Subordination Properties for Certain Analytic Functions

Abstract

The purpose of the present paper is to derive a subordination result for functions in the class 𝐻^∗_𝑛(𝛼,𝜆,𝑏) of normalized analytic functions in the open unit disk 𝕌. A number of interesting applications of the subordination result are also considered.

1. Introduction

Let 𝐴 denote the class of functions of the form 𝑓(𝑧)=𝑧+^∞_𝑛=2𝑎_𝑛𝑧^𝑛 (1.1) which are analytic in the unit disc 𝕌={𝑧∶|𝑧|<1}. We also denote by 𝐾 the class of functions 𝑓∈𝐴 that are convex in 𝕌.

Given two functions 𝑓,𝑔∈𝐴, where 𝑓 is given by (1.1) and 𝑔 is defined by 𝑔(𝑧)=𝑧+^∞_𝑛=2𝑏_𝑛𝑧^𝑛, (1.2) the Hadamard product (or convolution) 𝑓∗𝑔 is defined by (𝑓∗𝑔)(𝑧)=𝑧+^∞_𝑛=2𝑎_𝑛𝑏_𝑛𝑧^𝑛(𝑧∈𝕌). (1.3)

By using the Hadamard product, Ruscheweyh [1] defined 𝐷^𝛼𝑧𝑓(𝑧)=(1−𝑧)^𝛼+1∗𝑓(𝑧)(𝛼≥−1). (1.4) From the definition of (1.4), we observe that 𝐷^𝑛𝑓(𝑧)=𝑧(𝑧^𝑛−1𝑓(𝑧))^(𝑛)𝑛!, (1.5) when 𝑛=𝛼∈ℕ₀=ℕ∪{0}={0,1,2,…}. The symbol 𝐷^𝑛𝑓(𝑧) (𝑛∈ℕ₀) was called the 𝑛-th order Ruscheweyh derivative of 𝑓 by Al-Amiri [2]. We also note that 𝐷⁰𝑓(𝑧)=𝑓(𝑧) and 𝐷¹𝑓(𝑧)=𝑧𝑓^′(𝑧).

Definition 1.1. Suppose that 𝑓∈𝐴. Then the function 𝑓 is said to be a member of the class 𝐻_𝑛(𝛼,𝜆,𝑏) if it satisfies |𝜆(𝑛+1)(𝐷^𝑛+1𝑓(𝑧)/𝑧)+[1−𝜆(𝑛+1)](𝐷^𝑛𝑓(𝑧)/𝑧)−1𝜆(𝑛+1)(𝐷^𝑛+1𝑓(𝑧)/𝑧)+[1−𝜆(𝑛+1)](𝐷^𝑛𝑓(𝑧)/𝑧)+2𝑏(1−𝛼)−1|<1(𝑧∈𝕌;0≤𝛼<1;𝜆≥0;𝑏∈ℂ⧵{0};𝑛∈ℕ₀). (1.6) By specializing 𝛼, 𝜆, 𝑏, and 𝑛, one can obtain various subclasses studied by many authors (see, e.g., [3–11]).

Definition 1.2. Let 𝑔 be analytic and univalent in 𝕌. If 𝑓 is analytic in 𝕌, 𝑓(0)=𝑔(0), and 𝑓(𝕌)⊂𝑔(𝕌), then one says that 𝑓 is subordinate to 𝑔 in 𝕌, and one writes 𝑓≺𝑔 or 𝑓(𝑧)≺𝑔(𝑧). One also says that 𝑔 is superordinate to 𝑓 in 𝕌.

Definition 1.3. An infinite sequence {𝑏_𝑘}^∞_𝑘=1 of complex numbers will be called a subordinating factor sequence if for every univalent function 𝑓 in 𝐾, one has ^∞_𝑘=1𝑏_𝑘𝑎_𝑘𝑧^𝑘≺𝑓(𝑧)(𝑧∈𝕌;𝑎₁=1). (1.7)

Lemma 1.4. (see [12]). The sequence {𝑏_𝑘}^∞_𝑘=1 is a subordinating factor sequence if and only if Re{1+2^∞_𝑘=1𝑏_𝑘𝑧^𝑘}>0(𝑧∈𝕌). (1.8)

Now, we prove the following lemma which gives a sufficient condition for functions belonging to the class 𝐻_𝑛(𝛼,𝜆,𝑏).

