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Geometric Mappings under the Perturbed Extension Operators in Complex Systems Analysis.

1. Introduction

The theory of several complex variables derives from the theory of one complex variable. There are many excellent results in geometric function theories of one complex variable. It is natural to think that we can extend these results in several complex variables, while some basic theorems (such as the models of the coefficients of the homogeneous expansion for biholomorphic functions being bounded on the unit disk [1]) are found not to hold in several complex variables. In 1933, Cartan [2] suggested that we can consider the geometric constraint of biholomorphic mappings, such as star-likeness and convexity. So many scholars devoted themselves to the research of star-like mappings and convex mappings. Recently many subclasses or expansions of star-like and convex mappings are introduced. The properties of biholomorphic mappings with special geometric properties are important research objects in geometric function theories of several complex variables. It is easy to find specific examples of these new subclasses or expansions in C, while it is very difficult in [C.sup.n]. In order to study these subclasses better in several complex variables, we need the specific examples imminently.

In 1995, Roper and Suffridge [3] introduced an operator

[[phi].sub.n] (f) (z) = (f ([z.sub.1]), [square root of (f' ([z.sub.1]))] [z.sub.0])', (1)

where z = ([z.sub.1], [z.sub.0]) [member of] [B.sup.n], [z.sub.1] [member of] D, [z.sub.0] = ([z.sub.2], ..., [z.sub.n]) [epsilon] [C.sup.n-1], f([z.sub.1]) [member of] H(D), [square root of f' (0)] = 1. Roper and Suffridge proved the Roper-Suffridge operator preserves convexity and star-likeness on [B.sup.n]. Graham et al. generalized the Roper-Suffridge operator and discussed the generalized operators preserving star-likeness and the block property in [4, 5]. In 2002, Graham et al. extended the Roper-Suffridge operator on the unit ball in [C.sup.n] and proved the extended operator preserves star-likeness and convexity if and only if some conditions are satisfied in [6]. In 2003, Gong and Liu [7] generalized the Roper-Suffridge operator and obtained the generalized operator preserving [epsilon] star-likeness on the Reinhardt domain which leads to star-likeness and convexity in the cases of [epsilon] = 0 and [epsilon] = 1, respectively.

All of the above illustrate that the Roper-Suffridge operator has good properties. Through the Roper-Suffridge operator or its generalizations we can construct lots of convex mappings and star-like mappings in [C.sup.n] by corresponding functions on the unit disk D of C. That will promote the development of the research of biholomorphic mappings. So the Roper-Suffridge operator plays an important role in several complex variables. In recent years, there are lots of results about the Roper-Suffridge extension operator which was generalized and modified on different domains in different spaces to preserve the geometric characteristics of convex mappings, star-like mappings, and their subclasses. Graham and Kohr gave a survey about the Roper-Suffridge extension operator and the developments in the theory of biholomorphic mappings in several complex variables to which it had led in [8].

Muir and Suffridge [9] introduced the following generalized Roper-Suffridge extension operator:

F (z) = (f ([z.sub.1]) + f' ([z.sub.1]) P ([z.sub.0]), [square root of f' ([z.sub.1])] [z.sub.0])', (2)

where f is a normalized biholomorphic function on the unit disk D, z = ([z.sub.1], [z.sub.0])' [member of] [B.sup.n], [z.sub.1] [member of] D, [z.sub.0] = ([z.sub.2], ..., [z.sub.n])' [member of] [C.sup.n-1]. The branch of the power function is chosen such that [square root of f'(0)] = 1. P : [C.sup.n-1] [right arrow] C is a homogeneous polynomial of degree 2. The extended operator (2) was proved to preserve star-likeness on [parallel]P[parallel] [less than or equal to] 1/4 and convexity on [parallel]P[parallel] [less than or equal to] 1/2 in [9] and was proved to take the extreme points of normalized convex functions to extreme points of normalized convex mappings of the Euclidean ball in [C.sup.n] under precise conditions by Muir in [10]. Also the extended operator (2) was studied by Kohr and Muir in [11,12] with Loewner chains. Later the operator (2) was generalized by Elin and Levenshtein and the generalized operator was proved to preserve the spiral-likeness property in [13] and was concluded that it can be embedded in a Loewner chain on the unit ball in [C.sup.n] in [14]. Moreover, Elin and Levenshtein presented an extension operator for semigroup generators and concluded that the new one-dimensional covering results established in [13] are crucial. Furthermore, (2) was modified and discussed by Cui et al. in [15]. Elin introduced a general construction of the extension operators

