Sandwich-Type Theorems for a Class of Multiplier Transformations Associated with the Noor Integral Operators

Abstract

We obtain some subordination- and superordination-preserving properties for a class of multiplier transformations associated with Noor integral operators defined on the space of normalized analytic functions in the open unit disk. The sandwich-type theorems for these transformations are 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 and nonnegative integer 𝑛, letℋ[𝑎,𝑛]=𝑓∈ℋ∶𝑓(𝑧)=𝑎+𝑎_𝑛𝑧^𝑛+𝑎_𝑛+1𝑧^𝑛+1+⋯. (1.1) We also denote 𝒜 by the subclass of ℋ[𝑎,1] with the usual normalization 𝑓(0)=𝑓^′(0)−1=0.

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: 𝜙𝑝(𝑧),𝑧𝑝^′(𝑧)≺ℎ(𝑧)(𝑧∈𝕌), (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 𝑞 that satisfies 𝑞≺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)≺𝜑𝑝(𝑧),𝑧𝑝^′(𝑧)(𝑧∈𝕌), (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 𝑞 that satisfies 𝑞≺𝑞 for all subordinants 𝑞 of (1.3) is said to be the best subordinant.

Definition 1.3 (see [2]). We denote by 𝒬 the class of functions 𝑓 that are analytic and injective on 𝕌⧵𝐸(𝑓), where 𝐸𝑓=𝜁∈𝜕𝕌∶lim_𝑧→𝜁=∞, (1.4) and are such that 𝑓^′(𝜁)≠0 for 𝜁∈𝜕𝕌⧵𝐸(𝑓).
Following Komatu [3], we introduce the integral operator ℒ^𝜆(𝑐)∶𝒜→𝒜 defined by ℒ^𝜆(𝑐)𝑓(𝑧)𝑐∶=^𝜆Γ(𝜆)¹₀𝑡^𝑐−21log𝑡^𝜆−1𝑓(𝑡𝑧)𝑑𝑡Re{𝑐}>0;𝜆≥0;𝑓∈𝒜, (1.5) where the symbol Γ stands the Gamma function. We also note that the operator ℒ^𝜆(𝑐)𝑓(𝑧) defined by (1.5) can be expressed by the series expansion as follows: ℒ^𝜆(𝑐)𝑓(𝑧)=𝑧+^∞_𝑘=2𝑐𝑐+𝑘−1^𝜆𝑎_𝑘𝑧^𝑘. (1.6) Obviously, we have, for 𝜆,𝜈≥0, ℒ^𝜆(𝑐)ℒ^𝜈(𝑐)𝑓(𝑧)=ℒ^𝜆+𝜈(𝑐)𝑓(𝑧). (1.7) In particular, the operator ℒ^𝜆(2) is closely related to the multiplier transformation studied earlier by Flett [4]. Various interesting properties of the operator ℒ^𝜆(2) have been studied by Jung et al. [5] and Liu [6]. We also note from (1.6) that we can define the operator ℒ^𝜆(𝑐) for any real number 𝜆.
Let 𝑓^𝜆_𝑐(𝑧)=𝑧+^∞_𝑘=2𝑐𝑐+𝑘−1^𝜆𝑧^𝑘(Re{𝑐}>0;𝜆≥0) (1.8) and let 𝑓_𝑐^𝜆,𝜇 be defined such that 𝑓^𝜆_𝑐(𝑧)∗𝑓_𝑐^𝜆,𝜇(𝑧)=𝑧(1−𝑧)^𝜇𝜇>0;𝑧∈𝕌, (1.9) where the symbol ∗ stands for the Hadamard product (or convolution). Then, motivated essentially by the Noor integral operator [7] (see also [8–11]), we now introduce the operator 𝐼^𝜅_𝜆,𝜇∶𝒜→𝒜, which are defined here by 𝐼_𝑐^𝜆,𝜇𝑓(𝑧)=𝑓_𝑐^𝜆,𝜇∗𝑓(𝑧)𝑓∈𝒜;Re{𝑐}>0;𝜆≥0;𝜇>0. (1.10) In view of (1.9) and (1.10), we obtain the following relations: 𝑧𝐼_𝑐^𝜆,𝜇𝑓(𝑧)^′=𝑐𝐼_𝑐^𝜆+1,𝜇𝑓(𝑧)−(𝑐−1)𝐼_𝑐^𝜆,𝜇𝑓(𝑧),𝑧𝐼_𝑐^𝜆,𝜇𝑓(𝑧)^′=𝜇𝐼_𝑐^𝜆,𝜇+1𝑓(𝑧)−𝜇−1𝐼_𝑐^𝜆,𝜇𝑓(𝑧). (1.11)(1.12)
Making use of the principle of subordination between analytic functions, Miller et al. [12] investigated some subordination theorems involving certain integral operators for analytic functions in 𝕌 (see, also [13]). Moreover, Miller and Mocanu [2] considered differential superordinations, as the dual concept of differential subordinations (see also [14]). In the present paper, we obtain the subordination- and superordination-preserving properties of the multiplier transformations 𝐼_𝑐^𝜆,𝜇 defined by (1.10) with the sandwich-type theorems.
The following lemmas will be required in our present investigation.

