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ISRN Mathematical Analysis
Volume 2012 (2012), Article ID 909632, 17 pages
http://dx.doi.org/10.5402/2012/909632
Research Article

A Class of Integral Operators Preserving Subordination and Superordination for Analytic Functions

1Department of Mathematics, Faculty of Science, P.O. Box 838, Dammam 31113, Saudi Arabia
2Department of Mathematics, Rzeszów University of Technology, Ul. W. Pola 2, 35-959 Rzeszów, Poland

Received 8 May 2012; Accepted 26 June 2012

Academic Editors: J. Cui, J.-L. Wu, and C. Zhu

Copyright © 2012 H. A. Al-Kharsani et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The purpose of the paper is to investigate several subordination- and superordination-preserving properties of a class of integral operators, which are defined on the space of analytic functions in the open unit disk. The sandwich-type theorem for these integral operators is also presented. Moreover, we consider an application of the subordination and superordination theorem to the Gauss hypergeometric function.

1. Introduction

Let be the class of functions analytic in the unit disk 𝑈={𝑧|𝑧|<1}, and denote by 𝐴 the class of analytic functions in 𝑈 and usually normalized, that is, 𝐴={𝑓𝑓(0)=1,𝑓(0)=1}. 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 such that ||||𝑤(0)=0,𝑤(𝑧)<1(𝑧𝕌),𝑓(𝑧)=𝐹(𝑤(𝑧))(𝑧𝕌).(1.1) In this case, we write 𝑓𝐹(𝑧𝕌)or𝑓(𝑧)𝐹(𝑧)(𝑧𝕌).(1.2) If the function 𝐹 is univalent in 𝕌, then we have (cf. [1]) 𝑓𝐹(𝑧𝕌)𝑓(0)=𝐹(0),𝑓(𝕌)𝐹(𝕌).(1.3)

Definition 1.1 (Miller and Mocanu [1]). Let 𝜙2(1.4) and let be univalent in 𝕌. If 𝑝 is analytic in 𝕌 and satisfies the following differential subordination: 𝜙𝑝(𝑧),𝑧𝑝(𝑧)(𝑧)(𝑧𝕌),(1.5) then 𝑝 is called a solution of the differential subordination. A univalent function 𝑞 is called a dominant of the solutions of the differential subordination or, more simply, a dominant if 𝑝𝑞 for all 𝑝 satisfying the differential subordination (1.5). A dominant ̃𝑞 that satisfies ̃𝑞𝑞 for all dominants 𝑞 of (1.5) is said to be the best dominant.

Definition 1.2 (Miller and Mocanu [2]). Let 𝜑2 and let be analytic in 𝕌. If 𝑝 and 𝜑(𝑝(𝑧),𝑧𝑝(𝑧)) are univalent in 𝕌 and satisfy the following differential superordination: 𝑝(𝑧)𝜑(𝑧),𝑧𝑝(𝑧)(𝑧𝕌),(1.6) then 𝑝 is called a solution of the differential superordination. An analytic function 𝑞 is called a subordinat of the solutions of the differential superordination or, more simply, a subordinant if 𝑞𝑝 for all 𝑝 satisfying the differential superordination (1.6). A univalent subordinat ̃𝑞 that satisfies 𝑞̃𝑞 for all subordinats 𝑞 of (1.6) is said to be the best subordinat.

Definition 1.3 (Miller and Mocanu [2]). We denote by 𝒬 the class of functions 𝑓 that are analytic and injective on 𝕌𝐸(𝑓), where 𝐸(𝑓)𝜁𝜁𝜕𝕌,lim𝑧𝜁𝑓(𝑧)=,(1.7) and are such that 𝑓(𝜁)0,𝜁𝜕𝕌𝐸(𝑓).(1.8)
We define the family of integral operators 𝐼𝛽,𝛾𝑓(𝑧) as follows: 𝐼𝛽,𝛾𝑓(𝑧)=𝛾+𝛽𝑧𝛾𝑧0𝑓𝛽(𝑡)𝛾1(𝑡)(𝑡)𝑑𝑡1/𝛽,(1.9) where each of the functions 𝑓 and belong to the class A and the parmeters 𝛽𝐶{0}, 𝛾𝐶, Re(𝛾+𝛽)>0, were so constrained that the integral operators in (1.9) exist.
Throughout this paper, we will denote by 𝒜𝛽,𝛾 the following analytic function class: 𝒜𝛽,𝛾=𝑓(𝕌)𝑓(𝑧)𝑧𝐼0,𝛽,𝛾(𝑧)𝑧.0,(𝑧𝑈)(1.10)
This integral operator 𝐼𝛽,𝛾(𝑓) defined by (1.9) has been extensively studied by authors [36] with suitable restriction on the parameters 𝛽 and 𝛾.
In particular, if we take 𝛾=0 we get the integral operator defined by Bulboacă [712] and if we put (𝑡)=𝑡 in (1.9), we will get the results in [13].
In the present paper, we obtain the subordination- and superordination-preserving properties of the integral operator 𝐼𝛽,𝛾(𝑓) defined by (1.9) with the sandwich-type theorem. We also consider an interesting application of our main results to the Gauss hypergeometric function.
The following lemmas will be required in our present investigation.

