JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 10.1155/2014/608641 608641 Research Article Growth Theorems for a Subclass of Strongly Spirallike Functions Cui Yan-Yan Wang Chao-Jun Zhu Si-Feng Marcellán Francisco J. 1 College of Mathematics and Statistics Zhoukou Normal University Zhoukou, Henan 466001 China zknu.edu.cn 2014 1282014 2014 08 06 2014 22 07 2014 22 07 2014 12 8 2014 2014 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.

In this paper we consider a subclass of strongly spirallike functions on the unit disk D in the complex plane C , namely, strongly almost spirallike functions of type β and order α . We obtain the growth results for strongly almost spirallike functions of type β and order α on the unit disk D in C by using subordination principles and the geometric properties of analytic mappings. Furthermore we get the growth theorems for strongly almost starlike functions of order α and strongly starlike functions on the unit disk D of C . 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  obtained the following well-known growth and deviation theorem.

Theorem 1 (see [<xref ref-type="bibr" rid="B1">1</xref>]).

If f ( z ) is a normalized biholomorphic function on the unit disk D , then (1) | z | ( 1 + | z | ) 2 | f ( z ) | | z | ( 1 - | z | ) 2 , 1 - | z | ( 1 + | z | ) 3 | f ( z ) | 1 + | z | ( 1 - | z | ) 3 .

Many scholars tried to extend the beautiful results to the cases in several complex variables. However, Cartan  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.  obtained the growth theorems for starlike mappings on the unit ball B n in C n firstly. After that, there are a lot of followup studies. Gong et al.  extended the results to the cases on B n and obtained the growth theorems for starlike mappings on the bounded convex Reinhardt domains B p . Graham and Varolin  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  obtained the growth theorems for starlike mappings on the general bounded starlike and circular domains in C n . Liu and Lu  obtained the growth theorems for starlike mappings of order α on the bounded starlike and circular domains. Feng and Lu  obtained the growth theorems for almost starlike mappings of order α on the bounded starlike and circular domains. Honda  obtained the growth theorems for normalized biholomorphic k -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 .

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  gave the definition of spirallike mappings of type β in several complex variables. Hamada and Kohr  obtained the growth theorems for spirallike mappings on some domains. Later Feng  gave the definition of almost spirallike mappings of type β and order α on the unit ball B n in C n . Feng et al.  obtained the growth theorems for almost spirallike mappings of type β and order α 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 C firstly.

In , Cai and Liu gave the definition of strongly almost spirallike functions of type β and order α 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 β and order α on D , where D is the unit disk. Moreover we get the growth theorems for strongly almost starlike functions of order α and strongly starlike functions on D . At last, we obtain the deviation results of these functions.

Definition 2 (see [<xref ref-type="bibr" rid="B17">17</xref>]).

Suppose that f ( z ) is an analytic function on D , α [ 0,1 ) , β ( - π / 2 , π / 2 ) , c ( 0,1 ) , and (2) | - α + i tan β 1 - α + 1 - i tan β 1 - α · f ( z ) z f ( z ) - 1 + c 2 1 - c 2 | < 2 c 1 - c 2 , z D { 0 } . Then f ( z ) is called a strongly almost spirallike function of type β and order α on D .

We can get the definition of strongly spirallike functions of type β , strongly almost starlike functions of order α , and strongly starlike functions on D  by setting α = 0 , β = 0 , and α = β = 0 , respectively, in Definition 2.

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

Lemma 3 (see [<xref ref-type="bibr" rid="B1">1</xref>]).

Let g ( z ) be an univalent analytic function on D . Then f ( z ) g ( z ) if and only if f ( 0 ) = g ( 0 ) , f ( D ) g ( D ) .

Lemma 4 (see [<xref ref-type="bibr" rid="B20">20</xref>]).

| ( z - z 1 ) / ( z - z 2 ) | = k    ( 0 < k 1 , z 1 z 2 ) represents a circle whose center is z 0 and whose radius is ρ in C , where (3) z 0 = z 1 - k 2 z 2 1 - k 2 , ρ = k | z 1 - z 2 | 1 - k 2 .