Lemma 1.5. If the function 𝑓 which is defined by (1.1) satisfies the following condition: ^∞_𝑘=2[1+𝜆(𝑘−1)]𝐶(𝑛,𝑘)|𝑎_𝑘|≤(1−𝛼)|𝑏|(0≤𝛼<1;𝜆≥0;𝑏∈ℂ⧵{0};𝑛∈ℕ₀), (1.9) where 𝐶(𝑛,𝑘)=^𝑘_𝑗=2(𝑗+𝑛−1)(𝑘−1)!(𝑘=2,3,…), (1.10) then 𝑓∈𝐻_𝑛(𝛼,𝜆,𝑏).

Proof. Suppose that the inequality (1.9) holds. Using the identity 𝑧(𝐷^𝑛𝑓(𝑧))^′=(𝑛+1)𝐷^𝑛+1𝑓(𝑧)−𝑛𝐷^𝑛𝑓(𝑧), (1.11) we have for 𝑧∈𝕌, 𝐷|(1−𝜆)^𝑛𝑓(𝑧)𝑧+𝜆(𝐷^𝑛𝑓(𝑧))^′𝐷−1|−|2𝑏(1−𝛼)+(1−𝜆)^𝑛𝑓(𝑧)𝑧+𝜆(𝐷^𝑛𝑓(𝑧))^′−1|=|^∞_𝑘=2[1+𝜆(𝑘−1)]𝐶(𝑛,𝑘)𝑎_𝑘𝑧^𝑘−1|−|2𝑏(1−𝛼)+^∞_𝑘=2[1+𝜆(𝑘−1)]𝐶(𝑛,𝑘)𝑎_𝑘𝑧^𝑘−1|≤^∞_𝑘=2[1+𝜆(𝑘−1)]𝐶(𝑛,𝑘)|𝑎_𝑘||𝑧|^𝑘−1−{2|𝑏|(1−𝛼)−^∞_𝑘=2[1+𝜆(𝑘−1)]𝐶(𝑛,𝑘)|𝑎_𝑘||𝑧|^𝑘−1}≤2{^∞_𝑘=2[1+𝜆(𝑘−1)]𝐶(𝑛,𝑘)|𝑎_𝑘|−|𝑏|(1−𝛼)}≤0, (1.12) which shows that 𝑓 belongs to 𝐻_𝑛(𝛼,𝜆,𝑏).

Let 𝐻^∗_𝑛(𝛼,𝜆,𝑏) denote the class of functions 𝑓 in 𝐴 whose Taylor-Maclaurin coefficients 𝑎_𝑘 satisfy the condition (1.9).

We note that 𝐻^∗_𝑛(𝛼,𝜆,𝑏)⊆𝐻_𝑛(𝛼,𝜆,𝑏). (1.13)

Example 1.6. (i) For 0≤𝛼<1, 𝜆>0, 𝑏∈ℂ⧵{0}, and 𝑛∈ℕ₀, the following function defined by: 𝑓₀(𝑧)=𝑧+2𝑏(1−𝛼)𝑧(𝑛+1)(𝜆+1)²³𝐹₂1(1,2,1+𝜆1;2+𝜆,𝑛+2;𝑧)(𝑧∈𝕌), (1.14) is in the class 𝐻_𝑛(𝛼,𝜆,𝑏).
(ii) For 0≤𝛼<1, 𝜆>0, 𝑏∈ℂ⧵{0}, and 𝑛∈ℕ₀, the following functions defined by: 𝑓₁(𝑧)=𝑧±(1−𝛼)|𝑏|𝑧(𝜆+1)(𝑛+1)²𝑓(𝑧∈𝕌),₂(𝑧)=𝑧±(1−𝛼)|𝑏|𝑧(2𝜆+1)(𝑛+1)(𝑛+2)³𝑓(𝑧∈𝕌),₃1(𝑧)=𝑧±𝑧(𝜆+1)(𝑛+1)²±2[(1−𝛼)|𝑏|−1]𝑧(2𝜆+1)(𝑛+1)(𝑛+2)³(𝑧∈𝕌) (1.15) are in the class 𝐻^∗_𝑛(𝛼,𝜆,𝑏).

In this paper, we obtain a sharp subordination result associated with the class 𝐻^∗_𝑛(𝛼,𝜆,𝑏) by using the same techniques as in [13] (see also [14–16]). Some applications of the main result which give important results of analytic functions are also investigated.