Lemma 1.4 (Miller and Mocanu [14]). Suppose that the function 𝐻2(1.11) satisfies the following condition: 𝔢{𝐻(𝑖𝑠,𝑡)}0(1.12) for all real 𝑠 and for all 1𝑡2𝑛1+𝑠2(𝑛={1,2,3,}).(1.13) If the function 𝑝(𝑧)=1+𝑝𝑛𝑧𝑛+(1.14) is analytic in 𝕌 and 𝐻𝑝𝔢(𝑧),𝑧𝑝(𝑧)>0(𝑧𝕌),(1.15) then 𝔢{𝑝(𝑧)}>0(𝑧𝕌).(1.16)

Lemma 1.5 (Miller and Mocanu [15]). Let 𝛽,𝛾 with 𝛽0 and let with (0)=𝑐. If 𝔢{𝛽(𝑧)+𝛾}>0(𝑧𝕌),(1.17) then the solution 𝑞, 𝑞(0)=𝑐, of the following differential equation: 𝑞(𝑧)+𝑧𝑞(𝑧)𝛽𝑞(𝑧)+𝛾=(𝑧)(1.18) is analytic in 𝕌 and satisfies the inequality given by 𝔢{𝛽𝑞(𝑧)+𝛾}>0(𝑧𝕌).(1.19)

Lemma 1.6 (Miller and Mocanu [1]). Let 𝑝𝒬 with 𝑝(0)=𝑎 and let 𝑞(𝑧)=𝑎+𝑎𝑛𝑧𝑛+(1.20) be analytic in 𝕌 with 𝑞(𝑧)𝑎,𝑛.(1.21) If 𝑞 is not subordinate to 𝑝, then there exist points 𝑧0=𝑟0𝑒𝑖𝜃𝕌,𝜁0𝜕𝕌𝐸(𝑓),(1.22) for which 𝑞𝕌𝑟0𝑧𝑝(𝕌),𝑞0𝜁=𝑝0,𝑧0𝑞𝑧0=𝑚𝜁0𝑝𝜁0(𝑚𝑛).(1.23)

Our next lemmas deal with the notion of subordination chain. A function 𝐿(𝑧,𝑡) defined on 𝕌×[0,) is called the subordination chain (or Löwner chain) if 𝐿(𝑧,𝑡) is analytic and univalent in 𝕌 for all 𝑡[0,),𝐿(𝑧,𝑡) is continuously differentiable on [0,) for all 𝑧𝕌 and 𝐿(𝑧,𝑠)𝐿(𝑧,𝑡)(𝑧𝕌;0𝑠<𝑡).(1.24)

Lemma 1.7 (Miller and Mocanu [2]). Let [𝑎,1]={𝑓𝑓(0)=𝑎,𝑓(0)0} and []𝑞𝑎,1,𝜑2.(1.25) Also set 𝜑𝑞(𝑧),𝑧𝑞(𝑧)(𝑧)(𝑧𝕌).(1.26) If 𝐿𝑞(𝑧,𝑡)=𝜑(𝑧),𝑡𝑧𝑞(𝑧)(1.27) is a subordination chain and []𝑝𝑎,1𝒬,(1.28) then 𝑝(𝑧)𝜑(𝑧),𝑧𝑝(𝑧)(𝑧𝕌)(1.29) implies that 𝑞(𝑧)𝑝(𝑧)(𝑧𝕌).(1.30) Furthermore, if 𝜑𝑞(𝑧),𝑧𝑝(𝑧)=(𝑧)(1.31) has a univalent solution 𝑞𝒬, then 𝑞 is the best subordinat.

Lemma 1.8 (Pommerenke [16]). The function 𝐿(𝑧,𝑡)=𝑎1(𝑡)𝑧+,(1.32) with 𝑎1(𝑡)0,lim𝑡||𝑎1||(𝑡)=,(1.33) is a subordination chain if and only if 𝔢𝑧𝜕𝐿(𝑧,𝑡)/𝜕𝑧𝜕𝐿(𝑧,𝑡)/𝜕𝑡>0(𝑧𝕌;0𝑡<).(1.34)

2. Main Results

Our first subordination is contained in Theorem 2.1. To short the formulas in this section, let us denote 𝒥𝛽,𝛾(𝑓)=(𝑧)(𝑧)𝑧𝛾1𝑓(𝑧)𝑧𝛽.(2.1)

