Growth Theorems for a Subclass of Strongly Spirallike Functions

Abstract

In this paper we consider a subclass of strongly spirallike functions on the unit disk in the complex plane , namely, strongly almost spirallike functions of type and order . We obtain the growth results for strongly almost spirallike functions of type and order on the unit disk in by using subordination principles and the geometric properties of analytic mappings. Furthermore we get the growth theorems for strongly almost starlike functions of order and strongly starlike functions on the unit disk of . These growth results follow the deviation results of these functions.

1. Introduction

Growth theorems for univalent analytic functions are important parts in geometric function theories of one complex variable. In 1983, Duren [1] obtained the following well-known growth and deviation theorem.

Theorem 1 (see [1]). If is a normalized biholomorphic function on the unit disk , then

Many scholars tried to extend the beautiful results to the cases in several complex variables. However, Cartan [2] pointed out that the corresponding growth theorem does not hold in several complex variables. He suggested that we may consider the biholomorphic mappings with special geometrical characteristic, such as convex mappings and starlike mappings.

In 1991, Barnard et al. [3] obtained the growth theorems for starlike mappings on the unit ball in firstly. After that, there are a lot of followup studies. Gong et al. [4] extended the results to the cases on and obtained the growth theorems for starlike mappings on the bounded convex Reinhardt domains . Graham and Varolin [5] obtained the growth and covering theorems for normalized biholomorphic convex functions on the unit disk and also obtained the growth and covering theorems for normalized biholomorphic starlike functions on the unit disk by Alexander’s theorem. Liu and Ren [6] obtained the growth theorems for starlike mappings on the general bounded starlike and circular domains in . Liu and Lu [7] obtained the growth theorems for starlike mappings of order on the bounded starlike and circular domains. Feng and Lu [8] obtained the growth theorems for almost starlike mappings of order on the bounded starlike and circular domains. Honda [9] obtained the growth theorems for normalized biholomorphic -symmetric convex mappings on the unit ball in complex Banach spaces. In recent years, there are a lot of new results about the growth and covering theorems for the subclasses of biholomorphic mappings in several complex variables [10–12].

It can be seen that we can make a great breakthrough in the growth and covering theorems for the subclasses of biholomorphic mappings in several complex variables if we restrict the biholomorphic mappings with the geometrical characteristic. The mappings discussed focus on starlike mappings, convex mappings, and their subclasses.

In 1974, Suffridge extended starlike mappings and convex mappings and gave the definition of spirallike mappings. Gurganus [13] gave the definition of spirallike mappings of type in several complex variables. Hamada and Kohr [14] obtained the growth theorems for spirallike mappings on some domains. Later Feng [15] gave the definition of almost spirallike mappings of type and order on the unit ball in . Feng et al. [16] obtained the growth theorems for almost spirallike mappings of type and order on the unit ball in complex Banach spaces.

However, when we introduce the definition of the new subclasses of starlike mappings, convex mappings, and spirallike mappings, we always discuss them in firstly.

In [17], Cai and Liu gave the definition of strongly almost spirallike functions of type and order on the unit disk. They also discussed their coefficient estimates.

In this paper, we mainly discuss the growth theorems for strongly almost spirallike functions of type and order on , where is the unit disk. Moreover we get the growth theorems for strongly almost starlike functions of order and strongly starlike functions on . At last, we obtain the deviation results of these functions.

Definition 2 (see [17]). Suppose that is an analytic function on , , , , and Then is called a strongly almost spirallike function of type and order on .

We can get the definition of strongly spirallike functions of type [18], strongly almost starlike functions of order [19], and strongly starlike functions on [19] by setting , , and , respectively, in Definition 2.

In order to give the main results, we need the following lemmas.

Lemma 3 (see [1]). Let be an univalent analytic function on . Then if and only if , .

Lemma 4 (see [20]). represents a circle whose center is and whose radius is in , where

Lemma 5 (see [20]). Let be an analytic function on and . Then and for .

2. Main Results

Theorem 6. Let be a strongly almost spirallike function of type and order on and . Then where

Proof. Since is a strongly almost spirallike function of type and order on , we get Let Then so we have . Therefore we get that there exists an analytic function on which satisfies , where , . Then Immediately, we have It follows that From Lemma 3, we deduce that the image of the unit disk under the mapping is the disk whose center is and whose radius is , where So we have Then
On the one hand, in view of (14), we have Observing that and for and , we get for and . Thus, in view of (15), (16), and (17), we obtain Let Then we have This means that Let Obviously, we have Observing that and , we deduce that . So is a monotone decreasing function for . Also we have from Lemma 4. Then
On the other hand, by direct computations, we have It follows that This means that . By (14) we know that In view of (15) and (19), we have Let Then Let Immediately, we have Also, we can get for , . Moreover, it is obvious that and . So we obtain . Therefore is a monotone increasing function for . In addition, we have from Lemma 4. Hence
From the above results, we obtain This completes the proof.

Theorem 7. Suppose that is a strongly almost starlike function of order on and . Then

Proof. Let and in Theorem 6. Then (34) holds, so we can obtain the same result; that is, where Therefore we get the conclusion.

Let in Theorem 7; we can get the following result for strongly starlike functions.

Corollary 8. Let be a strongly starlike function on and . Then

Theorem 9. Let be a strongly almost spirallike function of type and order on and . Then where

Proof. From Theorem 6, we have Let . Since we get Thus Furthermore, It follows that Let ; we have Consequently, Observing that , we have This completes the proof.

Similar to Theorem 9, by Theorem 7, we can get the following results.

Theorem 10. Let be a strongly almost starlike function of order on and . Then

Theorem 11. Let be a strongly almost starlike function of order on and . Then

Remark 12. Let in Theorem 11. Then we have Let in Theorem 10. Then we have
Let in Theorem 11; we can get the following result.

Corollary 13. Let be a strongly starlike function on and . Then

Proof. According to Corollary 8, we obtain Let . Since we have Thus So we get Letting , it follows that Therefore we obtain
Also, we can get the conclusion by letting in Theorem 11. This completes the proof.

Theorem 14. Suppose that is a strongly starlike function on and ; then

Proof. On the one hand, from Corollary 13, we obtain .
On the other hand, by and in the proof of Theorem 6, we can obtain for . Let . Then we have Therefore is a monotone increasing function with respect to . Also we can know that from Lemma 4. Hence By (14) we obtain Furthermore, , so Let . Since , we have Therefore we obtain Then we have So Therefore we obtain This completes the proof.

From Theorems 6 and 9, we can get the following result.

Theorem 15. Let be a strongly almost spirallike function of type and order on and , . Then where

From Theorems 7 and 11, we can get the following result.

Theorem 16. Let be a strongly almost starlike function of order on and . Then

Let in Theorem 16; we can get the following result.

Corollary 17. Let be a strongly starlike function on and . Then

Conflict of Interests

The authors declare that they have no conflict of interests.

Acknowledgments

This work is supported by NSF of China (nos. 11271359 and U1204618) and Science and Technology Research Projects of Henan Provincial Education Department (nos. 14B110015 and 14B110016).