Lemma 1.4 (see [15]). Suppose that the function 𝐻∶ℂ²→ℂ satisfies the condition: Re{𝐻(𝑖𝑠,𝑡)}≤0, (1.13) 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 Re𝐻𝑝(𝑧),𝑧𝑝^′(𝑧)>0(𝑧∈𝕌), (1.14) then Re{𝑝(𝑧)}>0 in 𝕌.

Lemma 1.5 (see [16]). 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)(𝑧)(𝑧)(0)𝛽𝑞+𝛾=ℎ𝑧∈𝕌;𝑞=𝑐 (1.15) is analytic in 𝕌 and satisfies Re{𝛽𝑞(𝑧)+𝛾}>0(𝑧∈𝕌).

Lemma 1.6 (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), 𝑝𝑧₀=𝑞𝜁₀,𝑧₀𝑝^′𝑧₀=𝑚𝜁₀𝑞^′𝜁₀(𝑚≥𝑛). (1.16)

Lemma 1.7 (see [2]). Let 𝑞∈ℋ[𝑎,1], let 𝜑∶ℂ²→ℂ, and set 𝜑(𝑞(𝑧),𝑧𝑞^′(𝑧))≡ℎ(𝑧). If 𝐿(𝑧,𝑡)=𝜑(𝑞(𝑧),𝑡𝑧𝑞^′(𝑧)) is a subordination chain and 𝑝∈ℋ[𝑎,1]∩𝒬, then ℎ(𝑧)≺𝜑𝑝(𝑧),𝑧𝑝^′(𝑧)(𝑧∈𝕌) (1.17) implies that 𝑞(𝑧)≺𝑝(𝑧)(𝑧∈𝕌). (1.18) Furthermore, if 𝜑(𝑞(𝑧),𝑧𝑝^′(𝑧))=ℎ(𝑧) has a univalent solution 𝑞∈𝒬, then 𝑞 is the best subordinant.

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 𝑧∈𝕌 and 0≤𝑠<𝑡.

Lemma 1.8 (see [17]). The function 𝐿(𝑧,𝑡)=𝑎₁(𝑡)𝑧+⋯ with 𝑎₁(𝑡)≠0 and lim_𝑡→∞|𝑎₁(𝑡)|=∞. Suppose that 𝐿(⋅;𝑡) ia analytic in 𝕌 for all 𝑡≥0, 𝐿(𝑧;⋅) is continuously differentiable on [0,∞) for all 𝑧∈𝕌. If 𝐿(𝑧;𝑡) satisfies |𝐿(𝑧;𝑡)|≤𝐾₀𝑎₁(𝑡)|𝑧|<𝑟₀<1;0≤𝑡<∞ (1.19) for some positive constants 𝐾₀ and 𝑟₀ and ℜ𝑧𝜕𝐿(𝑧,𝑡)/𝜕𝑧()(),𝜕𝐿𝑧,𝑡/𝜕𝑡>0𝑧∈𝕌;0≤𝑡<∞ (1.20) then 𝐿(𝑧;𝑡) is a subordination chain.

2. Main Results

Firstly, we begin by proving the following subordination theorem involving the multiplier transformation 𝐼_𝑐^𝜆,𝜇 defined by (1.10).