Theorem 2.1. Let 𝑓,𝑔𝒜𝛽,𝛾. Suppose that 𝑧𝒥𝔢1+𝛽,𝛾(𝑔)𝒥𝛽,𝛾(𝑔)>𝛿(𝑧𝕌),(2.2) where 𝛿=1+|𝛾+𝛽|2||1(𝛾+𝛽)2||4𝔢(𝛾+𝛽)(𝔢(𝛾+𝛽)>0).(2.3) Then the following subordination relation: 𝒥𝛽,𝛾(𝑓)𝒥𝛽,𝛾(𝑔)(𝑧𝕌)(2.4) implies that 𝐼𝛽,𝛾(𝑓)(𝑧)𝑧𝛽𝐼𝛽,𝛾(𝑔)(𝑧)𝑧𝛽(𝑧𝕌),(2.5) where 𝐼𝛽,𝛾 is the integral operator defined by (1.9). Moreover, the function [𝐼𝛽,𝛾(𝑔)(𝑧)/𝑧]𝛽 is the best dominant.

Proof. Let us define the functions 𝐹 and 𝐺 by 𝐼𝐹(𝑧)=𝛽,𝛾(𝑓)(𝑧)𝑧𝛽𝐼,𝐺(𝑧)=𝛽,𝛾(𝑔)(𝑧)𝑧𝛽,(2.6) respectively. Then 𝐺(𝑧)0(|𝑧|<1).(2.7) We first show that, if the function 𝑞 is defined by 𝑞𝐺(𝑧)=1+𝑧(𝑧)𝐺(𝑧)(𝑧𝕌),(2.8) then 𝔢{𝑞(𝑧)}>0(𝑧𝕌).(2.9) In terms of the function 𝒥𝛽,𝛾(𝑔), the definition (1.9) readily yields 𝛽𝑧𝐼𝛽,𝛾(𝑔)(𝑧)𝐼𝛽,𝛾𝒥(𝑔)(𝑧)=𝛾+(𝛾+𝛽)𝛽,𝛾(𝑔).𝐺(𝑧)(2.10) We also have 𝛽𝑧𝐼𝛽,𝛾(𝑔)(𝑧)𝐼𝛽,𝛾(𝑔)(𝑧)=𝛽+𝑧𝐺(𝑧).𝐺(𝑧)(2.11) By a simple calculation in conjunction with (2.10) and (2.11), we obtain the following relationship: 1+𝑧[𝒥𝛽,𝛾(𝑔)][𝒥𝛽,𝛾(𝑔)]=𝑞(𝑧)+𝑧𝑞(𝑧)𝑞(𝑧)+𝛾+𝛽=(𝑧).(2.12) We also see from (2.2) that []𝔢(𝑧)+𝛾+𝛽>0(𝑧𝕌)(2.13) and, by using Lemma 1.5, we conclude that the differential equation (2.12) has a solution 𝑞(𝑈) with 𝑞(0)=(0)=1.(2.14) Let us put 𝑣𝐻(𝑢,𝑣)=𝑢+𝑢+𝛾+𝛽+𝛿,(2.15) where 𝛿 is given by (2.3). From (2.2), (2.12), and (2.15), we obtain 𝐻𝑞𝔢(𝑧),𝑧𝑞(𝑧)>0(𝑧𝕌).(2.16) Now we proceed to show that []1𝔢𝐻(𝑖𝑠,𝑡)0𝑠;𝑡21+𝑠2.(2.17) Indeed, from (2.15), we have []𝑡𝔢𝐻(𝑖𝑠,𝑡)=𝔢𝑖𝑠+=[]𝑖𝑠+𝛾+𝛽+𝛿𝑡𝔢𝛾+𝛽||||𝛾+𝛽+𝑖𝑠2𝐸+𝛿𝛿(𝑠)2||||𝛾+𝛽+𝑖𝑠2,(2.18) where 𝐸𝛿[]𝑠(𝑠)=𝔢(𝛾+𝛽)2𝛿2||||4𝛿𝔪(𝛾+𝛽)𝑠2𝛿𝛾+𝛽2+𝔢(𝛾+𝛽).