2. Main Theorem

Theorem 2.1. Let 𝑓∈𝐻^∗_𝑛(𝛼,𝜆,𝑏). Then (𝜆+1)(𝑛+1)2[(𝜆+1)(𝑛+1)+|𝑏|(1−𝛼)](𝑓∗𝑔)(𝑧)≺𝑔(𝑧)(𝑧∈𝕌) (2.1) for every function 𝑔 in 𝐾, and Re𝑓(𝑧)>−(𝜆+1)(𝑛+1)+|𝑏|(1−𝛼)(𝜆+1)(𝑛+1). (2.2) The constant (𝜆+1)(𝑛+1)/2[(𝜆+1)(𝑛+1)+|𝑏|(1−𝛼)] cannot be replaced by a larger one.

Proof. Let 𝑓∈𝐻^∗_𝑛(𝛼,𝜆,𝑏) and let 𝑔(𝑧)=𝑧+^∞_𝑘=2𝑐_𝑘𝑧^𝑘 (2.3) be any function in the class 𝐾. Then we readily have (𝜆+1)(𝑛+1)2[(𝜆+1)(𝑛+1)+|𝑏|(1−𝛼)](𝑓∗𝑔)(𝑧)=(𝜆+1)(𝑛+1)2[(𝜆+1)(𝑛+1)+|𝑏|(1−𝛼)](𝑧+^∞_𝑘=2𝑎_𝑘𝑐_𝑘𝑧^𝑘). (2.4) Thus, by Definition 1.2, the subordination result (2.1) will hold true if the sequence {(𝜆+1)(𝑛+1)𝑎_𝑘}2[(𝜆+1)(𝑛+1)+|𝑏|(1−𝛼)]^∞_𝑘=1 (2.5) is a subordinating factor sequence, with 𝑎₁=1. In view of Lemma 1.4, this is equivalent to the following inequality: Re{1+^∞_𝑘=1(𝜆+1)(𝑛+1)𝑎[(𝜆+1)(𝑛+1)+|𝑏|(1−𝛼)]^𝑘𝑧^𝑘}>0(𝑧∈𝕌). (2.6) Now, since [1+𝜆(𝑘−1)]𝐶(𝑛,𝑘)(𝜆≥0,𝑛∈ℕ₀) (2.7) is an increasing function of 𝑘, we have Re{1+^∞_𝑘=1(𝜆+1)(𝑛+1)𝑎[(𝜆+1)(𝑛+1)+|𝑏|(1−𝛼)]^𝑘𝑧^𝑘}=Re{1+(𝜆+1)(𝑛+1)𝑧+1[(𝜆+1)(𝑛+1)+|𝑏|(1−𝛼)][(𝜆+1)(𝑛+1)+|𝑏|(1−𝛼)]^∞_𝑘=2(𝜆+1)(𝑛+1)𝑎_𝑘𝑧^𝑘}>1−(𝜆+1)(𝑛+1)𝑟−1[(𝜆+1)(𝑛+1)+|𝑏|(1−𝛼)][(𝜆+1)(𝑛+1)+|𝑏|(1−𝛼)]^∞_𝑘=2[1+𝜆(𝑘−1)]𝐶(𝑛,𝑘)|𝑎_𝑘|𝑟^𝑘>1−(𝜆+1)(𝑛+1)[(𝜆+1)(𝑛+1)+|𝑏|(1−𝛼)]𝑟−|𝑏|(1−𝛼)[(𝜆+1)(𝑛+1)+|𝑏|(1−𝛼)]𝑟>0(|𝑧|=𝑟). (2.8) This proves the inequality (2.6), and hence also the subordination result (2.1) asserted by Theorem 2.1. The inequality (2.2) follows from (2.1) by taking 𝑧𝑔(𝑧)=1−𝑧∈𝐾. (2.9)
Next, we consider the function 𝑓₁(𝑧)=𝑧−|𝑏|(1−𝛼)𝑧(𝜆+1)(𝑛+1)²(0≤𝛼<1;𝜆≥0;𝑏∈ℂ⧵{0};𝑛∈ℕ₀) (2.10) which is a member of the class 𝐻^∗_𝑛(𝛼,𝜆,𝑏). Then by using (2.1), we have (𝜆+1)(𝑛+1)𝑓2[(𝜆+1)(𝑛+1)+|𝑏|(1−𝛼)]¹𝑧(𝑧)≺1−𝑧(𝑧∈𝕌). (2.11) It can be easily verified for the function 𝑓₁(𝑧) defined by (2.10) that inf_𝑧∈𝕌{Re((𝜆+1)(𝑛+1)𝑓2[(𝜆+1)(𝑛+1)+|𝑏|(1−𝛼)]¹1(𝑧))}=−2(𝑧∈𝕌) (2.12) which completes the proof of Theorem 2.1.