Theorem 2.1. Let 𝑓,𝑔∈𝒜. Suppose that Re1+𝑧𝜙^′′(𝑧)𝜙^′(𝑧)>−𝛿𝜙(𝑧)∶=(1−𝛼)𝐼_𝑐^𝜆+1,𝜇𝑔(𝑧)+𝛼𝐼_𝑐^𝜆,𝜇𝑔(𝑧);0≤𝛼<1;𝑧∈𝕌, (2.1) where ()𝛿=1−𝛼²+|𝑐−1+𝛼|²−(1−𝛼)²−(𝑐−1+𝛼)²4(){}({}).1−𝛼Re𝑐−1+𝛼Re𝑐−1+𝛼>0 (2.2) Then the subordination: (1−𝛼)𝐼_𝑐^𝜆+1,𝜇𝑓(𝑧)+𝛼𝐼_𝑐^𝜆,𝜇𝑓(𝑧)≺𝜙(𝑧)(𝑧∈𝕌) (2.3) implies that 𝐼_𝑐^𝜆,𝜇𝑓(𝑧)≺𝐼_𝑐^𝜆,𝜇𝑔(𝑧)(𝑧∈𝕌). (2.4) Moreover, the function 𝐼_𝑐^𝜆,𝜇𝑔(𝑧) is the best dominant.