[[phi].sub.r] [h](x, y) = (h(x), [GAMMA] ((h, x)y)), (3)

where (x, y) is on the unit ball of the product X * Y of two Banach spaces and [GAMMA](h, x) is an operator-valued mapping which satisfies some natural conditions. [[phi].sub.[GAMMA]][h](x, y) was proved to preserve star-likeness and spiral-likeness under some conditions in [16].

Now, we introduce a new extension operator

[mathematical expression not reproducible] (4)

on the Bergman-Hartogs domain

[mathematical expression not reproducible], (5)

where [mathematical expression not reproducible] is a normalized univalent holomorphic function on D, G is a holomorphic function in [C.sup.n-1] with G(0) = 0, DG(0) = 1, [gamma] [greater than or equal to] 0, and the power functions take the branches such that [mathematical expression not reproducible]. The homogeneous expansion of G(z) is [[summation].sup.[infinity].sub.j=0] [P.sub.j](z), where [P.sub.j](z) is a homogeneous polynomial of degree j. Equation (4) is the modification of the extension operators discussed in [17].

In the case of [w.sub.(1)] = ... = [w.sub.(s)] = 0, (4) leads to the following operator:

[mathematical expression not reproducible], (6)

which can also be seen as the modification of the following generalized Roper-Suffridge extension operator introduced by Muir on the unit ball in complex Banach spaces,

[mathematical expression not reproducible], (7)

where [mathematical expression not reproducible]. Muir proved [[[PHI].sub.G,[gamma]] (f)](z) is a Loewner chain preserving extension operator provided that G satisfies some conditions in [12].

In this paper, we mainly discuss the invariance of several biholomorphic mappings under the generalized Roper-Suffridge extension operators (4) on the Bergman-Hartogs domains [mathematical expression not reproducible] which is based on the unit ball [B.sup.n]. In Section 2, we give some definitions and lemmas that are used to derive the main results. In Sections 3-5, we detailedly discuss the perturbed Extension Operator (4) preserving the geometric properties of strong and almost spiral-like mappings of type [beta] and order [alpha], [S.sup.*.sub.[OMEGA]] ([beta], A, B), as well as almost spiral-like mappings of type and order under different conditions on Bergman-Hartogs domains and thus generalize the conclusions on the unit ball [B.sup.n] in [C.sup.n]. At last, we derive that the generalized Roper-Suffridge operators preserve the properties of subclasses of the three kinds of biholomorphic mappings mentioned above. The conclusions include and promote some known results.

2. Definitions and Lemmas

In the following, let D denote the unit disk in C and [B.sup.n] denote the unit ball in [C.sup.n]. Let DF(z) denote the Frechet derivative of F at z.

To get the main results, we need the following definitions and lemmas.

Definition 1 (see [18]). Let [OMEGA] be a bounded star-like circular domain in [C.sup.n]. The Minkowski functional [rho](z) of [OMEGA] is [C.sup.1] except for a lower-dimensional manifold. If f(z) is a normalized locally biholomorphic mapping on [OMEGA], let [alpha] [member of] [0,1), [beta] [member of] (-[pi]/2, [pi]/2), and

[mathematical expression not reproducible]. (8)

Then f(z) is called a strong and almost spiral-like mapping of type [beta] and order [alpha] on [OMEGA].

Setting [alpha] = 0, [beta] = 0, and [alpha] = [beta] = 0, Definition 1 reduces to the definition of strong spiral-like mappings of type [beta], strong and almost star-like mappings of order [alpha], and strong star-like mappings, respectively.

