Re{1+𝑧𝜙^′′(𝑧)𝜙^′(𝑧)}>−𝛿(𝑧∈𝕌;𝜙(𝑧)∶=𝑧(𝑧𝑔(𝑧))^𝛽), (3.26) where 𝛿 is given by (3.2), 𝑧(𝑧𝑓(𝑧))^𝛽 is univalent in 𝕌, and 𝑧(𝑧𝐼_𝛽,𝛾(𝑓)(𝑧))^𝛽∈𝒬, where 𝐼_𝛽,𝛾 is the integral operator defined by (1.6). Then, the superordination
𝑧(𝑧𝑔(𝑧))^𝛽≺𝑧(𝑧𝑓(𝑧))^𝛽 (3.27) implies that
𝑧(𝑧𝐼_𝛽,𝛾(𝑔)(𝑧))^𝛽≺𝑧(𝑧𝐼_𝛽,𝛾(𝑓)(𝑧))^𝛽. (3.28) Moreover, the function 𝑧(𝑧𝐼_𝛽,𝛾(𝑔)(𝑧))^𝛽 is the best subordinant.

Proof. The first part of the proof is similar to that of Theorem 3.1 and so we will use the same notation as in the proof of Theorem 3.1. Now let us define the functions 𝐹 and 𝐺, respectively, by (3.5). We first note that by using (3.9) and (3.10), we obtain𝜙(𝑧)=𝛾−𝛽1𝛾−𝛽+1𝐺(𝑧)+𝛾−𝛽+1𝑧𝐺^′(𝑧)∶=𝜑(𝐺^′(𝑧),𝑧𝐺^′(𝑧)). (3.29) After a simple calculation, (3.29) yields the relationship1+𝑧𝜙^′′(𝑧)𝜙^′(𝑧)=𝑞(𝑧)+𝑧𝑞^′(𝑧),𝑞(𝑧)+𝛾−𝛽 (3.30) where the function 𝑞 is defined by (3.7). Then, by using the same method as in the proof of Theorem 3.1, we can prove that
Re{𝑞(𝑧)}>0(𝑧∈𝕌), (3.31) that is, 𝐺 defined by (3.5) is convex(univalent) in 𝕌.
Next, we prove that the superordination condition (3.27) implies that
𝐹(𝑧)≺𝐺(𝑧)(𝑧∈𝕌). (3.32) Now, consider the function 𝐿(𝑧,𝑡) defined by
𝐿(𝑧,𝑡)∶=𝛾−𝛽𝑡𝛾−𝛽+1𝐺(𝑧)+𝛾−𝛽+1𝑧𝐺^′(𝑧)(𝑧∈𝕌;0≤𝑡<∞). (3.33) Since 𝐺 is convex and Re{𝛾−𝛽}>0, we can prove easily that 𝐿(𝑧,𝑡) is a subordination chain as in the proof of Theorem 3.1. Therefore, according to Lemma 2.4, we conclude that the superordination condition (3.27) must imply the superordination given by (3.32). Furthermore, since the differential equation (3.29) has the univalent solution 𝐺, it is the best subordinant of the given differential superordination. Therefore, we complete the proof of Theorem 3.3.

If we combine Theorems 3.1 and 3.3, then we obtain the following sandwich-type theorem.

Theorem 3.4. Let 𝑓,𝑔_𝑘∈Σ_𝛽,𝛾(𝑘=1,2). Suppose that
Re{1+𝑧𝜙_𝑘^′′(𝑧)𝜙^′_𝑘(𝑧)}>−𝛿(𝑧∈𝕌;𝜙_𝑘(𝑧)∶=𝑧(𝑧𝑔_𝑘(𝑧))^𝛽;𝑘=1,2), (3.34) where 𝛿 is given by (3.2), 𝑧(𝑧𝑓(𝑧))^𝛽 is univalent in 𝕌, and 𝑧(𝑧𝐼_𝛽,𝛾(𝑓)(𝑧))^𝛽∈𝒬, where 𝐼_𝛽,𝛾 is the integral operator defined by (1.6). Then, the subordination
𝑧(𝑧𝑔₁(𝑧))^𝛽≺𝑧(𝑧𝑓(𝑧))^𝛽≺𝑧(𝑧𝑔₂(𝑧))^𝛽 (3.35) implies that
𝑧(𝑧𝐼_𝛽,𝛾(𝑔₁)(𝑧))^𝛽≺𝑧(𝑧𝐼_𝛽,𝛾(𝑓)(𝑧))^𝛽≺𝑧(𝑧𝐼_𝛽,𝛾(𝑔₂)(𝑧))^𝛽. (3.36) Moreover, the functions 𝑧(𝑧𝐼_𝛽,𝛾(𝑔₁)(𝑧))^𝛽 and 𝑧(𝑧𝐼_𝛽,𝛾(𝑔₂)(𝑧))^𝛽 are the best subordinant and the best dominant, respectively.

The assumption of Theorem 3.4, that the functions 𝑧(𝑧𝑓(𝑧))^𝛽 and 𝑧(𝑧𝐼_𝛽,𝛾(𝑓)(𝑧))^𝛽 need to be univalent in 𝕌, may be replaced by another condition in the following result.