(2.19) For 𝛿 given by (2.3), we note that the coefficient of 𝑠2 is in the quadratic expression for 𝐸𝛿(𝑠) defined by (2.19) is greater than or equal to zero. Moreover, the discriminant Δ of 𝐸𝛿(𝑠) in (2.19) is represented by 14Δ=4𝛿2[]𝔢(𝛾+𝛽)2||||+2𝛿1+𝛾+𝛽2[]𝔢(𝛾+𝛽)𝔢(𝛾+𝛽)2,(2.20) which, for the assumed value of 𝛿 given by (2.3), yields Δ=0,(2.21) and so the quadratic expression for 𝑠 in 𝐸𝛿(𝑠) given by (2.19) is a perfect square. We thus see from (2.18) that []1𝔢𝐻(𝑖𝑠,𝑡)0𝑠;𝑡21+𝑠2.(2.22) Hence, by using Lemma 1.4, we conclude that []𝔢𝑞(𝑧)>0(𝑧𝕌),(2.23) that is, the function 𝐺 defined by (2.6) is convex in 𝕌.
Next, we prove that the subordination condition (2.4) implies that 𝐹(𝑧)𝐺(𝑧)(𝑧𝕌)(2.24) for the functions 𝐹 and 𝐺 defined by (2.6). For this purpose, we consider the function 𝐿(𝑧,𝑡) given by 𝐿(𝑧,𝑡)=𝐺(𝑧)+1+𝑡𝛾+𝛽𝑧𝐺(𝑧)(𝑧𝕌;00𝑥00086𝑡<).(2.25) Since 𝐺 is convex in 𝕌 and 𝔢(𝛾+𝛽)>0, we obtain 𝜕𝐿(𝑧,𝑡)|||𝜕𝑧𝑧=0=𝐺(0)1+1+𝑡𝛾+𝛽0(𝑧𝕌;0𝑡<),𝔢𝑧𝜕𝐿(𝑧,𝑡)/𝜕𝑧𝜕𝐿(𝑧,𝑡)/𝜕𝑡=𝔢𝛾+𝛽+(1+𝑡)1+𝑧𝐺(𝑧)𝐺(𝑧)>0(𝑧𝕌).(2.26) Therefore, by virtue of Lemma 1.8, 𝐿(𝑧,𝑡) is a subordination chain. We observe from the definition of a subordination chain that 𝒥𝛽,𝛾(𝑔)(𝑧)=𝐺(𝑧)+𝑧𝐺(𝑧)𝛾+𝛽=𝐿(𝑧,0),𝐿(𝑧,0)𝐿(𝑧,𝑡)(𝑧𝕌;0𝑡<).(2.27) This implies that 𝐿(𝜁,𝑡)𝐿(𝕌,0)=𝒥𝛽,𝛾(𝑔)(𝕌)(𝜁𝜕𝕌;0𝑡<).(2.28)
Now suppose that 𝐹 is not subordinate to 𝐺. Then, by Lemma 1.6, there exist points 𝑧𝕌 and 𝜁𝜕𝕌 such that 𝐹𝑧0𝜁=𝐺0,𝑧0𝐹𝑧0=(1+𝑡)𝜁0𝐺𝜁0(0𝑡<).(2.29) Hence, we have 𝐿𝜁0𝜁,𝑡=𝐺0+1+𝑡𝜁𝛾+𝛽0𝐺𝜁0𝑧=𝐹0+𝑧0𝐹𝑧0=𝑧𝛾+𝛽0𝑧0𝛾1𝑧0𝑓𝑧0𝑧0𝛽𝒥𝛽,𝛾(𝑔)𝕌,(2.30) by virtue of the subordination condition (2.4). This contradicts the above observation that 𝐿𝜁0,𝑡𝒥𝛽,𝛾(𝑔)(𝕌).(2.31) Therefore, the subordination condition (2.4) must imply the subordination given by (2.24). 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 (2.3) in Theorem 2.1 satisfies the following inequality 0<𝛿1/2.