Definition 2 (see [19]). Let [OMEGA] be a bounded star-like circular domain in [C.sup.n]. The Minkowski functional [rho](z) of [OMEGA] is [C.sup.1] except for a lower-dimensional manifold. Let F(z) be a normalized locally biholomorphic mapping on [OMEGA]. If

[mathematical expression not reproducible], (9)

where -1 [less than or equal to] A < B < 1 and [beta] [member of] (-[pi]/2, [pi]/2), then we call F [member of] [S.sup.*.sub.[OMEGA]] ([beta], A, B).

Setting A = -1 = -B - 2[alpha], A = -B = -[alpha], and B [right arrow] [1.sup.-] in Definition 2, respectively, we obtain the corresponding definitions of spiral-like mappings of type [beta] and order [alpha], strong spiral-like mappings of type [beta] and order [alpha], and almost spiral-like mappings of type [beta] and order [alpha] on [OMEGA].

Definition 3 (see [20]). Let [OMEGA] be a bounded star-like circular domain in [C.sup.n]. The Minkowski functional [rho](z) of [OMEGA] is [C.sup.1] except for a lower-dimensional manifold. Let f(z) be a normalized locally biholomorphic mapping on [OMEGA]. If

[mathematical expression not reproducible], (10)

where [alpha] [member of] [0,1) and [beta] [member of] (-[pi]/2, [pi]/2), then we call f(z) an almost spiral-like mapping of type [beta] and order [alpha].

Setting [alpha] = 0, [beta] = 0 in Definition 3, we obtain the definition of spiral-like mappings of type [beta] and almost star-like mappings of order [alpha] on [B.sup.n], respectively.

Lemma 4 (see [21]). Let [OMEGA] be a bounded star-like circular domain in [C.sup.n]. The Minkowski functional [rho](z) of [OMEGA] is [C.sup.1] except for a lower-dimensional manifold [[OMEGA].sub.0]. Then we have

[mathematical expression not reproducible], (11)

Lemma 5 (see [12]). Let P(z) be a homogeneous polynomial of degree m and let DP(z) be the Frechet derivative of P at z. Then

DP (z) z = mP (z). (12)

Lemma 6 (see [17]). Let [rho](w, z) be the Minkowski functional of [mathematical expression not reproducible]. Let [mathematical expression not reproducible]. Then [rho](w, z) = 1 and

[mathematical expression not reproducible], (13)

where

[mathematical expression not reproducible]. (14)

Lemma 7 (see [1]). If f(z) is a normalized biholomorphic function on the unit disk D, then

[mathematical expression not reproducible]. (15)

Lemma 8 (see [22]). Let f(z) be a strong spiral-like mapping of type [alpha] on bounded and balanced domain [OMEGA] with [alpha] [member of] (-[pi]/2, [pi]/2) and c [member of] (0,1). Let the Minkowski functional of [OMEGA] be [rho](z). Then

[rho](z)/[(1 + c [rho] (z)).sup.2] [less than or equal to] [rho] (f (z)) [less than or equal to] [rho](z)/[(1 + c [rho] (z)).sup.2]. (16)

Let f(z) be a strong spiral-like function of type [alpha] on D, then

[absolute value of z]/[(1 + c [absolute value of z]).sup.2] [less than or equal to] [absolute value of f (z)] [less than or equal to] [absolute value of z]/[(1 - c [absolute value of z]).sup.2]. (17)

Lemma 9 (see [23]). Let [OMEGA] [subset] [C.sup.n] be a bounded star-like circular domain and the Minkowski functional [rho](z) of [OMEGA] be [C.sup.1] except for some submanifolds of lower dimensions. Let f(z) [member of] [S.sup.*.sub.[OMEGA]] (A, B) be k-fold symmetric. Then

[mathematical expression not reproducible], (18)

or, equivalently,

[mathematical expression not reproducible]. (19)

The above estimates are all accurate.