3. Some Applications

Taking 𝑛=0 in Theorem 2.1, we obtain the following.

Corollary 3.1. If the function 𝑓 defined by (1.1) satisfies ^∞_𝑘=2[1+𝜆(𝑘−1)]|𝑎_𝑘|≤𝑚(𝜆≥0,𝑚>0) (3.1) then for every function 𝑔 in 𝐾, one has (𝜆+1)𝑚2(𝜆+𝑚+1)(𝑓∗𝑔)(𝑧)≺𝑔(𝑧)(𝑧∈𝕌),Re𝑓(𝑧)>−(1+𝜆+1). (3.2) The constant (𝜆+1)/2(𝜆+𝑚+1) cannot be replaced by larger one.

Putting 𝜆=0 in Theorem 2.1, we have the following corollary.

Corollary 3.2. If the function 𝑓 defined by (1.1) satisfies ^∞_𝑘=2𝐶(𝑛,𝑘)|𝑎_𝑘|≤𝑚,𝑚>0,𝑛∈ℕ₀, (3.3) where 𝐶(𝑛,𝑘) is defined by (1.10), then for every function 𝑔 in 𝐾, one has (𝑛+1)𝑚2(𝑛+𝑚+1)(𝑓∗𝑔)(𝑧)≺𝑔(𝑧)(𝑧∈𝕌),Re𝑓(𝑧)>−(1+𝑛+1). (3.4) The constant (𝑛+1)/2(𝑛+𝑚+1) cannot be replaced by larger one.

Next letting 𝜆=1 and 𝑛=0, in Theorem 2.1, we obtain the following corollary.

Corollary 3.3. If the function 𝑓 satisfies ^∞_𝑘=2𝑘|𝑎_𝑘|≤𝑚(𝑚>0), (3.5) then for every function 𝑔 in 𝐾, one has 1𝑚(𝑚+2)(𝑓∗𝑔)(𝑧)≺𝑔(𝑧)(𝑧∈𝕌),Re𝑓(𝑧)>−(1+2). (3.6) The constant 1/(𝑚+2) cannot be replaced by a larger one.

Remark 3.4. Putting 𝜆=1, 𝑚=1, and 𝑛=0, in Theorem 2.1, we obtain the result due to Singh [17].

Also, by taking 𝜆=0 and 𝑛=0, in Theorem 2.1, we have the following.

Corollary 3.5. If the function 𝑓 satisfies ^∞_𝑘=2|𝑎_𝑘|≤𝑚(𝑚>0), (3.7) then for every function 𝑔 in 𝐾, one has 12(𝑚+1)(𝑓∗𝑔)(𝑧)≺𝑔(𝑧)(𝑧∈𝕌),Re𝑓(𝑧)>−(1+𝑚). (3.8) The constant 1/2(𝑚+1) cannot be replaced by a larger one.

It is clearly from the proof of Theorem 2.1 that the function 𝑓(𝑧)=𝑧−𝑚𝑧² (𝑚>0,𝑧∈𝕌) is the extremal function of Corollary 3.5. Also, the following example gives a nonpolynomial extremal function for the same corollary.

Example 3.6. Let the function ℎ be defined by ℎ(𝑧)=(𝑚+1)𝑧(𝑚+1)+𝑚𝑧(𝑚>0,𝑧∈𝕌), (3.9) the above function is analytic in 𝕌 and it is equivalent to ℎ(𝑧)=𝑧+^∞_𝑘=2(−𝑚)𝑚+1^𝑘−1𝑧^𝑘. (3.10) Then we have ^∞_𝑘=2|(−𝑚)𝑚+1^𝑘−1|=𝑚. (3.11) Therefore, the Taylor-Maclaurin coefficients of the function ℎ satisfy the condition in Corollary 3.5. Moreover, it can be easily verified that inf_𝑧∈𝕌Reℎ(𝑧)=ℎ(−1)=−(𝑚+1). (3.12) Then, the constant −(𝑚+1) cannot be replaced by a larger one. Therefore, the function ℎ is the extremal function of Corollary 3.5.