Certain Subordination Properties for Subclasses of Analytic Functions Involving Complex Order

Abstract

We derive several subordination results for a certain class of analytic functions defined by the Sălăgean operator In the present investigation.

1. Introduction and Preliminaries

Let 𝒜 denote the class of functions 𝑓 of the form 𝑓(𝑧)=𝑧+^∞_𝑘=2𝑎_𝑘𝑧^𝑘, (1.1) which are analytic in the open unit disk 𝕌={𝑧∶|𝑧|<1}. (1.2) Further, by 𝒮 we will denote the class of all functions in 𝒜 which are univalent in 𝕌.

Also let 𝒮^∗(𝑏),𝒦(𝑏) denote, respectively, the subclasses of 𝒜 consisting of functions that are starlike of complex order 𝑏(𝑏∈ℂ⧵{0}), convex of complex order 𝑏(𝑏∈ℂ⧵{0}) in 𝕌. In particular, the classes 𝒮^∗∶=𝒮^∗(1) and 𝒦∶=𝒦(1) are the familiar classes of starlike and convex functions in 𝕌.

Sălăgean [1] introduced the following operator which is popularly known as the Sălăgean derivative operator: 𝐷⁰𝑓(𝑧)(𝑧),𝐷=𝑓₁𝑓(𝑧)=𝑧𝑓^′(𝑧), (1.3) and, in general, 𝐷^𝑛𝑓(𝑧)=𝐷𝐷^𝑛−1𝑓(𝑧)(𝑛∈ℕ={1,2,3,…}). (1.4)It is easy to see that from (1.1), 𝐷^𝑛𝑓(𝑧)=𝑧+^∞_𝑘=2𝑘^𝑛𝑎_𝑘𝑧^𝑘. (1.5) Let 𝐺_𝑛(𝜆,𝑏,𝐴,𝐵) denote the subclass of 𝒜 consisting of functions 𝑓(𝑧) which satisfy 11+𝑏)𝐷(1−𝜆^𝑛𝑓(𝑧)𝑧+𝜆𝐷^𝑛𝑓(𝑧)′−1≺1+𝐴𝑧()1+𝐵𝑧−1≤𝐵<𝐴≤1. (1.6) Equivalently, (1−𝜆)𝐷^𝑛𝑓(𝑧)/𝑧+𝜆𝐷^𝑛𝑓(𝑧)′−1𝑏(𝐴−𝐵)−𝐵(1−𝜆)𝐷^𝑛𝑓(𝑧)/𝑧+𝜆𝐷^𝑛𝑓(𝑧)′−1<1. (1.7) We note that, for 𝑧∈𝕌,(1)𝐺_𝑛(𝜆,𝑏,1,−1)=𝐺_𝑛(𝜆,𝑏)={𝑓∈𝒜∶Re(1+1/𝑏[(1−𝜆)(𝐷^𝑛𝑓(𝑧)/𝑧)+𝜆(𝐷^𝑛𝑓(𝑧))′−1])}>0, (2)𝐺_𝑛(0,𝑏,1,−1)=𝐺_𝑛(𝑏)={𝑓∈𝒜∶Re(1+1/𝑏[𝐷^𝑛𝑓(𝑧)/𝑧−1])}>0, (3)𝐺_𝑛(1,𝑏,1,−1)=𝑅_𝑛(𝑏)={𝑓∈𝒜∶Re(1+1/𝑏[(𝐷^𝑛𝑓(𝑧))′−1])}>0, (4)𝐺₀(0,𝑏,1,−1)=𝐺(𝑏)={𝑓∈𝒜∶Re(1+1/𝑏[𝑓(𝑧)/𝑧−1])}>0, (5)𝐺₀(1,𝑏,1,−1)=𝑅(𝑏)={𝑓∈𝒜∶Re(1+1/𝑏[𝑓^′(𝑧)−1])}>0, (6)𝐺₀(0,1−𝛼,1,−1)=𝐺(𝛼)={𝑓∈𝒜∶Re(𝑓(𝑧)/𝑧)>𝛼,0≤𝛼<1}, (7)𝐺₀(1,1−𝛼,1,−1)=𝑅(𝛼)={𝑓∈𝒜∶Re(𝑓^′(𝑧))>𝛼,0≤𝛼<1}.

The class 𝑅(𝑏) was studied by Abdul Halim [2], while the class 𝐺(𝛼) was studied by Chen [3, 4] and the class 𝑅(𝛼) was studied by Èzrohi [5] (see also the works of Altintas and Özkan [6], Aouf et al. [7], Attiya [8], Kamali and Akbulut [9], Özkan [10], and Shanmugam et al. [11]). A systematic investigation of the starlike and convex functions involving Sălăgean derivative was done by Aouf et al. very recently [7].