Theorem 2.3. Let 𝑓,𝑔𝒜𝛽,𝛾. Suppose that 𝑧𝒥𝔢1+𝛽,𝛾(𝑔)𝒥𝛽,𝛾(𝑔)>𝛿(𝑧𝕌),(2.32) where 𝛿 is given by (2.3), and that the function 𝒥𝛽,𝛾(𝑓) is univalent in 𝕌 and such that [𝐼𝛽,𝛾(𝑓)(𝑧)/𝑧]𝛽𝒬, where 𝐼𝛽,𝛾 is the integral operator defined by (1.9). Then the following superordination relation: 𝒥𝛽,𝛾(𝑔)𝒥𝛽,𝛾(𝑓)(𝑧𝕌)(2.33) implies that 𝐼𝛽,𝛾(𝑔)(𝑧)𝑧𝛽𝐼𝛽,𝛾(𝑓)(𝑧)𝑧𝛽(𝑧𝕌).(2.34) Moreover, the function [𝐼𝛽,𝛾(𝑔)(𝑧)/𝑧]𝛽 is the best subordinat.

Proof. The first part of the proof is similar to that of Theorem 2.1 and so we will use the same notation as in the proof of Theorem 2.1. Now let us define the functions 𝐹 and 𝐺, as before, by (2.6). We first note that, by using (2.3) and (2.11), we obtain 𝒥𝛽,𝛾(𝑔)=𝐺(𝑧)+𝑧𝐺(𝑧)𝐺𝛾+𝛽=𝜑(𝑧),𝑧𝐺.(𝑧)(2.35) After a simple calculation, (2.35) yields the following relationship: 𝑧𝒥1+𝛽,𝛾(𝑔)𝒥𝛽,𝛾(𝑔)=𝑞(𝑧)+𝑧𝑞(𝑧)𝑞(𝑧)+𝛾+𝛽,(2.36) where function 𝑞 is defined by (2.8). Then, by using the same method as in the proof of Theorem 2.1, we can prove that 𝔢{𝑞(𝑧)}>0(𝑧𝕌),(2.37) that is, 𝐺 defined by (2.6) is convex (univalent) in 𝕌.
Next, we prove that the superordination condition (2.33) implies that 𝐹(𝑧)𝐺(𝑧)(𝑧𝕌).(2.38) For this purpose, we consider the function 𝐿(𝑧,𝑡) defined by 𝑡𝐿(𝑧,𝑡)=𝐺(𝑧)+𝛾+𝛽𝑧𝐺(𝑧)(𝑧𝕌;0𝑡<).(2.39) Since 𝐺 is convex and 𝔢(𝛾+𝛽)>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 (2.33) must imply the superordination given by (2.38). Furthermore, since the differential equation (2.35) has the univalent solution 𝐺, it is the best subordinat of the given differential subordination. We thus complete the proof of Theorem 2.3.

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

Theorem 2.4. Let 𝑓,𝑔1,𝑔2𝒜𝛽,𝛾. Suppose that 𝑧𝒥𝔢1+𝛽,𝛾𝑔1𝒥𝛽,𝛾𝑔1𝑧𝒥>𝛿,𝔢1+𝛽,𝛾𝑔2𝒥𝛽,𝛾𝑔2>𝛿(𝑧𝕌),(2.40) where 𝛿 is given by (2.3), and that the function 𝒥𝛽,𝛾(𝑓) is univalent in 𝕌 and such that [𝐼𝛽,𝛾(𝑓)(𝑧)/𝑧]𝛽𝒬, where 𝐼𝛽,𝛾 is the integral operator defined by (1.9). Then the following subordination relation: 𝒥𝛽,𝛾𝑔1𝒥𝛽,𝛾(𝑓)𝒥𝛽,𝛾𝑔2(𝑧𝕌)(2.41) implies that 𝐼𝛽,𝛾𝑔1(𝑧)𝑧𝛽𝐼𝛽,𝛾(𝑓)(𝑧)𝑧𝛽𝐼𝛽,𝛾𝑔2(𝑧)𝑧𝛽(𝑧𝕌).(2.42) Moreover, the functions [𝐼𝛽,𝛾(𝑔1)(𝑧)/𝑧]𝛽 and [𝐼𝛽,𝛾(𝑔2)(𝑧)/𝑧]𝛽 are the best subordinat and the best dominant, respectively.

The assumption of Theorem 2.4, that the functions 𝒥𝛽,𝛾(𝑓) and [𝐼𝛽,𝛾(𝑓)(𝑧)/𝑧]𝛽 need to be univalent in 𝑈, may be replaced by another condition in the following result.

Corollary 2.5. Let 𝑓,𝑔1,𝑔2𝒜𝛽,𝛾. Suppose that the condition (2.49) is satisfied and that 𝑓(𝑧)/𝑧𝒬 and 𝑧𝒥𝔢1+𝛽,𝛾(𝑓)𝒥𝛽,𝛾(𝑓)>𝛿(𝑧𝕌),(2.43) where 𝛿 is given by (2.3). Then the following subordination relation: 𝒥𝛽,𝛾𝑔1𝒥𝛽,𝛾(𝑓)𝒥𝛽,𝛾𝑔2(𝑧𝕌)(2.44) implies that 𝐼𝛽,𝛾𝑔1(𝑧)𝑧𝛽𝐼𝛽,𝛾(𝑓)(𝑧)𝑧𝛽𝐼𝛽,𝛾𝑔2(𝑧)𝑧𝛽(𝑧𝕌),(2.45) where 𝐼𝛽,𝛾 is the integral operator defined by (1.9). Moreover, the functions [𝐼𝛽,𝛾(𝑔1)(𝑧)/𝑧]𝛽 and [𝐼𝛽,𝛾((𝑔2)(𝑧)/𝑧)]𝛽 are the best subordinat and the best dominant, respectively.

Proof. In order to prove Corollary 2.5, we have to show that the condition (2.43) implies the univalence of each of the functions 𝒥𝛽,𝛾(𝑓) and 𝐹(𝑧)=[𝐼𝛽,𝛾(𝑓)(𝑧)/𝑧]𝛽.
Since 0<𝛿1/2, just as in Remark 2.2, the condition (2.43) means that 𝜓 is a close-to-convex function in 𝕌 (see [17]), and hence 𝒥𝛽,𝛾(𝑓) 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 are being omitted here. Thus, by applying Theorem 2.4, we readily obtain Corollary 2.5.

By setting 𝛾+𝛽=2 in Theorem 2.4, so that 𝛿=1/4, we deduce the following consequence of Theorem 2.4.

