Abstract

We introduce a certain subclass of multivalent analytic functions by making use of the principle of subordination between these functions and Cătas operator. Such results as subordination and superordination properties, convolution properties, inclusion relationships, distortion theorems, inequality properties, and sufficient conditions for multivalent starlikeness are provide. The results presented here would provide extensions of those given in earlier works. Several other new results are also obtained.

1. Introduction

Let denote the class of functions of the following form: which are analytic in the open unit disk .

For simplicity, we write

A function is said to be in the class of -valent starlike functions of order in if it satisfies the following inequality: Let be the class of analytic functions of the following form: Let , where is given by (1.1) and is defined by Then the Hadanard product (or convolution) of the functions and is defined by We consider the following multiplier transformations.

Definition 1.1 (see [1]). Let . For , define the multiplier transformations on by the following infinite series: It is easily verified from (1.7), that
It should be remarked that the class of multiplier transforms is a generalization of several other linear operators considered, in earlier investigations (see [212]).
If is given by (1.1), then we have where In particular, we set
For two functions and , analytic in , we say that the function is subordinate to in , and write if there exists a Schwarz function , which is analytic in with such that Indeed, it is known that Furthermore, if the function is univalent in , then we have the following equivalence: By making use of the linear operator and the above-mentioned principle of subordination between analytic functions, we introduce and investigate the following subclass of the class of -valent analytic functions.

Definition 1.2. A function is said to be in the class if it satisfies the following subordination condition: where (and throughout this paper unless otherwise mentioned) the parameters , and are constrained as follows:

For simplicity, we write Clearly, the class is a subclass of the familiar class of Bazilevič functions of type .

If we set in the class , which was studied by Liu [13]. In particular, Zhu [14] determined the sufficient conditions such that .

Ctas [1, 5, 15], Cho and Srivastava [6], Cho and Kim [7], and Kumar et al. [10] obtained many interesting results associated with the multiplier operator.

In the present paper, we aim at proving such results as subordination and superordination properties, convolution properties, inclusion relationships, distortion theorems, inequality properties, and sufficient conditions for multivalent starlikeness of the class . The results presented here would provide extensions of those given in earlier works. Several other new results are also obtained.

2. Preliminary Results

In order to establish our main results, we need the following definition and lemmas.

Definition 2.1 (see [16]). Denote by the set of all functions that are analytic and injective on , where and such that for .

Lemma 2.2 (see [17]). Let the function be analytic and univalent (convex) in with . Suppose also that the function given by is analytic in . If then and is the best dominant of (2.3).

Lemma 2.3 (see [18]). Let be a convex univalent function in and let with If the function is analytic in and then and is the best dominant.

Lemma 2.4 (see [16]). Let be convex univalent in and . Further assume that if and is univalent in , then implies that and is the best subdominant.

Lemma 2.5 (Jach’s Lemma [19]). Let be a noncostant analytic function in with . If attains its maximum value on the circle at , then where is a real number.

Lemma 2.6 (see [20]). Let be analytic and convex in . If and ; then

Lemma 2.7 (see [21, 22]). Let . Suppose also that m is convex and univalent in with If is analytic in with , then the following subordination: implies that

Lemma 2.8 (see [23]). Let analytic in and be analytic and convex in . If , then .

Lemma 2.9 (see [24]). Let , and , where If satisfies the following subordination condition: then

3. Main Results

We begin by presenting our first subordination property given by Theorem 3.1 below.

Theorem 3.1. Let with . Then

Proof. Define the function by Then is analytic in with . By taking the derivatives in the both sides in equality (3.2) and using (1.8), we get An application of Lemma 2.2 to (3.3) yields where The proof of Theorem 3.1 is thus completed.

Theorem 3.2. Let be univalent in , . Suppose also that satisfies If satisfying the following subordination: then and is the best dominant.

Proof. Let the function be defined by (3.2). We know that (3.3) holds true. Combining (3.3) and (3.7), we find that

By Lemma 2.3 and (3.9), we easily get the assertion of Theorem 3.2.

Taking in Theorem 3.2, we get the following result.