Proof. Let us define the functions 𝐹 and 𝐺, respectively, by 𝐹(𝑧)∶=𝐼_𝑐^𝜆,𝜇𝑓(𝑧),𝐺(𝑧)∶=𝐼_𝑐^𝜆,𝜇𝑔(𝑧), (2.5)
We first show that if the function 𝑞 is defined by 𝑞(𝑧)∶=1+𝑧𝐺^′′(𝑧)𝐺^′(𝑧)(𝑧∈𝕌), (2.6) then Re𝑞(𝑧)>0(𝑧∈𝕌). (2.7) Taking the logarithmic differentiation on both sides of the second equation in (2.5) and using (1.11) for 𝑔∈𝒜, we obtain 𝑐𝜙(𝑧)=(𝑐−1+𝛼)𝐺(𝑧)+(1−𝛼)𝑧𝐺^′(𝑧), (2.8) which, in conjunction with (2.8), yields the relationship: 1+𝑧𝜙^′′(𝑧)𝜙^′(𝑧)=1+𝑧𝐺^′′(𝑧)𝐺^′(𝑧)+𝑧𝑞^′(𝑧)𝑞(𝑧)+()/()(𝑧)+𝑐−1+𝛼1−𝛼=𝑞𝑧𝑞^′(𝑧)𝑞(𝑧)+()/()(𝑧).𝑐−1+𝛼1−𝛼≡ℎ (2.9) From (2.1), we have Reℎ(𝑧)+𝑐−1+𝛼(),1−𝛼>0𝑧∈𝕌 (2.10) and by using Lemma 1.5, we conclude that the differential equation (2.9) has a solution 𝑞∈ℋ(𝕌) with 𝑞(0)=ℎ(0)=1.
Let us put 𝐻()𝑣𝑢,𝑣=𝑢+𝑢+(𝑐−1+𝛼)/(1−𝛼)+𝛿, (2.11) where 𝛿 is given by (2.2). From (2.1), (2.9), and (2.11), we obtain Re𝐻𝑞(𝑧),𝑧𝑞^′(𝑧)>0(𝑧∈𝕌). (2.12) Now we proceed to show that Re{𝐻(𝑖𝑠,𝑡)}≤0 for all real 𝑠 and 𝑡≤−(1+𝑠²)/2. From (2.11), we have {𝐻(𝑡Re𝑖𝑠,𝑡)}=Re𝑖𝑠+()/()=𝑡()/()𝑖𝑠+𝑐−1+𝛼1−𝛼+𝛿𝑐−1+𝛼1−𝛼|(𝑐−1+𝛼)/(1−𝛼)+𝑖𝑠|²𝐸+𝛿≤−^𝛿(𝑠)2|(𝑐−1+𝛼)/(1−𝛼)+𝑖𝑠|², (2.13) where 𝐸_𝛿(𝑠){}∶=Re𝑐−1+𝛼()𝑠1−𝛼−2𝛿²−4𝛿𝑠Im{𝑐−1+𝛼}()()1−𝛼−2𝛿𝑐−1+𝛼(1−𝛼)²+Re{𝑐−1+𝛼}().1−𝛼 (2.14) For 𝛿 given by (2.2), we can prove easily that the expression 𝐸_𝛿(𝑠) given by (2.14) is positive or equal to zero. Moreover, the quadratic expression by 𝑠 in (2.14) is a perfect square for the assumed value of 𝛿. Hence from (2.13), we see that Re{𝐻(𝑖𝑠,𝑡)}≤0 for all real 𝑠 and 𝑡≤−(1+𝑠²)/2. Thus, by using Lemma 1.4, we conclude that Re{𝑞(𝑧)}>0 for all 𝑧∈𝕌. That is, 𝑞 is convex in 𝕌.
Next, we prove that the subordination condition (2.3) implies that 𝐹(𝑧)≺𝐺(𝑧)(𝑧∈𝕌) (2.15) for the functions 𝐹 and 𝐺 defined by (2.5). Without loss of generality, we can assume that 𝐺 is analytic and univalent on 𝕌 and that 𝐺^′(𝜁)≠0(|𝜁|=1). Now we consider the function 𝐿(𝑧,𝑡) given by 𝐿(𝑧,𝑡)∶=𝑐−1+𝛼𝑐𝐺(𝑧)+(1−𝛼)(1+𝑡)𝑐𝑧𝐺^′(𝑧)(𝑧∈𝕌;0≤𝑡<∞). (2.16) We note that 𝜕𝐿(𝑧,𝑡)𝜕𝑧_𝑧=0=𝐺^′(0)(𝑐−1+𝛼)+(1−𝛼)(1+𝑡)𝑐≠0(0≤𝑡<∞;Re{𝑐−1+𝛼}>0). (2.17) This shows that the function 𝐿(𝑧,𝑡)=𝑎₁(𝑡)𝑧+⋯ (2.18) satisfies the condition 𝑎₁(𝑡)≠0 for all 𝑡∈[0,∞). By using the well-known growth and distortion theorems for convex functions, it is easy to check that the first part of Lemma 1.8 is satisfied. Furthermore, we have ()Re𝑧𝜕𝐿𝑧,𝑡/𝜕𝑧𝜕𝐿(𝑧,𝑡)/𝜕𝑡=Re𝑐−1+𝛼+()1−𝛼1+𝑡1+𝑧𝐺^′′(𝑧)𝐺^′(𝑧)>0, (2.19) since 𝐺 is convex and Re{𝑐−1+𝛼}>0. Therefore, by virtue of Lemma 1.8, 𝐿(𝑧,𝑡) is a subordination chain. We observe from the definition of a subordination chain that 𝜙(𝑧)=𝑐−1+𝛼𝑐𝐺(𝑧)+1−𝛼𝑐𝑧𝐺^′(𝑧)(),𝐿()().=𝐿𝑧,0𝑧,0≺𝐿𝑧,𝑡)(𝑧∈𝕌;0≤𝑡<∞ (2.20) This implies that 𝐿(𝜁,𝑡)∉𝐿(𝕌,0)=𝜙(𝕌)(𝜁∈𝜕𝕌;0≤𝑡<∞). (2.21)
Now suppose that 𝐹 is not subordinate to 𝐺, then by Lemma 1.6, there exists points 𝑧₀∈𝕌 and 𝜁₀∈𝜕𝕌 such that 𝐹𝑧₀=𝐺𝜁₀,𝑧₀𝐹𝑧₀=(1+𝑡)𝜁₀𝐺^′𝜁₀(0≤𝑡<∞). (2.22) Hence we have 𝐿𝜁₀,𝑡=𝑐−1+𝛼𝑐𝐺𝜁₀+(1−𝛼)(1+𝑡)𝑐𝜁₀𝐺^′𝜁₀=𝑐−1+𝛼𝑐𝐹𝑧₀+1−𝛼𝑐𝑧₀𝐹^′𝑧₀=(1−𝛼)𝐼_𝑐^𝜆+1,𝜇𝑓𝑧₀+𝛼𝐼_𝑐^𝜆,𝜇𝑓𝑧₀∈𝜙(𝕌), (2.23) by virtue of the subordination condition (2.3). This contracts the above observation that 𝐿(𝜁₀,𝑡)∉𝜙(𝕌). Therefore, the subordination condition (2.3) must imply the subordination given by (2.15). Considering 𝐹(𝑧)=𝐺(𝑧), we see that the function 𝐺(𝑧) is the best dominant. This evidently completes the proof of Theorem 2.1.