Lemma 5 (see [<xref ref-type="bibr" rid="B20">20</xref>]).

Let f ( z ) : D D be an analytic function on D and f ( 0 ) = 0 . Then | f ( 0 ) | 1 and | f ( z ) | | z | for z D .

2. Main Results Theorem 6.

Let f ( z ) be a strongly almost spirallike function of type β and order α on D and α [ 1 / 2,1 ) , β ( - π / 2 , π / 2 ) , c ( 0,1 ) . Then (4) 1 - c 2 | z | 2 1 + m 1 | z | 2 + n | z | | z f ( z ) f ( z ) | 1 + m 1 | z | 2 + n | z | 1 - m 2 | z | 2 , where (5) m 1 = c 2 [ 2 ( 1 - α ) cos β ( | sin β | + cos β ) - 1 ] , n = 2 c ( 1 - α ) cos β , m 2 = c 2 [ 1 - 4 α ( 1 - α ) cos 2 β ] .

Proof.

Since f ( z ) is a strongly almost spirallike function of type β and order α on D , we get (6) | - α + i tan β 1 - α + 1 - i tan β 1 - α · f ( z ) z f ( z ) - 1 + c 2 1 - c 2 | < 2 c 1 - c 2 . Let (7) p ( z ) = - α + i tan β 1 - α + 1 - i tan β 1 - α · f ( z ) z f ( z ) . Then (8) p ( 0 ) = 1 , | p ( z ) - 1 + c 2 1 - c 2 | < 2 c 1 - c 2 , so we have p ( z ) ( 1 + c z ) / ( 1 - c z ) . Therefore we get that there exists an analytic function w ( z ) on D which satisfies p ( z ) = ( 1 + c w ( z ) ) / ( 1 - c w ( z ) ) , where w ( 0 ) = 0 , | w ( z ) | < 1 . Then (9) - α + i tan β 1 - α + 1 - i tan β 1 - α · f ( z ) z f ( z ) = 1 + c w ( z ) 1 - c w ( z ) . Immediately, we have (10) c w ( z ) = ( f ( z ) / z f ( z ) ) - 1 ( f ( z ) / z f ( z ) ) + ( ( 1 - 2 α + i tan β ) / ( 1 - i tan β ) ) . It follows that (11) | [ f ( z ) z f ( z ) - 1 ] [ f ( z ) z f ( z ) - 2 α - 1 - i tan β 1 - i tan β ] - 1 | = c | w ( z ) | . From Lemma 3, we deduce that the image of the unit disk D under the mapping f ( z ) / z f ( z ) is the disk whose center is a and whose radius is ρ , where (12) a = [ 1 - c 2 | w ( z ) | 2 · 2 α - 1 - i tan β 1 - i tan β ] 1 1 - c 2 | w ( z ) | 2 = 1 1 - c 2 | w ( z ) | 2 { 1 - c 2 | w ( z ) | 2 [ 2 α cos 2 β - cos ( 2 β ) = 1 - c 2 | w ( z ) | 2 + 1 - c 2 | w ( z ) | 2 | w ( z ) | 2 + i ( α - 1 ) sin ( 2 β ) ] } , ρ = c | w ( z ) | · 2 ( 1 - α ) cos β 1 - c 2 | w ( z ) | 2 . So we have (13) | f ( z ) z f ( z ) - a | ρ . Then (14) | a | - ρ | f ( z ) z f ( z ) | | a | + ρ .