Corollary 2.6. Let 𝑓,𝑔1,𝑔2𝒜𝛽,2𝛽. Suppose that 𝑧𝒥𝔢1+𝛽,𝛾𝑔1𝒥𝛽,𝛾𝑔11>4𝑧𝒥,𝔢1+𝛽,𝛾𝑔2𝒥𝛽,𝛾𝑔21>4(𝑧𝕌),(2.46) and that the function 𝒥𝛽,𝛾(𝑓) is univalent in 𝕌 and [𝐼𝛽,2𝛽(𝑓)(𝑧)/𝑧]𝛽𝒬, where 𝐼𝛽,2𝛽 is the integral operator defined by (1.9) with 𝛾=𝛽+2. Then the following subordination relation: 𝒥𝛽,𝛾𝑔1𝒥𝛽,𝛾(𝑓)𝒥𝛽,𝛾𝑔2(𝑧𝕌)(2.47) implies that 𝐼𝛽,2𝛽𝑔1(𝑧)𝑧𝛽𝐼𝛽,2𝛽(𝑓)(𝑧)𝑧𝛽𝐼𝛽,2𝛽𝑔2(𝑧)𝑧𝛽(𝑧𝕌).(2.48) Moreover, the functions [𝐼𝛽,2𝛽(𝑔1)(𝑧)/𝑧]𝛽 and [𝐼𝛽,2𝛽(𝑔2)(𝑧)/𝑧]𝛽 are the best subordinat and the best dominant, respectively.

If we take 𝛾+𝛽=1+𝑖 in Theorem 2.4, then we are easily led to the following result.

Corollary 2.7. Let 𝑓,𝑔1,𝑔2𝒜𝛽,1+𝑖𝛽. Suppose that 𝑧𝒥𝔢1+𝛽,𝛾𝑔1𝒥𝛽,𝛾𝑔1>534𝑧𝒥,𝔢1+𝛽,𝛾𝑔2𝒥𝛽,𝛾𝑔2>534(𝑧𝕌),(2.49) and that the function 𝒥𝛽,𝛾(𝑓) is univalent in 𝕌 and [𝐼𝛽,1+𝑖𝛽(𝑓)(𝑧)/𝑧]𝛽𝒬, where 𝐼𝛽,1+𝑖𝛽 is the integral operator defined by (1.9) with 𝛾=𝛽+2. Then the following subordination relation: 𝒥𝛽,𝛾𝑔1𝒥𝛽,𝛾(𝑓)𝒥𝛽,𝛾𝑔2(𝑧𝕌)(2.50) implies that 𝐼𝛽,1+𝑖𝛽𝑔1(𝑧)𝑧𝛽𝐼𝛽,1+𝑖𝛽(𝑓)(𝑧)𝑧𝛽𝐼𝛽,1+𝑖𝛽𝑔2(𝑧)𝑧𝛽(𝑧𝕌).(2.51) Moreover, the functions [𝐼𝛽,1+𝑖𝛽(𝑔1)(𝑧)/𝑧]𝛽𝑎𝑛𝑑[𝐼𝛽,1+𝑖𝛽(𝑔2)(𝑧)/𝑧]𝛽 are the best subordinat, respectively.

3. Application to the Gauss Hypergeometric Function

We begin by recalling that the Gauss hypergeometric function 2𝐹1(𝑎,𝑏;𝑐;𝑧) is defined by (see, for details, [18] and [19, Chapter 14]) 2𝐹1(𝑎,𝑏;𝑐;𝑧)=𝑛=0(𝑎)𝑛(𝑏)𝑛(𝑐)𝑛𝑧𝑛𝑛!𝑧𝕌;𝑎,𝑏;𝑐0;0,={0,1,2,}(3.1) where (𝜆)𝑣 denotes the Pochhammer symbol (or the shifted factorial) defined (for 𝜆,𝑣 and in terms of the Gamma function) by (𝜆)𝑣=Γ(𝜆+𝑣)=1Γ(𝜆)(𝜈=0;𝜆{0})𝜆(𝜆+1)(𝜆+𝑣1)(𝜈=𝑛;𝜆).(3.2) For this useful special function, the following Eurlerian integral representation is fairly well known [19, page. 293]: 2𝐹1(𝑎,𝑏;𝑐;𝑧)=Γ(𝑐)Γ(𝑎)Γ(𝑐𝑎)10𝑡𝑎1(1𝑡)𝑐𝑎1(1𝑧𝑡)𝑏.𝑑𝑡𝔢(𝑐)>𝔢(𝑎)>0;arg(1𝑧)𝜋𝜀;0<𝜀<𝜋(3.3) In view of (3.3), we set 𝑧𝑔(𝑧)=(1𝑧)𝑘(𝑘>0),(3.4)𝛾=0 and (𝑧)=𝑧𝑒𝑧, so that the definition (1.9) yields 𝐼𝛽=𝛽𝑧0𝑡𝛽1(1𝑡)1𝑘𝛽𝑑𝑡1/𝛽𝛽=𝑧10𝑢𝛽1(1𝑧𝑢)1𝑘𝛽𝑑𝑢1/𝛽=𝑧2𝐹1(𝛽,𝑘𝛽1,𝛽+1;𝑧)1/𝛽.(3.5) Moreover, we note that 𝑔(𝑧)𝑧=1(1𝑧)𝑘0(𝑧𝕌)(3.6) and for 𝛽>0 the condition (2.2) becomes 𝑘𝛽2(𝛿+1). Therefore, by applying Theorem 2.1 with 𝑔(𝑧)=𝑧/(1𝑧)𝑘, (𝑘>0), we obtain the following result.