Remark 2.2. We note that 𝛿 given by (2.2) in Theorem 2.1 satisfies the inequality 0<𝛿≤1/2.

We next prove a dual problem of Theorem 2.1, in the sense that the subordinations are replaced by superordinations.

Theorem 2.3. Let 𝑓,𝑔∈𝒜. Suppose that Re1+𝑧𝜙^′′(𝑧)𝜙^′(𝑧)>−𝛿𝜙(𝑧)∶=(1−𝛼)𝐼_𝑐^𝜆+1,𝜇𝑔(𝑧)+𝛼𝐼_𝑐^𝜆,𝜇𝑔(𝑧);0≤𝛼<1;𝑧∈𝕌, (2.24) where 𝛿 is given by (2.2), and the function 𝐼_𝑐^𝜆+1,𝜇𝑓(𝑧) is univalent in 𝕌 and 𝐼_𝑐^𝜆,𝜇𝑓(𝑧)∈𝒬. Then the superordination: 𝜙(𝑧)≺(1−𝛼)𝐼_𝑐^𝜆+1,𝜇𝑓(𝑧)+𝛼𝐼_𝑐^𝜆,𝜇𝑓(𝑧)(𝑧∈𝕌) (2.25) implies that 𝐼_𝑐^𝜆,𝜇𝑔(𝑧)𝑧≺𝐼_𝑐^𝜆,𝜇𝑓(𝑧)(𝑧∈𝕌). (2.26) Moreover, the function 𝐼_𝑐^𝜆,𝜇𝑔(𝑧) is the best subordinant.

Proof. Let us define the functions 𝐹 and 𝐺, respectively, by (2.5). We first note that, if the function 𝑞 is defined by (2.6), by using (2.8), then we obtain 𝜙(𝑧)=𝑐−1+𝛼𝑐𝐺(𝑧)+1−𝛼𝑐𝑧𝐺^′(𝑧)=∶𝜑𝐺(𝑧),𝑧𝐺^′(𝑧). (2.27) After a simple calculation, (2.27) yields the relationship: 1+𝑧𝜙^′′(𝑧)𝜙^′(𝑧)=𝑞(𝑧)+𝑧𝑞^′(𝑧)𝑞(𝑧)+()/().𝑐−1+𝛼1−𝛼 (2.28) Then by using the same method as in the proof of Theorem 2.1, we can prove that Re{𝑞(𝑧)}>0 for all 𝑧∈𝕌. That is, 𝐺 defined by (2.6) is convex (univalent) in 𝕌.
Next, we prove that the subordination condition (2.27) implies that 𝐹(𝑧)≺𝐺(𝑧)(𝑧∈𝕌) (2.29) for the functions 𝐹 and 𝐺 defined by (2.5). Now consider the function 𝐿(𝑧,𝑡) defined by 𝐿(𝑧,𝑡)∶=𝑐−1+𝛼𝑐𝐺(𝑧)+(1−𝛼)𝑡𝑐𝑧𝐺^′(𝑧)(𝑧∈𝕌;0≤𝑡<∞). (2.30) Since 𝐺 is convex and Re{𝑐−1+𝛼}>0, we can prove easily that 𝐿(𝑧,𝑡) is a subordination chain as in the proof of Theorem 2.1. Therefore according to Lemma 1.7, we conclude that the superordination condition (2.27) must imply the superordination given by (2.29). Furthermore, since the differential equation (2.27) has the univalent solution 𝐺, it is the best subordinant of the given differential superordination. Therefore we complete the proof of Theorem 2.3.

If we combine this Theorems 2.1 and 2.3, then we obtain the following sandwich-type theorem.