In our proposed investigation of functions in these subclasses of the normalized analytic function class 𝒜, we need the following definitions and results.

Definition 1.1 (Hadamard product or convolution). For functions 𝑓 and 𝑔 in the class 𝒜, where 𝑓(𝑧) of the form (1.1) and 𝑔(𝑧) is given by 𝑔(𝑧)=𝑧+^∞_𝑘=2𝑐_𝑘𝑧^𝑘, (1.8) the Hadamard product (or convolution) (𝑓∗𝑔)(𝑧) is defined by 𝑓∗𝑔(𝑧)=𝑧+^∞_𝑘=2𝑎_𝑘𝑏_𝑘𝑧^𝑘=𝑔∗𝑓(𝑧),𝑧∈𝕌. (1.9)

Definition 1.2 (subordination principle). For analytic functions 𝑔 and ℎ with 𝑔(0)=ℎ(0),𝑔 is said to be subordinate to ℎ, denoted by 𝑔≺ℎ, if there exists an analytic function 𝑤 such that 𝑤(0)|𝑤(𝑧𝑔(𝑧)(𝑤(𝑧,=0,)|<1,=ℎ)) (1.10) for all 𝑧∈𝕌.

Definition 1.3 (see [12], subordinating factor sequence). A sequence {𝑏_𝑘}^∞_𝑘=1 of complex numbers is said to be a subordinating sequence if, whenever 𝑓(𝑧) is of the form (1.1) is analytic, univalent, and convex in 𝕌, one has the subordination given by^∞_𝑘=1𝑏_𝑘𝑎_𝑘𝑧^𝑘≺𝑓(𝑧)=^∞_𝑘=1𝑎_𝑘𝑧^𝑘,𝑧∈𝕌,𝑎₁=1. (1.11)

Lemma 1.4 (see [12]). The sequence {𝑏_𝑘}^∞_𝑘=1 is a subordinating factor sequence for the class 𝒦 of convex univalent functions if and only if ℜ𝔢1+2^∞_𝑘=1𝑏_𝑘𝑧^𝑘>0,𝑧∈𝕌. (1.12)

2. Main Results

Theorem 2.1. Let the function 𝑓 of the form (1.1) satisfy the following condition: ^∞_𝑛=2(1+|𝐵|)[(1+𝜆(𝑘−1))]𝑘^𝑛𝑎_𝑘≤(𝐴−𝐵)|𝑏|. (2.1) Then 𝑓∈𝐺_𝑛(𝜆,𝑏,𝐴,𝐵).

Proof. Suppose the inequality (2.1) holds. Then we have for 𝑧∈𝕌)𝐷(1−𝜆^𝑛𝑓(𝑧)𝑧+𝜆𝐷^𝑛𝑓(𝑧)−)()𝐷′−1(𝐴−𝐵𝑏−𝐵1−𝜆^𝑛𝑓(𝑧)𝑧+𝜆𝐷^𝑛𝑓(𝑧)=′−1_∞𝑘=2[(1+𝜆(𝑘−1))]𝑘^𝑛𝑎_𝑘𝑧^𝑘−1−(𝐴−𝐵)𝑏−𝐵^∞_𝑘=2[(1+𝜆(𝑘−1))]𝑘^𝑛𝑎_𝑘𝑧^𝑘−1≤^∞_𝑘=2[(1+𝜆(𝑘−1))]𝑘^𝑛𝑎_𝑘|𝑧|^𝑘−1−(𝐴−𝐵)|𝑏|−|𝐵|^∞_𝑘=2[(1+𝜆(𝑘−1))]𝑘^𝑛𝑎_𝑘|𝑧|^𝑘−1≤^∞_𝑘=2(1+|𝐵|)[(1+𝜆(𝑘−1))]𝑘^𝑛𝑎_𝑘−(𝐴−𝐵)|𝑏|≤0 (2.2) which shows that 𝑓 belongs to the class 𝐺_𝑛(𝜆,𝑏,𝐴,𝐵).

In view of Theorem 2.1, we now introduce the subclass 𝐺^∗_𝑛(𝜆,𝑏,𝐴,𝐵) which consists of functions 𝑓∈𝒜 whose Taylor-Maclaurin coefficients satisfy the inequality (2.1). We note that 𝐺^∗_𝑛(𝜆,𝑏,𝐴,𝐵)⊂𝐺_𝑛(𝜆,𝑏,𝐴,𝐵).

In this work, we prove several subordination relationships involving the function class 𝐺^∗_𝑛(𝜆,𝑏,𝐴,𝐵) employing the technique used earlier by Attiya [13] and Srivastava and Attiya [14].