Theorem 3.1. Let 𝑓𝒜𝛽,𝛾 with 𝛾=0,  𝛽>0,  (𝑧)=𝑧𝑒𝑧. Suppose that 𝑘𝛽2(𝛿+1)(𝑘>0),(3.7) where 𝛿=1+𝛽2||1𝛽2||.4𝛽(3.8) Then the following subordination relation: (1𝑧)𝑓(𝑧)𝑧𝛽1(1𝑧)𝑘𝛽1(𝑧𝕌)(3.9) implies that 𝐼𝛽(𝑓)(𝑧)𝑧𝛽2𝐹1(𝛽,𝑘𝛽1;𝛽+1;𝑧)(𝑧𝕌),(3.10) where 𝐼𝛽 is the integral operator defined by (1.9). Moreover, the function 2𝐹1(𝛽,𝑘𝛽1;𝛽+1;𝑧) is the best dominant.

For 𝛽=1 we get 𝛿=1/2 and Theorem 3.1 becomes the following Corollary.

Corollary 3.2. Let 𝑓𝒜1,0 and 0<𝑘3. Then the following subordination relation: (1𝑧)𝑓(𝑧)𝑧1(1𝑧)𝑘1(𝑧𝕌)(3.11) implies that 𝐼1(𝑓)(𝑧)𝑧2𝐹1(1,𝑘1;2,𝑧)(𝑧𝕌),(3.12) where the integral operator 𝐼1 is defined by (1.9).

We state the following result as the dual result of Theorem 3.1, which can be obtained by similarly applying Theorem 2.3.

Theorem 3.3. Under the assumption of Theorem 3.1, suppose also that the function (1𝑧)[𝑓(𝑧)/𝑧]𝛽 is univalent in 𝕌 and that [𝐼𝛽(𝑓)(𝑧)/𝑧]𝛽𝒬, where 𝐼𝛽 is the integral operator defined by (1.9). Then the following superordination relation: 1(1𝑧)𝑘𝛽1(1𝑧)𝑓(𝑧)𝑧𝛽(𝑧𝕌)(3.13) implies that 2𝐹1𝐼(𝛽,𝑘𝛽1;𝛽+1;𝑧)𝛽(𝑓)(𝑧)𝑧𝛽(𝑧𝕌).(3.14) Moreover, the function 2𝐹1(𝛽,𝑘𝛽1;𝛽1;𝑧) is the best subordinat.

If  we set in (1.9) 𝑔(𝑧)=𝑧/(1𝑧)𝑘(𝑘>0),𝛾=0 and (𝑧)=𝑧/(1𝑧), then we get 𝐼𝛽=𝛽𝑧0𝑡𝛽1(1𝑡)1𝑘𝛽𝑑𝑡1/𝛽𝛽=𝑧10𝑢𝛽1(1𝑧𝑢)1𝑘𝛽𝑑𝑢1/𝛽=𝑧2𝐹1(𝛽,𝑘𝛽+1;𝛽+1;𝑧)1/𝛽.(3.15) Therefore by applying Theorem 2.1, we obtain the following result.

Theorem 3.4. Let 𝑓𝒜𝛽,𝛾 with 𝛾=0,  𝛽>0,  (𝑧)=𝑧/(1𝑧). Suppose that 𝑘>0 and 𝑘𝛽2𝛿,(3.16) where 𝛿=1+𝛽2||1𝛽2||.4𝛽(3.17) Then the following subordination relation: 11𝑧𝑓(𝑧)𝑧𝛽1(1𝑧)𝑘𝛽+1(𝑧𝕌)(3.18) implies that 𝐼𝛽(𝑓)(𝑧)𝑧𝛽2𝐹1(𝛽,𝑘𝛽+1;𝛽+1;𝑧)(𝑧𝕌),(3.19) where 𝐼𝛽 is the integral operator defined by (1.9). Moreover, the function 2𝐹1(𝛽,𝑘𝛽+1;𝛽+1;𝑧) is the best dominant.

By taking 𝛽=1 in Theorem 3.1, we are led to Corollary 3.5.