On the one hand, in view of (14), we have (15) | f ( z ) z f ( z ) | 1 1 - c 2 | w ( z ) | 2 × { | 1 - c 2 | w ( z ) | 2 ( 2 α cos 2 β - cos ( 2 β ) ) | + c 2 | w ( z ) | 2 ( 1 - α ) | sin ( 2 β ) | + c | w ( z ) | · 2 ( 1 - α ) cos β | w ( z ) | 2 } . Observing that (16) 2 α cos 2 β - cos ( 2 β ) = 2 α cos 2 β - 2 cos 2 β + 1 = 2 ( α - 1 ) cos 2 β + 1 = 1 - 2 ( 1 - α ) cos 2 β and 1 - 2 ( 1 - α ) cos 2 β < 1 for α [ 1 / 2,1 ) and β ( - π / 2 , π / 2 ) , we get (17) 1 - c 2 | w ( z ) | 2 ( 2 α cos 2 β - cos ( 2 β ) ) > 0 for c ( 0,1 ) and | w ( z ) | < 1 . Thus, in view of (15), (16), and (17), we obtain (18) | f ( z ) z f ( z ) | 1 1 - c 2 | w ( z ) | 2 × { 1 - c 2 | w ( z ) | 2 ( 2 α cos 2 β - cos ( 2 β ) ) + c 2 | w ( z ) | 2 ( 1 - α ) | sin ( 2 β ) | + c | w ( z ) | · 2 ( 1 - α ) cos β | w ( z ) | 2 } = 1 1 - c 2 | w ( z ) | 2 × { 1 + c 2 | w ( z ) | 2 × [ ( 1 - α ) | sin ( 2 β ) | ( 2 α cos 2 β - cos ( 2 β ) ) - ( 2 α cos 2 β - cos ( 2 β ) ) ] + c | w ( z ) | · 2 ( 1 - α ) cos β [ ( 1 - α ) | sin ( 2 β ) | - ( 2 α cos 2 β - cos ( 2 β ) ) ] } = 1 1 - c 2 | w ( z ) | 2 × { 1 + c 2 | w ( z ) | 2 × [ 2 ( 1 - α ) | sin β | cos β ( 1 - 2 ( 1 - α ) cos 2 β ) - ( 1 - 2 ( 1 - α ) cos 2 β ) ] + c | w ( z ) | · 2 ( 1 - α ) cos β c 2 | w ( z ) | 2 } = 1 1 - c 2 | w ( z ) | 2 × { 1 + c 2 | w ( z ) | 2 × [ 2 ( 1 - α ) cos β ( | sin β | + cos β ) - 1 ] + c | w ( z ) | · 2 ( 1 - α ) cos β | w ( z ) | 2 } . Let (19) c 2 [ 2 ( 1 - α ) cos β ( | sin β | + cos β ) - 1 ] = m 1 , 2 c ( 1 - α ) cos β = n . Then we have (20) | f ( z ) z f ( z ) | 1 + m 1 | w ( z ) | 2 + n | w ( z ) | 1 - c 2 | w ( z ) | 2 . This means that (21) | z f ( z ) f ( z ) | 1 - c 2 | w ( z ) | 2 1 + m 1 | w ( z ) | 2 + n | w ( z ) | . Let (22) | w ( z ) | = x , 1 - c 2 x 2 1 + m 1 x 2 + n x = g ( x ) . Obviously, we have (23) g ( x ) = - n c 2 x 2 + 2 ( m 1 + c 2 ) x + n ( 1 + m 1 x 2 + n x ) 2 . Observing that (24) m 1 + c 2 = c 2 · 2 ( 1 - α ) cos β ( | sin β | + cos β ) > 0 and n > 0 , x = | w ( z ) | 0 , we deduce that g ( x ) < 0 . So g ( x ) is a monotone decreasing function for x [0,1) . Also we have | w ( z ) | | z | from Lemma 4. Then (25) | z f ( z ) f ( z ) | 1 - c 2 | w ( z ) | 2 1 + m 1 | w ( z ) | 2 + n | w ( z ) | 1 - c 2 | z | 2 1 + m 1 | z | 2 + n | z | .