Theorem 2.4. Let 𝑓,𝑔_𝑘∈𝒜(𝑘=1,2). Suppose that Re1+𝑧𝜙_𝑘^′′(𝑧)𝜙^′_𝑘(𝑧)>−𝛿𝜙_𝑘(𝑧)∶=(1−𝛼)𝐼_𝑐^𝜆+1,𝜇𝑔_𝑘(𝑧)+𝛼𝐼_𝑐^𝜆,𝜇𝑔_𝑘(𝑧);𝑘=1,2;0≤𝛼<1;𝑧∈𝕌, (2.31) where 𝛿 is given by (2.2), and the function 𝐼_𝑐^𝜆+1,𝜇𝑓(𝑧) is univalent in 𝕌 and 𝐼_𝑐^𝜆,𝜇𝑓(𝑧)∈𝒬. Then the subordination relation: 𝜙₁(𝑧)≺𝐼_𝑐^𝜆+1,𝜇𝑓(𝑧)≺𝜙₂(𝑧)(𝑧∈𝕌) (2.32) implies that 𝐼_𝑐^𝜆,𝜇𝑔_𝑘(𝑧)≺𝐼_𝑐^𝜆,𝜇𝑓(𝑧)≺𝐼_𝑐^𝜆,𝜇𝑔₂(𝑧)(𝑧∈𝕌). (2.33) Moreover, the functions 𝐼_𝑐^𝜆,𝜇𝑔₁(𝑧) and 𝐼_𝑐^𝜆,𝜇𝑔₂(𝑧) are the best subordinant and the best dominant, respectively.

The assumption of Theorem 2.4, that the functions 𝐼_𝑐^𝜆+1,𝜇𝑓(𝑧)and𝐼_𝑐^𝜆,𝜇𝑓(𝑧) need to be univalent in 𝕌, may be replaced by another conditions in the following result.

Corollary 2.5. Let 𝑓,𝑔_𝑘∈𝒜(𝑘=1,2). Suppose that the condition (2.31) is satisfied and Re1+𝑧𝜓^′′(𝑧)𝜓^′(𝑧)>−𝛿𝜓(𝑧)∶=(1−𝛼)𝐼_𝑐^𝜆+1,𝜇𝑓(𝑧)+𝛼𝐼_𝑐^𝜆,𝜇𝑓(𝑧);0≤𝛼<1;𝑧∈𝕌;𝑓∈𝒬, (2.34) where 𝛿 is given by (2.2). Then the subordination relation: 𝜙₁(𝑧)≺𝜓(𝑧)≺𝜙₂(𝑧)(𝑧∈𝕌) (2.35) implies that 𝐼_𝑐^𝜆,𝜇𝑔_𝑘(𝑧)≺𝐼_𝑐^𝜆,𝜇𝑓(𝑧)≺𝐼_𝑐^𝜆,𝜇𝑔₂(𝑧)(𝑧∈𝕌). (2.36) Moreover, the functions 𝐼_𝑐^𝜆,𝜇𝑔₁(𝑧) and 𝐼_𝑐^𝜆,𝜇𝑔₂(𝑧) are the best subordinant and the best dominant, respectively.

Proof. In order to prove Corollary 2.5, we have to show that the condition (2.34) implies the univalence of 𝜓(𝑧) and 𝐹(𝑧)∶=𝐼_𝑐^𝜆,𝜇𝑓(𝑧). Since 0<𝛿≤1/2 from Remark 2.2, the condition (2.34) means that 𝜓 is a close-to-convex function in 𝕌 (see [18]) and hence 𝜓 is univalent in 𝕌. Furthermore, by using the same techniques as in the proof of Theorem 2.4, we can prove the convexity (univalence) of 𝐹 and so the details may be omitted. Therefore, by applying Theorem 2.4, we obtain Corollary 2.5.

By setting 𝑐=2 and 𝛼=0 in Theorem 2.4, so that 𝛿=1/2, we deduce the following consequence of Theorem 2.4.