Corollary 3.5. Let 𝑓𝒜𝛽,𝛾 with 𝛾=0,  𝛽=1,  (𝑧)=𝑧/(1𝑧). Then the following subordination relation: 1𝑧1(1𝑧)𝑓(𝑧)(1𝑧)2(𝑧𝕌)(3.20) implies that 𝐼1(𝑓)(𝑧)𝑧2𝐹1(1,2;2;𝑧)(𝑧𝕌),(3.21) where the integral operator 𝐼1 is defined by (1.9).

We state the following result as the dual result of Theorem 3.4, which can be obtained by similarly applying Theorem 2.3.

Theorem 3.6. Under the assumption of Theorem 3.4, suppose also that the function (1/(1𝑧))[𝑓(𝑧)/𝑧]𝛽 is univalent in 𝕌 and that [𝐼𝛽(𝑓)(𝑧)/𝑧]𝛽𝒬, where 𝐼𝛽 is the integral defined by (1.9) if we take 𝛾=0. Then the following superordination relation: 1(1𝑧)𝑘𝛽+111𝑧𝑓(𝑧)𝑧𝛽(𝑧𝕌)(3.22) implies that 2𝐹1𝐼(𝛽,𝑘𝛽+1;𝛽+1;𝑧)𝛽(𝑓)(𝑧)𝑧𝛽(𝑧𝕌).(3.23) Moreover, the function 2𝐹1(𝛽,𝑘𝛽+1,𝛽+1;𝑧) is the best subordinat.

Acknowledgment

This paper was supported by the Science Research Program in Science College in Dammam University, Saudi Arabia.

References

  1. S. S. Miller and P. T. Mocanu, Differential Subordinations, Theory and Applications, vol. 225 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2000.
  2. S. S. Miller and P. T. Mocanu, “Subordinants of differential superordinations,” Complex Variables: Theory and Application, vol. 48, no. 10, pp. 815–826, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. J. W. Alexander, “Functions which map the interior of the unit circle upon simple regions,” Annals of Mathematics Second Series, vol. 17, no. 1, pp. 12–22, 1915. View at Publisher · View at Google Scholar
  4. S. D. Bernardi, “Convex and starlike univalent functions,” Transactions of the American Mathematical Society, vol. 135, pp. 429–446, 1969. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. B. C. Carlson and D. B. Shaffer, “Starlike and prestarlike hypergeometric functions,” SIAM Journal on Mathematical Analysis, vol. 15, no. 4, pp. 737–745, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. R. J. Libera, “Some classes of regular univalent functions,” Proceedings of the American Mathematical Society, vol. 16, pp. 755–758, 1965. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. T. Bulboacă, “On a special integral subordination,” Indian Journal of Pure and Applied Mathematics, vol. 28, no. 3, pp. 361–369, 1997.
  8. T. Bulboacă, “Integral operators that preserve the subordination,” Bulletin of the Korean Mathematical Society, vol. 34, no. 4, pp. 627–636, 1997. View at Zentralblatt MATH
  9. T. Bulboacă, “On a class of integral operators that preserve the subordination,” Pure Mathematics and Applications, vol. 13, no. 1-2, pp. 87–96, 2002, Algebraic systems (Felix-Oradea, 2001).
  10. T. Bulboacă, “A class of superordination-preserving integral operators,” Koninklijke Nederlandse Akademie van Wetenschappen. Indagationes Mathematicae New Series, vol. 13, no. 3, pp. 301–311, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. T. Bulboacă, “A class of double subordination-preserving integral operators,” Pure Mathematics and Applications, vol. 15, no. 2-3, pp. 87–106, 2004. View at Zentralblatt MATH
  12. T. Bulboacă, “Sandwich-type theorems for a class of integral operators,” Bulletin of the Belgian Mathematical Society Simon Stevin, vol. 13, no. 3, pp. 537–550, 2006. View at Zentralblatt MATH
  13. N. E. Cho and S. Owa, “Double subordination-preserving properties for certain integral operators,” Journal of Inequalities and Applications, vol. 2007, Article ID 83073, 10 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. S. S. Miller and P. T. Mocanu, “Differential subordinations and univalent functions,” The Michigan Mathematical Journal, vol. 28, no. 2, pp. 157–172, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. S. S. Miller and P. T. Mocanu, “Univalent solutions of Briot-Bouquet differential equations,” Journal of Differential Equations, vol. 56, no. 3, pp. 297–309, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. C. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, Germany, 1975.
  17. W. Kaplan, “Close-to-convex schlicht functions,” The Michigan Mathematical Journal, vol. 1, no. 2, pp. 169–185, 1952. View at Zentralblatt MATH
  18. S. Owa and H. M. Srivastava, “Univalent and starlike generalized hypergeometric functions,” Canadian Journal of Mathematics, vol. 39, no. 5, pp. 1057–1077, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, an Introduction to the General Theory of Infinite Processes and of Analytic Functions, with an Account of the Principal Transcendental Functions, Cambridge University Press, Cambridge, UK, 4th edition, 1927.