On the other hand, by direct computations, we have (26) | a | 2 = 1 ( 1 - c 2 | w ( z ) | 2 ) 2 × { [ 1 - c 2 | w ( z ) | 2 ( 1 + 2 ( α - 1 ) cos 2 β ) ] 2 + + + [ 1 - c 2 | w ( z ) | 2 ( 1 + 2 ( α - 1 ) cos 2 β ) ] 2 + c 4 | w ( z ) | 4 [ 2 ( α - 1 ) cos β sin β ] 2 } = 1 ( 1 - c 2 | w ( z ) | 2 ) 2 × { 1 - 2 c 2 | w ( z ) | 2 [ 1 + 2 ( α - 1 ) cos 2 β ] + + + + c 4 | w ( z ) | 4 [ 1 + 4 α ( α - 1 ) cos 2 β ] } , | ρ | 2 = 1 ( 1 - c 2 | w ( z ) | 2 ) 2 · 4 c 2 | w ( z ) | 2 ( 1 - α ) 2 cos 2 β . It follows that (27) ( | a | 2 - ρ 2 ) ( 1 - c 2 | w ( z ) | 2 ) 2 = 1 - 2 c 2 | w ( z ) | 2 [ 1 + 2 ( α - 1 ) cos 2 β + 2 ( 1 - α ) 2 cos 2 β ] + c 4 | w ( z ) | 4 [ 1 + 4 α ( α - 1 ) cos 2 β ] = 1 - 2 c 2 | w ( z ) | 2 [ 1 + 2 α ( α - 1 ) cos 2 β ] + c 4 | w ( z ) | 4 [ 1 + 4 α ( α - 1 ) cos 2 β ] = [ 1 - 2 c 2 | w ( z ) | 2 + c 4 | w ( z ) | 4 ] + c 4 | w ( z ) | 4 · 4 α ( α - 1 ) cos 2 β - 2 c 2 | w ( z ) | 2 · 2 α ( α - 1 ) cos 2 β = ( 1 - c 2 | w ( z ) | 2 ) 2 + 4 α ( α - 1 ) cos 2 β · c 2 | w ( z ) | 2 ( c 2 | w ( z ) | 2 - 1 ) > 0 . This means that | a | > ρ . By (14) we know that (28) | f ( z ) z f ( z ) | | a | - ρ . In view of (15) and (19), we have (29) | z f ( z ) f ( z ) | 1 | a | - ρ = | a | + ρ | a | 2 - ρ 2 = ( ( 1 - c 2 | w ( z ) | 2 ) 2 ( | a | + ρ ) ) × ( ( 1 - c 2 | w ( z ) | 2 ) 2 + 4 α ( α - 1 ) cos 2 β ( 1 - c 2 | w ( z ) | 2 ) 2 · c 2 | w ( z ) | 2 ( c 2 | w ( z ) | 2 - 1 ) ) - 1 1 + m 1 | w ( z ) | 2 + n | w ( z ) | 1 - c 2 | w ( z ) | 2 + 4 α ( 1 - α ) cos 2 β · c 2 | w ( z ) | 2 = 1 + m 1 | w ( z ) | 2 + n | w ( z ) | 1 + [ 4 α ( 1 - α ) cos 2 β - 1 ] c 2 | w ( z ) | 2 . Let (30) c 2 [ 1 - 4 α ( 1 - α ) cos 2 β ] = m 2 . Then (31) | z f ( z ) f ( z ) | 1 + m 1 | w ( z ) | 2 + n | w ( z ) | 1 - m 2 | w ( z ) | 2 . Let (32) | w ( z ) | = x , 1 + m 1 x 2 + n x 1 - m 2 x 2 = h ( x ) . Immediately, we have (33) h ( x ) = ( 2 m 1 x + n ) ( 1 - m 2 x 2 ) + ( 1 + m 1 x 2 + n x ) · 2 m 2 x ( 1 - m 2 x 2 ) 2 = n m 2 x 2 + 2 ( m 1 + m 2 ) x + n ( 1 - m 2 x 2 ) 2 = n m 2 ( 1 - m 2 x 2 ) 2 × [ ( x + m 1 + m 2 n m 2 ) 2 + n 2 m 2 - ( m 1 + m 2 ) 2 n 2 m 2 2 ] . Also, we can get (34) n 2 m 2 - ( m 1 + m 2 ) 2 = - 4 c 2 ( 1 - α ) 2 cos 2 β · 2 ( 1 - 2 α ) | sin β | cos β 0 for α [ 1 / 2,1 ) , β ( - π / 2 , π / 2 ) . Moreover, it is obvious that m 2 > 0 and n > 0 . So we obtain h ( x ) > 0 . Therefore h ( x ) is a monotone increasing function for x [0,1) . In addition, we have | w ( z ) | | z | from Lemma 4. Hence (35) | z f ( z ) f ( z ) | 1 + m 1 | w ( z ) | 2 + n | w ( z ) | 1 - m 2 | w ( z ) | 2 1 + m 1 | z | 2 + n | z | 1 - m 2 | z | 2 .

