Research Article | Open Access

# Some Properties of Certain Multivalent Analytic Functions Involving the Cătas Operator

**Academic Editor:**Marco Squassina

#### 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 [2–12]).

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