Research Article | Open Access

# Growth Theorems for a Subclass of Strongly Spirallike Functions

**Academic Editor:**Francisco J. Marcellán

#### 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).

#### References

- P. L. Duren,
*Univalent Functions*, Springer, Berlin, Germany, 1983. View at: MathSciNet - H. Cartan, “Sur la possibilite d'entendre aux fonctions de plusieurs variables complexes la theorie des fonctions univalents,” in
*Lecons sur les Fonctions Univalents on Mutivalents*, P. Montel, Ed., pp. 129–155, Gauthier-Villar, 1933. View at: Google Scholar - R. W. Barnard, C. H. FitzGerald, and S. Gong, “The growth and 1/4-theorems for starlike mappings in ${\mathbb{C}}^{n}$,”
*Pacific Journal of Mathematics*, vol. 150, no. 1, pp. 13–22, 1991. View at: Publisher Site | Google Scholar | MathSciNet - S. Gong, S. K. Wang, and Q. H. Yu, “The growth and $1/4$-theorem for starlike mappings on ${B}_{p}$,”
*Chinese Annals of Mathematics B*, vol. 11, no. 1, pp. 100–104, 1990. View at: Google Scholar | MathSciNet - I. Graham and D. Varolin, “Bloch constants in one and several variables,”
*Pacific Journal of Mathematics*, vol. 174, no. 2, pp. 347–357, 1996. View at: Google Scholar | MathSciNet - T. Liu and G. Ren, “The growth theorem for starlike mappings on bounded starlike circular domains,”
*Chinese Annals of Mathematics B*, vol. 19, no. 4, pp. 401–408, 1998. View at: Google Scholar | MathSciNet - H. Liu and K. P. Lu, “Two subclasses of starlike mappings in several complex variables,”
*Chinese Annals of Mathematics*, vol. 21, no. 5, pp. 533–546, 2000. View at: Google Scholar | MathSciNet - S. X. Feng and K. P. Lu, “The growth theorem for almost starlike mappings of order $\alpha $ on bounded starlike circular domains,”
*Chinese Quarterly Journal of Mathematics. Shuxue Jikan*, vol. 15, no. 2, pp. 50–56, 2000. View at: Google Scholar | MathSciNet - T. Honda, “The growth theorem for $k$-fold symmetric convex mappings,”
*The Bulletin of the London Mathematical Society*, vol. 34, no. 6, pp. 717–724, 2002. View at: Publisher Site | Google Scholar | MathSciNet - N. I. Mahmudov and M. Eini Keleshteri, “$q$-extensions for the apostol type polynomials,”
*Journal of Applied Mathematics*, vol. 2014, Article ID 868167, 8 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet - E. Merkes and M. Salmassi, “Subclasses of uniformly starlike functions,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 15, no. 3, pp. 449–454, 1992. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Singh, “A subordination theorem for spirallike functions,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 24, no. 7, pp. 433–435, 2000. View at: Publisher Site | Google Scholar | MathSciNet - K. R. Gurganus, “
*ϕ*-like holomorphic functions in ${\mathbb{C}}^{n}$ and Banach spaces,”*Transactions of the American Mathematical Society*, vol. 205, pp. 389–406, 1975. View at: Google Scholar | MathSciNet - H. Hamada and G. Kohr, “Subordination chains and the growth theorem of spirallike mappings,”
*Mathematica*, vol. 42(65), no. 2, pp. 153–161 (2001), 2000. View at: Google Scholar | Zentralblatt MATH | MathSciNet - S. X. Feng,
*Some classes of holomorphic mappings in several complex variables [Ph.D. thesis]*, University of Science and Technology of China, Hefei, China, 2004. - S. X. Feng, T. S. Liu, and G. B. Ren, “The growth and covering theorems for several mappings on the unit ball in complex Banach spaces,”
*Chinese Annals of Mathematics A*, vol. 28, no. 2, pp. 215–230, 2007. View at: Google Scholar | MathSciNet - R. H. Cai and X. S. Liu, “The third and fourth coefficient estimations for the subclasses of strongly spirallike functions,”
*Journal of Zhanjiang Normal College*, vol. 31, pp. 38–43, 2010. View at: Google Scholar - H. Hamada and G. Kohr, “The growth theorem and quasiconformal extension of strongly spirallike mappings of type
*α*,”*Complex Variables*, vol. 44, no. 4, pp. 281–297, 2001. View at: Google Scholar | MathSciNet - M. Chuaqui, “Applications of subordination chains to starlike mappings in ${\mathbb{C}}^{n}$,”
*Pacific Journal of Mathematics*, vol. 168, no. 1, pp. 33–48, 1995. View at: Publisher Site | Google Scholar | MathSciNet - L. V. Ahlfors,
*Complex Analysis*, McGraw-Hill, New York, NY, USA, 3rd edition, 1978. View at: MathSciNet

#### Copyright

Copyright © 2014 Yan-Yan Cui et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.