From the above results, we obtain (36) 1 - c 2 | z | 2 1 + m 1 | z | 2 + n | z | | z f ( z ) f ( z ) | 1 + m 1 | z | 2 + n | z | 1 - m 2 | z | 2 . This completes the proof.

Theorem 7.

Suppose that f ( z ) is a strongly almost starlike function of order α on D and α [ 0,1 ) , c ( 0,1 ) . Then (37) 1 - c 2 | z | 2 1 + c 2 ( 1 - 2 α ) | z | 2 + 2 c ( 1 - α ) | z | | z f ( z ) f ( z ) | 1 + c 2 ( 1 - 2 α ) | z | 2 + 2 c ( 1 - α ) | z | 1 - c 2 ( 1 - 2 α ) 2 | z | 2 .

Proof.

Let β = 0 and α [ 0,1 ) in Theorem 6. Then (34) holds, so we can obtain the same result; that is, (38) 1 - c 2 | z | 2 1 + m 1 | z | 2 + n | z | | z f ( z ) f ( z ) | 1 + m 1 | z | 2 + n | z | 1 - m 2 | z | 2 , where (39) m 1 = c 2 ( 1 - 2 α ) , n = 2 c ( 1 - 2 α ) , m 2 = c 2 ( 1 - 2 α ) 2 . Therefore we get the conclusion.

Let α = 0 in Theorem 7; we can get the following result for strongly starlike functions.

Corollary 8.

Let f ( z ) be a strongly starlike function on D and c ( 0,1 ) . Then (40) 1 - c | z | 1 + c | z | | z f ( z ) f ( z ) | 1 + c | z | 1 - c | z | .

Theorem 9.

Let f ( z ) be a strongly almost spirallike function of type β and order α on D and α ( 1 / 2,1 ) , β ( - π / 2 , π / 2 ) , c ( 0,1 ) . Then (41) | f ( z ) | | z | ( 1 + m 2 | z | ) ( m 1 + m 2 - n m 2 ) / - 2 m 2 | f ( z ) | · ( 1 - m 2 | z | ) ( m 1 + m 2 + n m 2 ) / - 2 m 2 , where (42) m 1 = c 2 [ 2 ( 1 - α ) cos β ( | sin β | + cos β ) - 1 ] , n = 2 c ( 1 - α ) cos β , m 2 = c 2 [ 1 - 4 α ( 1 - α ) cos 2 β ] .

Proof.

From Theorem 6, we have (43) Re ( z f ( z ) f ( z ) ) | z f ( z ) f ( z ) | 1 + m 1 | z | 2 + n | z | 1 - m 2 | z | 2 . Let z = r e i θ . Since (44) Re ( z f ( z ) f ( z ) ) = r ln | f ( z ) | r , we get (45) r ln | f ( z ) | r 1 + m 1 | z | 2 + n | z | 1 - m 2 | z | 2 . Thus (46) ɛ | z | ln | f ( z ) | r d r ɛ | z | 1 + m 1 r 2 + n r ( 1 - m 2 r 2 ) r d r . Furthermore, (47) ɛ | z | 1 + m 1 r 2 + n r ( 1 - m 2 r 2 ) r d r = ( m 1 + m 2 ) ɛ | z | r 1 - m 2 r 2 d r + n ɛ | z | d r 1 - m 2 r 2 + ɛ | z | d r r = m 1 + m 2 - 2 m 2 ln | 1 - m 2 r 2 | | r = ɛ r = | z | + n 2 m 2 ln | - 2 m 2 r - 2 m 2 - 2 m 2 r + 2 m 2 | | r = ɛ r = | z | + ln r | r = ɛ r = | z | . It follows that (48) ln | f ( r e i θ ) | | r = ɛ r = | z | m 1 + m 2 - 2 m 2 ln | 1 - m 2 r 2 | | r = ɛ r = | z | + n 2 m 2 ln | m 2 r + 1 m 2 r - 1 | | r = ɛ r = | z | + ln r | r = ɛ r = | z | . Let ɛ 0 ; we have (49) ln | f ( z ) | m 1 + m 2 - 2 m 2 ln | 1 - m 2 | z | 2 | + n 2 m 2 ln | m 2 | z | + 1 m 2 | z | - 1 | + ln | z | . Consequently, (50) | f ( z ) | | z | · | 1 - m 2 | z | 2 | ( m 1 + m 2 ) / - 2 m 2 · | m 2 | z | + 1 m 2 | z | - 1 | n / 2 m 2 . Observing that m 2 < 1 , we have (51) | f ( z ) | | z | · ( 1 - m 2 | z | 2 ) ( m 1 + m 2 ) / - 2 m 2 · ( 1 + m 2 | z | 1 - m 2 | z | ) n / 2 m 2 = | z | ( 1 + m 2 | z | ) ( ( m 1 + m 2 ) / - 2 m 2 ) + ( n / 2 m 2 ) · ( 1 - m 2 | z | ) ( ( m 1 + m 2 ) / - 2 m 2 ) - ( n / 2 m 2 ) = | z | ( 1 + m 2 | z | ) ( m 1 + m 2 - n m 2 ) / - 2 m 2 · ( 1 - m 2 | z | ) ( m 1 + m 2 + n m 2 ) / - 2 m 2 . This completes the proof.