Corollary 2.6. Let 𝑓,𝑔_𝑘∈𝒜(𝑘=1,2). Suppose that Re1+𝑧𝜙_𝑘^′′(𝑧)𝜙^′_𝑘(𝑧)1>−2𝜙_𝑘(𝑧)∶=𝐼₂^𝜆+1,𝜇𝑔_𝑘(𝑧);𝑘=1,2;𝑧∈𝕌, (2.37) and the function 𝐼₂^𝜆+1,𝜇𝑓(𝑧) is univalent functions in 𝕌 and 𝐼₂^𝜆,𝜇𝑓(𝑧)∈𝒬. Then the subordination relation: 𝐼₂^𝜆+1,𝜇𝑔₁(𝑧)≺𝐼₂^𝜆+1,𝜇𝑓(𝑧)≺𝐼₂^𝜆+1,𝜇𝑔₂(𝑧)(𝑧∈𝕌) (2.38) implies that 𝐼₂^𝜆,𝜇𝑔₁(𝑧)≺𝐼₂^𝜆,𝜇𝑓(𝑧)≺𝐼₂^𝜆,𝜇𝑔₂(𝑧)(𝑧∈𝕌) (2.39) Moreover, the functions 𝐼₂^𝜆,𝜇𝑔₁(𝑧) and 𝐼₂^𝜆,𝜇𝑔₂(𝑧) are the best subordinant and the best dominant, respectively.

If we take 𝑐=2+𝑖 and 𝛼=0 in Theorem 2.4, then we easily to lead to the following result.

Corollary 2.7. Let 𝑓,𝑔_𝑘∈𝒜(𝑘=1,2). Suppose that Re1+𝑧𝜙_𝑘^′′(𝑧)𝜙^′_𝑘(𝑧)√>−3−54𝜙_𝑘(𝑧)∶=𝐼^𝜆,𝜇_2+𝑖𝑔_𝑘(𝑧);𝑘=1,2;𝑧∈𝕌, (2.40) and the function 𝐼^𝜆+1,𝜇_2+𝑖𝑓(𝑧) is univalent functions in 𝕌 and 𝐼^𝜆,𝜇_2+𝑖𝑓(𝑧)∈𝒬. Then the subordination relation: 𝐼^𝜆+1,𝜇_2+𝑖𝑔₁(𝑧)≺𝐼^𝜆+1,𝜇_2+𝑖𝑓(𝑧)≺𝐼^𝜆+1,𝜇_2+𝑖𝑔₂(𝑧)(𝑧∈𝕌) (2.41) implies that 𝐼^𝜆,𝜇_2+𝑖𝑔₁(𝑧)≺𝐼^𝜆,𝜇_2+𝑖𝑓(𝑧)≺𝐼^𝜆,𝜇_2+𝑖𝑔₂(𝑧)(𝑧∈𝕌). (2.42) Moreover, the functions 𝐼^𝜆,𝜇_2+𝑖𝑔₁(𝑧) and 𝐼^𝜆,𝜇_2+𝑖𝑔₂(𝑧) are the best subordinant and the best dominant, respectively.

The proof of Theorem 2.8 below is similar to that of Theorem 2.4 by using (1.12) and so the details may be omitted.

Theorem 2.8. Let 𝑓,𝑔_𝑘∈𝒜(𝑘=1,2). Suppose that Re1+𝑧𝜙_𝑘^′′(𝑧)𝜙^′_𝑘(𝑧)>−𝛿𝜙_𝑘(𝑧)∶=(1−𝛼)𝐼_𝑐^𝜆,𝜇+1𝑔_𝑘(𝑧)+𝛼𝐼_𝑐^𝜆,𝜇𝑔_𝑘(𝑧);𝑘=1,2;0≤𝛼<1;𝑧∈𝕌, (2.43) where 𝛿 is given by (2.2) with 𝑐=𝜇, and the function 𝐼_𝑐^𝜆,𝜇+1𝑓(𝑧) is univalent in 𝕌 and 𝐼_𝑐^𝜆,𝜇𝑓(𝑧)∈𝒬. Then the subordination relation: 𝜙₁(𝑧)≺𝐼_𝑐^𝜆,𝜇+1𝑓(𝑧)≺𝜙₂(𝑧)(𝑧∈𝕌) (2.44) implies that 𝐼_𝑐^𝜆,𝜇𝑔_𝑘(𝑧)≺𝐼_𝑐^𝜆,𝜇𝑓(𝑧)≺𝐼_𝑐^𝜆,𝜇𝑔₂(𝑧)(𝑧∈𝕌). (2.45) Moreover, the functions 𝐼_𝑐^𝜆,𝜇𝑔₁(𝑧) and 𝐼_𝑐^𝜆,𝜇𝑔₂(𝑧) are the best subordinant and the best dominant, respectively.