Theorem 2.2. Let 𝑓∈𝐺^∗_𝑛(𝜆,𝑏,𝐴,𝐵), and let 𝑔(𝑧) be any function in the usual class of convex functions 𝒦, then (1+𝜆)(1+|𝐵|)2^𝑛2[(1+𝜆)(1+|𝐵|)2^𝑛+(𝑏(𝑧)(𝑧)𝐴−𝐵)||]𝑓∗𝑔≺𝑔 (2.3) for every function 𝑔∈𝒦. Further, 𝑓(𝑧|𝐵2Re)>−(1+𝜆)(1+|)^𝑛+(𝐴−𝐵)|𝑏|(1+𝜆)(1+|𝐵|)2^𝑛,𝑧∈𝕌. (2.4) The constant factor (1+𝜆)(1+|𝐵|)2^𝑛/2[(1+𝜆)(1+|𝐵|)2^𝑛+(𝐴−𝐵)|𝑏|] in (2.3) cannot be replaced by a larger number.

Proof. Let 𝑓∈𝐺^∗_𝑛(𝜆,𝑏,𝐴,𝐵), and suppose that 𝑔(𝑧)=𝑧+∑^∞_𝑘=2𝑐_𝑘𝑧^𝑘∈𝒦. Then (1+𝜆)(1+|𝐵|)2^𝑛2[(1+𝜆)(1+|𝐵|)2^𝑛+(𝑏(𝑧)=(|𝐵2𝐴−𝐵)||]𝑓∗𝑔1+𝜆)(1+|)^𝑛2[(1+𝜆)(1+|𝐵|)2^𝑛+(𝐴−𝐵)|𝑏|]𝑧+^∞_𝑘=2𝑐_𝑘𝑎_𝑘𝑧^𝑘. (2.5) Thus, by Definition 1.3, the subordination result holds true if (1+𝜆)(1+|𝐵|)2^𝑛2[(1+𝜆)(1+|𝐵|)2^𝑛+(𝑏𝑎𝐴−𝐵)||]^𝑘∞_𝑘=1 (2.6) is a subordinating factor sequence, with 𝑎₁=1. In view of Lemma 1.4, this is equivalent to the following inequality: Re1+2^∞_𝑘=1(1+𝜆)(1+|𝐵|)2^𝑛2[(1+𝜆)(1+|𝐵|)2^𝑛+(𝑏𝑎𝐴−𝐵)||]^𝑘𝑧^𝑘>0,𝑧∈𝕌. (2.7) Since Ψ(𝑘)=(1+|𝐵|)[(1+𝜆(𝑘−1))]𝑘^𝑛 is an increasing function of 𝑘(𝑘≥2), we have, for |𝑧|=𝑟<1, Re1+^∞_𝑘=12(1+𝜆)(1+|𝐵|)2^𝑛2[(1+𝜆)(1+|𝐵|)2^𝑛+(𝑏𝑎𝐴−𝐵)||]^𝑘𝑧^𝑘(|𝐵2=Re1+1+𝜆)(1+|)^𝑛[(1+𝜆)(1+|𝐵|)2^𝑛+(𝐴−𝐵)|𝑏|]𝑧+^∞_𝑘=2(1+𝜆)(1+|𝐵|)2^𝑛[(1+𝜆)(1+|𝐵|)2^𝑛+(𝑏𝑎𝐴−𝐵)||]^𝑘𝑧^𝑘(|𝐵2≥1−1+𝜆)(1+|)^𝑛[(1+𝜆)(1+|𝐵|)2^𝑛+(𝑏1𝐴−𝐵)||]𝑟−[(1+𝜆)(1+|𝐵|)2^𝑛+(𝑏×𝐴−𝐵)||]_∞𝑘=2(1+|𝐵|)[1+𝜆(𝑘−1)]𝑘^𝑛𝑎_𝑘𝑟^𝑘(|𝐵2≥1−1+𝜆)(1+|)^𝑛[(1+𝜆)(1+|𝐵|)2^𝑛+(𝑏𝑟−|𝑏)𝐴−𝐵)||]|(𝐴−𝐵[(1+𝜆)(1+|𝐵|)2^𝑛+(𝑏|𝑧|𝐴−𝐵)||]𝑟=1−𝑟>0,=𝑟<1, (2.8) where we have also made use of the assertion (2.1) of Theorem 2.1. This evidently proves the inequality (2.3) and hence also the subordination result (2.3) asserted by Theorem 2.2. The inequality (2.4) follows from (2.3) by taking 𝑔(𝑧)=𝑧1−𝑧=𝑧+^∞_𝑘=2𝑧^𝑘∈𝒦. (2.9) To prove the sharpness of the constant (1+𝜆)(1+|𝐵|)2^𝑛/[(1+𝜆)(1+|𝐵|)2^𝑛+(𝐴−𝐵)|𝑏|], we consider the function 𝐹∈𝐺^∗_𝑛(𝜆,𝑏,𝐴,𝐵) defined by 𝐹(𝑧)(𝑏|=𝑧−𝐴−𝐵)|(1+𝜆)(1+|𝐵|)2^𝑛. (2.10) Thus, from (2.3), we have (1+𝜆)(1+|𝐵|)2^𝑛2[(1+𝜆)(1+|𝐵|)2^𝑛+(𝑏𝐹(𝑧)≺𝑧𝐴−𝐵)||]1−𝑧. (2.11) It is easily verified that (|𝐵2minRe1+𝜆)(1+|)^𝑛2[(1+𝜆)(1+|𝐵|)2^𝑛+(𝑏𝐹(𝑧)1𝐴−𝐵)||]=−2,𝑧∈𝕌. (2.12) This shows that the constant (1+𝜆)(1+|𝐵|)2^𝑛/2[(1+𝜆)(1+|𝐵|)2^𝑛+(𝐴−𝐵)|𝑏|] cannot be replaced by any larger one.