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

Theorem 10.

Let f ( z ) be a strongly almost starlike function of order 1 / 2 on D and c ( 0,1 ) . Then (52) | f ( z ) | e c | z | .

Theorem 11.

Let f ( z ) be a strongly almost starlike function of order α on D and α [ 0,1 ) { 1 / 2 } , c ( 0,1 ) . Then (53) | f ( z ) | | z | · [ 1 + c | 1 - 2 α | | z | ] ( ( 1 - α ) / ( 2 α - 1 ) ) + ( ( 1 - α ) / | 1 - 2 α | ) · [ 1 - c | 1 - 2 α | | z | ] ( ( 1 - α ) / ( 2 α - 1 ) ) - ( ( 1 - α ) / | 1 - 2 α | ) .

Remark 12.

Let 1 / 2 < α < 1 in Theorem 11. Then we have (54) | f ( z ) | | z | · [ 1 + c ( 2 α - 1 ) | z | ] ( 2 ( 1 - α ) ) / ( 2 α - 1 ) . Let 0 < α < 1 / 2 in Theorem 10. Then we have (55) | f ( z ) | | z | · [ 1 - c ( 1 - 2 α ) | z | ] ( 2 ( 1 - α ) ) / ( 2 α - 1 ) .

Let α = 0 in Theorem 11; we can get the following result.

Corollary 13.

Let f ( z ) be a strongly starlike function on D and c ( 0,1 ) . Then (56) | f ( z ) | | z | ( 1 - c | z | ) 2 .

Proof.

According to Corollary 8, we obtain (57) Re ( z f ( z ) f ( z ) ) | z f ( z ) f ( z ) | 1 + c | z | 1 - c | z | . Let z = r e i θ . Since (58) Re ( z f ( z ) f ( z ) ) = r ln | f ( z ) | r , we have (59) r ln | f ( z ) | r 1 + c | z | 1 - c | z | . Thus (60) ɛ | z | ln | f ( z ) | r d r ɛ | z | 1 + c r ( 1 - c r ) r d r = ɛ | z | 2 c 1 - c r d r + ɛ | z | d r r . So we get (61) ln | f ( r e i θ ) | | r = ɛ r = | z | 2 c ln ( 1 - c r ) - c | r = ɛ r = | z | + ln | r | | r = ɛ r = | z | . Letting ɛ 0 , it follows that (62) ln | f ( z ) | - 2 ln ( 1 - c | z | ) + ln | z | . Therefore we obtain (63) | f ( z ) | | z | ( 1 - c | z | ) 2 .

Also, we can get the conclusion by letting α = 0 in Theorem 11. This completes the proof.

Theorem 14.