By using a similar method given in the proof of Theorems 2.4 and 2.8, we have the corresponding two theorems below.

Theorem 2.9. Let 𝑓,𝑔_𝑘∈𝒜(𝑘=1,2) with the additional condition 𝑔_𝑘^′′(𝑧)≠0. Suppose that Re1+𝑧𝜙_𝑘^′′(𝑧)𝜙^′_𝑘(𝑧)>−𝛿𝜙_𝑘(𝑧)()𝐼∶=1−𝛼_𝑐^𝜆+1,𝜇𝑔_𝑘(𝑧)+𝛼𝐼_𝑐^𝜆,𝜇𝑔_𝑘(𝑧)𝑧;𝑘=1,2;0≤𝛼<1;𝑧∈𝕌, (2.46) where 𝛿 is given by ()𝛿=1−𝛼²+|𝑐|²−(1−𝛼)²−𝑐²4(){𝑐}({𝑐}),1−𝛼ReRe>0 (2.47) and the function 𝐼_𝑐^𝜆+1,𝜇𝑓(𝑧) is univalent in 𝕌 and 𝐼_𝑐^𝜆,𝜇𝑓(𝑧)∈𝒬. Then the subordination relation: 𝜙₁(𝑧)≺𝐼_𝑐^𝜆+1,𝜇𝑓(𝑧)𝑧≺𝜙₂(𝑧)(𝑧∈𝕌) (2.48) implies that 𝐼_𝑐^𝜆,𝜇𝑔₁(𝑧)𝑧≺𝐼_𝑐^𝜆,𝜇𝑓(𝑧)𝑧≺𝐼_𝑐^𝜆,𝜇𝑔₂(𝑧)𝑧(𝑧∈𝕌). (2.49) Moreover, the functions 𝐼_𝑐^𝜆,𝜇𝑔₁(𝑧)/𝑧 and 𝐼_𝑐^𝜆,𝜇𝑔₂(𝑧)/𝑧 are the best subordinant and the best dominant, respectively.

Theorem 2.10. Let 𝑓,𝑔_𝑘∈𝒜(𝑘=1,2) with the additional condition 𝑔_𝑘^′′(𝑧)≠0. Suppose that Re1+𝑧𝜙_𝑘^′′(𝑧)𝜙^′_𝑘(𝑧)>−𝛿𝜙_𝑘(𝑧)()𝐼∶=1−𝛼_𝑐^𝜆,𝜇+1𝑔_𝑘(𝑧)+𝛼𝐼_𝑐^𝜆,𝜇𝑔_𝑘(𝑧)𝑧;𝑘=1,2;0≤𝛼<1;𝑧∈𝕌, (2.50) where 𝛿 is given by (2.47) with 𝑐=𝜇, and the function 𝐼_𝑐^𝜆,𝜇+1𝑓(𝑧)/𝑧 is univalent in 𝕌 and 𝐼_𝑐^𝜆,𝜇𝑓(𝑧)/𝑧∈𝒬. Then the subordination relation: 𝜙₁(𝑧)≺𝐼_𝑐^𝜆,𝜇+1𝑓(𝑧)𝑧≺𝜙₂(𝑧)(𝑧∈𝕌) (2.51) implies that 𝐼_𝑐^𝜆,𝜇𝑔₁(𝑧)𝑧≺𝐼_𝑐^𝜆,𝜇𝑓(𝑧)𝑧≺𝐼_𝑐^𝜆,𝜇𝑔₂(𝑧)𝑧(𝑧∈𝕌). (2.52) Moreover, the functions 𝐼_𝑐^𝜆,𝜇𝑔₁(𝑧)/𝑧 and 𝐼_𝑐^𝜆,𝜇𝑔₂(𝑧)/𝑧 are the best subordinant and the best dominant, respectively.

Acknowledgment

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (no. 2011–0007037).