For the choices of 𝐴−1=0 and 𝐵+1=0, we get the following corollary.

Corollary 2.3. Let 𝑓∈𝐺^∗_𝑛(𝜆,𝑏) let and 𝑔(𝑧) be any function in the usual class of convex functions 𝒦, then (1+𝜆)2^𝑛[(1+𝜆)(1+|𝐵|)2^𝑛+|𝑏(𝑧)(𝑧)|]𝑓∗𝑔≺𝑔, (2.13) where 𝑏∈ℂ⧵{0} and 0≤𝜆<1, 𝑓(𝑧2)2Re)>−(1+𝜆^𝑛+|𝑏|(1+𝜆)2^𝑛,𝑧∈𝕌. (2.14) The constant factor (1+𝜆)2^𝑛/2[(1+𝜆)2^𝑛+|𝑏|] in (2.13) cannot be replaced by a larger number.

For the choices of 𝐴−1=𝐵+1=0 and 𝑛=0, one gets the following.

Corollary 2.4. Let 𝑓∈𝐺^∗(𝜆,𝑏), and let 𝑔(𝑧) be any function in the usual class of convex functions 𝒦, then (1+𝜆)2[|𝑏(𝑧)(𝑧)1+𝜆+|]𝑓∗𝑔≺𝑔, (2.15) where 𝑏∈ℂ⧵{0} and 0≤𝜆<1, 𝑓(𝑧[|𝑏Re)>−1+𝜆+|](1+𝜆),𝑧∈𝕌. (2.16) The constant factor (1+𝜆)/2[1+𝜆+|𝑏|] in (2.15) cannot be replaced by a larger number.

For the choices of 𝐴−1=𝐵+1=0,𝑛=0, and 𝜆=0, one gets the following.

Corollary 2.5. Let 𝑓∈𝐺^∗(𝑏), and 𝑔(𝑧) let be any function in the usual class of convex functions 𝒦, then 12[|𝑏(𝑧)(𝑧)1+|]𝑓∗𝑔≺𝑔, (2.17) where 𝑏∈ℂ⧵{0} and 0≤𝜆<1, Re𝑓(𝑧)>−[1+|𝑏|],𝑧∈𝕌. (2.18) The constant factor 1/2[1+|𝑏|] in (2.17) cannot be replaced by a larger number.

For the choices of 𝐴−1=𝐵+1=0,𝑛=0,𝜆=0, and 𝑏=1−𝛼, one gets the following.

Corollary 2.6. Let 𝑓∈𝐺^∗(𝛼), and let 𝑔(𝑧) be any function in the usual class of convex functions 𝒦, then 12()(𝑧)(𝑧)2−𝛼𝑓∗𝑔≺𝑔, (2.19) where 0≤𝛼<1, Re𝑓(𝑧)>−(2−𝛼),𝑧∈𝕌. (2.20) The constant factor 1/2(2−𝛼) in (2.19) cannot be replaced by a larger number.

For the choices of 𝐴−1=𝐵+1=0,𝑛=0,𝜆=1, and 𝑏=1−𝛼(0≤𝛼<1), one gets the following.

Corollary 2.7. Let 𝑓∈𝑅^∗(𝛼), and 𝑔(𝑧) let be any function in the usual class of convex functions 𝒦, then 22()(𝑧)(𝑧)3−𝛼𝑓∗𝑔≺𝑔 (2.21) where 0≤𝛼<1, Re𝑓(𝑧)>−3−𝛼2,𝑧∈𝕌. (2.22) The constant factor 1/(3−𝛼) in (2.21) cannot be replaced by a larger number.

Acknowledgment

The authors thank the referees for their insightful suggestions in the paper. The second and third authors were partially supported by MOHE: UKM-ST-06-FRGS0244-2010.