Suppose that f ( z ) is a strongly starlike function on D and c ( 0,1 ) ; then (64) e ( 4 ( 2 c 2    | z | 2 - 1 ) ) / ( 1 + c | z | ) 2 · | z | ( 1 + c | z | ) 2 < | f ( z ) | | z | ( 1 - c | z | ) 2 .

Proof.

On the one hand, from Corollary 13, we obtain | f ( z ) | ( | z | ) / ( 1 - c | z | ) 2 .

On the other hand, by a and ρ in the proof of Theorem 6, we can obtain (65) Re a - ρ ( | a | + ρ ) 2 = ( 1 - c | w ( z ) | ) 3 ( 1 + c | w ( z ) | ) 3 for α = β = 0 . Let λ ( x ) = ( 1 - c x ) 3 / ( 1 + c x ) 3 . Then we have (66) λ ( x ) = - 6 c ( 1 - c x ) 2 ( 1 + c x ) 4 < 0 . Therefore ( 1 - c | w ( z ) | ) 3 / ( 1 + c | w ( z ) | ) 3 is a monotone increasing function with respect to | w ( z ) | . Also we can know that | w ( z ) | | z | from Lemma 4. Hence (67) Re a - ρ ( | a | + ρ ) 2 = ( 1 - c | w ( z ) | ) 3 ( 1 + c | w ( z ) | ) 3 > ( 1 - c | z | ) 3 ( 1 + c | z | ) 3 . By (14) we obtain (68) Re f ( z ) z f ( z ) Re a - ρ . Furthermore, | f ( z ) / z f ( z ) | | a | + ρ , so (69) Re z f ( z ) f ( z ) = Re ( f ( z ) / z f ( z ) ) | f ( z ) / z f ( z ) | 2 > Re a - ρ ( | a | + ρ ) 2 . Let z = r e i θ . Since Re ( z f ( z ) / f ( z ) ) = r ( ln | f ( z ) | ) / r , we have (70) r ln | f ( z ) | r > ( 1 - c | z | ) 3 ( 1 + c | z | ) 3 . Therefore we obtain (71) ɛ | z | ln | f ( z ) | r d r > ɛ | z | ( 1 - c r ) 3 ( 1 + c r ) 3 · d r r . Then we have (72) ln | f ( z ) | > - 4 c | z | · 1 ( 1 + c | z | ) 2 + 4 c [ 1 | z | + c 1 + c | z | - c + 2 c ln | z | - 2 c ln ( 1 + c | z | ) ] . So (73) | f ( z ) | > e ( 4 ( 2 c 2 | z | 2 - 1 ) ) / ( 1 + c | z | ) 2 · | z | ( 1 + c | z | ) 2 . Therefore we obtain (74) e ( 4 ( 2 c 2 | z | 2 - 1 ) ) / ( 1 + c | z | ) 2 · | z | ( 1 + c | z | ) 2 < | f ( z ) | | z | ( 1 - c | z | ) 2 . This completes the proof.

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

Theorem 15.

Let f ( z ) be a strongly almost spirallike function of type β and order α on D and α ( 1 / 2,1 ) , β ( - π / 2 , π / 2 ) , c ( 0,1 ) . Then (75) | f ( z ) | ( 1 + m 1 | z | 2 + n | z | ) ( 1 + m 2 | z | ) ( m 1 + 3 m 2 - n m 2 ) / - 2 m 2    · ( 1 - m 2 | z | ) ( m 1 + 3 m 2 + n m 2 ) / - 2 m 2 , where (76) m 1 = c 2 [ 2 ( 1 - α ) cos β ( | sin β | + cos β ) - 1 ] , m 2 = c 2 [ 1 - 4 α ( 1 - α ) cos 2 β ] , n = 2 c ( 1 - α ) cos β .

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

Theorem 16.

Let f ( z ) be a strongly almost starlike function of order α on D and α [ 0,1 ) { 1 / 2 } , c ( 0,1 ) . Then (77) | f ( z ) | [ 1 + c 2 ( 1 - 2 α ) | z | 2 + 2 c ( 1 - α ) | z | ] × [ 1 + c | 1 - 2 α | | z | ] ( ( 2 - 3 α ) / ( 2 α - 1 ) ) + ( ( 1 - α ) / | 1 - 2 α | )    · [ 1 - c | 1 - 2 α | | z | ] ( ( 2 - 3 α ) / ( 2 α - 1 ) ) - ( ( 1 - α ) / | 1 - 2 α | ) .