Corollary 3.3. Let and . Suppose also that satisfies the condition (3.6). If satisfies the following subordination: then and is the best dominant.
If is subordinate to , then is superordinate to . We now derive the following superordination result for the class .

Theorem 3.4. Let be convex univalent in , with . Also let be univalent in . If then and is the best subdominant.

Proof. Let the function be defined by (3.2). Then An application of Lemma 2.4 yields the assertion of Theorem 3.4.

Taking in Theorem 3.4, we get the following corollary.

Corollary 3.5. Let be convex univalent in and with . Also let be univalent in . If then and is best subdominant.

Combining the above results of subordination and superordination. We easily get the following “Sandwich-type result”.

Corollary 3.6. Let be convex univalent and let be univalent in , , . Let satisfies (3.6). If is univalent in , also then and are, respectively, the best subordinate, and dominant.

Theorem 3.7. Let , and . Also let the function be defined by If satisfies one of the following conditions: or then

Proof. We define the function by It is easy to see that the function is analytic in with .
Differentiating both sides of (3.26) with respect to logarithmically, we get We now consider the function defined by Assume that there exists a point such that by Lemma 2.5, we know that If follows from (3.28) and (3.30) that But the inequalities in (3.31) and (3.32) contradict, respectively, the inequalities in (3.23) and (3.24). Therefore, we can conclude that which implies that We thus complete the proof of Theorem 3.7.

From Theorem 3.7, we easily get the following result for the class ß of Bazilevič functions of type .

Corollary 3.8. Let , and . Also let the function be defined by (3.22). If satisfies one of the following conditions: then .

Theorem 3.9. Let , and . Then for , where The bound is the best possible.

Proof. Suppose that where is analytic and has a positive real part in . By taking the derivatives in the both sides in equality (3.37) and using (1.9), we get By making use of the following well-known estimate (see [25]): in (3.38), we obtain that for , where is given by (3.36).
To show that the bound is the best possible, we consider the function defined by By noting that for , we conclude that the bound is the best possible. Theorem 3.9 is thus proved.

Theorem 3.10. Let with . Then where is analytic in with and .

Proof. Suppose that . It follows from (3.1) that where is analytic in with and .
By virtue of (3.44), we easily find that Combining (1.10), (1.16), and (3.45), we have The assertion (3.43) of Theorem 3.10 can now easily be derived from (3.46).

Theorem 3.11. Let with . Then

Proof. Suppose that with . We know that (3.1) holds true, which implies that It is easy to see that the condition (3.48) can be written as follows: Combining (1.9), (1.10), and (3.49), we easily get the convolution property (3.47) asserted by Theorem 3.11.

Theorem 3.12. Let and . Then

Proof. Suppose that . We know that Since , we easily find that that is, . Thus the assertion of Theorem 3.12 holds for .
If , by Theorem 3.1 and (3.52), we know that , that is, At the same time, we have Moreover, since and , is analytic and convex in . Combining (3.52)–(3.54) and Lemma 2.6, we find that that is, , which implies that the assertion (3.50) of Theorem 3.12 holds.
Let denote the class of functions of the following form: which are analytic and convex in and satisfy the following condition: By making use of the principle of subordination between analytic functions, we introduce the subclasses and of the class : Next, by using the operator defined by (1.7), we define the following two subclasses and of the class : Clearly, we know that
We now derive some inclusion relationships for the classes and , by similarly applying the method of proof of Proposition 1 obtained by Cho et al. [26] and Wang et al. [27].

Theorem 3.13. Let , and with Then

Theorem 3.14. Let and with (3.61) holds. Then

Proof. By virtue of (3.60) and Theorem 3.13, we observe that From (3.64), we conclude that the assertion of Theorem 3.14 holds true.

Taking in Theorems 3.13 and 3.14, we get the following results.

Corollary 3.15. Let , and . Then

Theorem 3.16. Let with and . Then The extremal function of (3.66) is defined by

Proof. Let with . From Theorem 3.1, we know that (3.1) holds, which implies that Combining both equations of (3.68), we get (3.66). By noting that the function