Corollary 3.5. Let 𝑓,𝑔_𝑘∈Σ_𝛽,𝛾(𝑘=1,2). Suppose that the condition (3.34) is satisfied and
Re{1+𝑧𝜓^′′(𝑧)𝜓^′(𝑧)}>−𝛿(𝑧∈𝕌;𝜓(𝑧)∶=𝑧(𝑧𝑓(𝑧))^𝛽;𝑧𝑓(𝑧)∈𝒬), (3.37) where 𝛿 is given by (3.2). Then, the subordination
𝑧(𝑧𝑔₁(𝑧))^𝛽≺𝑧(𝑧𝑓(𝑧))^𝛽≺𝑧(𝑧𝑔₂(𝑧))^𝛽 (3.38) implies that
𝑧(𝑧𝐼_𝛽,𝛾(𝑔₁)(𝑧))^𝛽≺𝑧(𝑧𝐼_𝛽,𝛾(𝑓)(𝑧))^𝛽≺𝑧(𝑧𝐼_𝛽,𝛾(𝑔₂)(𝑧))^𝛽, (3.39) where 𝐼_𝛽,𝛾 is the integral operator defined by (1.6). Moreover, the functions 𝑧(𝑧𝐼_𝛽,𝛾(𝑔₁)(𝑧))^𝛽 and 𝑧(𝑧𝐼_𝛽,𝛾(𝑔₂)(𝑧))^𝛽 are the best subordinant and the best dominant, respectively.

Proof. In order to prove Corollary 3.5, we have to show that the condition (3.37) implies the univalence of 𝜓(𝑧) and
𝐹(𝑧)∶=𝑧(𝑧𝐼_𝛽,𝛾(𝑓)(𝑧))^𝛽. (3.40) Since 0<𝛿≤1/2 from Remark 3.2, the condition (3.37) means that 𝜓 is a close-to-convex function in 𝕌 (see [19]) and hence 𝜓 is univalent in 𝕌. Furthermore, by using the same techniques as in the proof of Theorem 3.4, we can prove the convexity (univalence) of 𝐹 and so the details may be omitted. Therefore, by applying Theorem 3.4, we obtain Corollary 3.5.

By setting 𝛾−𝛽=1 in Theorem 3.4, so that 𝛿=1/2, we deduce the following consequence of Theorem 3.4.

Corollary 3.6. Let 𝑓,𝑔_𝑘∈Σ_𝛽,𝛽+1(𝑘=1,2). Suppose that
Re{1+𝑧𝜙_𝑘^′′(𝑧)𝜙^′_𝑘1(𝑧)}>−2(𝑧∈𝕌;𝜙_𝑘(𝑧)∶=𝑧(𝑧𝑔_𝑘(𝑧))^𝛽;𝑘=1,2), (3.41)𝑧(𝑧𝑓(𝑧))^𝛽 is univalent in 𝕌, and 𝑧(𝑧𝐼_𝛽,𝛽+1(𝑓)(𝑧))^𝛽∈𝒬, where 𝐼_𝛽,𝛽+1 is the integral operator defined by (1.6) with 𝛾=𝛽+1. Then, the subordination
𝑧(𝑧𝑔₁(𝑧))^𝛽≺𝑧(𝑧𝑓(𝑧))^𝛽≺𝑧(𝑧𝑔₂(𝑧))^𝛽 (3.42) implies that
𝑧(𝑧𝐼_𝛽,𝛽+1(𝑔₁)(𝑧))^𝛽≺𝑧(𝑧𝐼_𝛽,𝛽+1(𝑓)(𝑧))^𝛽≺𝑧(𝑧𝐼_𝛽,𝛽+1(𝑔₂)(𝑧))^𝛽. (3.43) Moreover, the functions 𝑧(𝑧𝐼_𝛽,𝛽+1(𝑔₁)(𝑧))^𝛽 and 𝑧(𝑧𝐼_𝛽,𝛽+1(𝑔₂)(𝑧))^𝛽 are the best subordinant and the best dominant, respectively.

If we take 𝛾−𝛽=1+𝑖 in Theorem 3.4, then we are easily led to the following result.

Corollary 3.7. Let 𝑓,𝑔_𝑘∈Σ_{𝛽,𝛽+1+𝑖}(𝑘=1,2). Suppose that
Re{1+𝑧𝜙_𝑘^′′(𝑧)𝜙^′_𝑘√(𝑧)}>−3−54(𝑧∈𝕌;𝜙_𝑘(𝑧)∶=𝑧(𝑧𝑔_𝑘(𝑧))^𝛽;𝑘=1,2), (3.44)(𝑧𝑓(𝑧))^𝛽 is univalent in 𝕌, and (𝑧𝐼_{𝛽,𝛽+1+𝑖}(𝑓)(𝑧))^𝛽∈𝒬, where 𝐼_{𝛽,𝛽+1+𝑖} is the integral operator defined by (1.6) with 𝛾=𝛽+1+𝑖. Then, the subordination:
𝑧(𝑧𝑔₁(𝑧))^𝛽≺𝑧(𝑧𝑓(𝑧))^𝛽≺𝑧(𝑧𝑔₂(𝑧))^𝛽 (3.45) implies that
𝑧(𝑧𝐼_{𝛽,𝛽+1+𝑖}(𝑔₁)(𝑧))^𝛽≺𝑧(𝑧𝐼_{𝛽,𝛽+1+𝑖}(𝑓)(𝑧))^𝛽≺𝑧(𝑧𝐼_{𝛽,𝛽+1+𝑖}(𝑔₂)(𝑧))^𝛽. (3.46) Moreover, the functions 𝑧(𝑧𝐼_{𝛽,𝛽+1+𝑖}(𝑔₁)(𝑧))^𝛽 and 𝑧(𝑧𝐼_{𝛽,𝛽+1+𝑖}(𝑔₂)(𝑧))^𝛽 are the best subordinant and the best dominant, respectively.

Acknowledgments

The authors would like to express their gratitude to Professor S. Stević for his valuable suggestions. This research was supported by Kyungsung University Grants in 2008.