Let α = 0 in Theorem 16; we can get the following result.

Corollary 17.

Let f ( z ) be a strongly starlike function on D and c ( 0,1 ) . Then (78) | f ( z ) | 1 + c | z | ( 1 - c | z | ) 3 .

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).

Duren P. L. Univalent Functions 1983 Berlin, Germany Springer MR708494 Cartan H. Montel P. Sur la possibilite d'entendre aux fonctions de plusieurs variables complexes la theorie des fonctions univalents Lecons sur les Fonctions Univalents on Mutivalents 1933 Gauthier-Villar 129 155 Barnard R. W. FitzGerald C. H. Gong S. The growth and 1/4-theorems for starlike mappings in C n Pacific Journal of Mathematics 1991 150 1 13 22 10.2140/pjm.1991.150.13 MR1120709 Gong S. Wang S. K. Yu Q. H. The growth and 1 / 4 -theorem for starlike mappings on B p Chinese Annals of Mathematics B 1990 11 1 100 104 MR1048976 Graham I. Varolin D. Bloch constants in one and several variables Pacific Journal of Mathematics 1996 174 2 347 357 MR1405592 2-s2.0-0002257301 Liu T. Ren G. The growth theorem for starlike mappings on bounded starlike circular domains Chinese Annals of Mathematics B 1998 19 4 401 408 MR1667426 2-s2.0-22444452512 Liu H. Lu K. P. Two subclasses of starlike mappings in several complex variables Chinese Annals of Mathematics 2000 21 5 533 546 MR1802453 Feng S. X. Lu K. P. The growth theorem for almost starlike mappings of order α on bounded starlike circular domains Chinese Quarterly Journal of Mathematics. Shuxue Jikan 2000 15 2 50 56 MR1832848 Honda T. The growth theorem for k -fold symmetric convex mappings The Bulletin of the London Mathematical Society 2002 34 6 717 724 10.1112/S0024609302001388 MR1924199 2-s2.0-0036853256 Mahmudov N. I. Eini Keleshteri M. q -extensions for the apostol type polynomials Journal of Applied Mathematics 2014 2014 8 868167 10.1155/2014/868167 MR3219435 Merkes E. Salmassi M. Subclasses of uniformly starlike functions International Journal of Mathematics and Mathematical Sciences 1992 15 3 449 454 10.1155/S0161171292000607 MR1169809 ZBL0758.30008 Singh S. A subordination theorem for spirallike functions International Journal of Mathematics and Mathematical Sciences 2000 24 7 433 435 10.1155/S0161171200004634 MR1781509 Gurganus K. R. ϕ-like holomorphic functions in C n and Banach spaces Transactions of the American Mathematical Society 1975 205 389 406 MR0374470 Hamada H. Kohr G. Subordination chains and the growth theorem of spirallike mappings Mathematica 2000 42(65) 2 153 161 (2001) MR1988620 ZBL1027.46094 Feng S. X. Some classes of holomorphic mappings in several complex variables [Ph.D. thesis] 2004 Hefei, China University of Science and Technology of China Feng S. X. Liu T. S. Ren G. B. The growth and covering theorems for several mappings on the unit ball in complex Banach spaces Chinese Annals of Mathematics A 2007 28 2 215 230 MR2317599 Cai R. H. Liu X. S. The third and fourth coefficient estimations for the subclasses of strongly spirallike functions Journal of Zhanjiang Normal College 2010 31 38 43 Hamada H. Kohr G. The growth theorem and quasiconformal extension of strongly spirallike mappings of type α Complex Variables 2001 44 4 281 297 MR1909520 Chuaqui M. Applications of subordination chains to starlike mappings in C n Pacific Journal of Mathematics 1995 168 1 33 48 10.2140/pjm.1995.168.33 MR1331993 Ahlfors L. V. Complex Analysis 1978 3rd New York, NY, USA McGraw-Hill MR510197