Lemma 10 (see [24]). Let f(x) be an almost spiral-like mapping of type [beta] and order [alpha] on the unit ball B in complex Banach spaces with [alpha] [member of] [0,1) and [beta] [member of] (-[pi]/2, [pi]/2). Then

[mathematical expression not reproducible]. (20)

For n = 1 we get

[mathematical expression not reproducible]. (21)

3. The Invariance of Strong and Almost Spiral-Like Mappings of Type [beta] and Order [alpha]

For simplicity, let [OMEGA] denote [mathematical expression not reproducible]. In this section we will show that the perturbed Roper-Suffridge extension operator (4) preserves the geometric characteristics of strong and almost spiral-like mappings of type [beta] and order [alpha] on [OMEGA], and thus we obtain the conclusion on [B.sup.n]; also we get the invariance of some subclasses.

Theorem 11. Let f([z.sub.1]) be a strong and almost spiral-like function of type [beta] and order [alpha] on D with [alpha] [member of] [0,1) and fi [member of] (-[pi]/2, [pi]/2), c [member of] (0,1). Let F(w,z) be the mapping denoted by (4) with [mathematical expression not reproducible]. Then F (w,z) is a strong and almost spiral-like mapping of type [beta] and order [alpha] on [OMEGA] provided that [P.sub.j] = 0 for j < 2/(1 - 4[gamma]) and

[mathematical expression not reproducible], (22)

where [delta] = max{[p.sub.1] [[delta].sub.1], ..., [p.sub.s] [[delta].sub.s]}.

Proof. By Definition 1, we need to prove

[mathematical expression not reproducible]. (23)

It is obvious that (23) holds for w = [z.sub.0] = 0. Otherwise, setting (w, z) = [zeta]([xi], [eta]) = [absolute value of [zeta]][e.sup.i[theta]] ([xi], [eta]) where ([xi], [eta]) [member of] [partial derivative][OMEGA] and [zeta] [member of] [bar.D]\ {0}, by Lemma 4, we obtain

[mathematical expression not reproducible]. (24)

Fix [xi], [eta], then (2[partial derivative][rho]/[partial derivative](w,z))([xi], [eta])[((DF(([zeta] [xi], [zeta] [eta])).sup.-1]F(([zeta] [xi], [zeta] [eta])/[zeta]) is holomorphic with respect to [zeta]. Due to the maximum modulus principle of holomorphic functions, the left side of (23) gets the maximum value at [absolute value of [zeta]] = 1. So we need only to prove that (23) holds for (w, z) [member of] [partial derivative][OMEGA], which follows [rho](w, z) = 1.

Let

[mathematical expression not reproducible]. (25)

Then q([z.sub.1]) [member of] H(D), [absolute value of q([z.sub.1])] < 1 and q(0) = -c. Letting g([z.sub.1]) = (2c/(1 - [c.sup.2]))(q([z.sub.1]) + c), we have g ([z.sub.1]) [member of] H(D), g(0) = 0 and [absolute value of q([z.sub.1])] = [absolute value of ((1 - [c.sup.2])/2c)g([z.sub.1]) - c] < 1. In view of ((1 - [c.sup.2])/2c) [absolute value of g([z.sub.1])] - c < [absolute value of ((1 - [c.sup.2])/2c) g ([z.sub.1]) - c], we obtain [absolute value of g([z.sub.1])] < 2c/(1 - c) < 1 provided that c [less than or equal to] 1/3. By Schwarz Lemma we get [absolute value of g([z.sub.1])] [less than or equal to] [z.sub.1]] which implies that

[absolute value of (q ([z.sub.1]) + c)] [less than or equal to] 1 - [c.sup.2]/2c [absolute value of [z.sub.1]]. (26)

In addition, from (4) we have

[mathematical expression not reproducible], (27)

where

[mathematical expression not reproducible]. (28)

Let [(DF(w, z)).sup.-1] F(w,z) = H(w,z) = ([h.sub.1], ..., [h.sub.s+n])', which follows DF(w, z)([h.sub.1], ..., [h.sub.s+n])' = F(w, z). Thus

[mathematical expression not reproducible]. (29)

Lemma 5 and a straightforward calculation show that

[mathematical expression not reproducible]. (30)

As a consequence of Lemma 6 we get

[mathematical expression not reproducible], (31)

where

[mathematical expression not reproducible]. (32)

Let

[mathematical expression not reproducible]. (33)

Taking into account (25) and (31) we obtain the following equation string:

[mathematical expression not reproducible]. (34)

where [mathematical expression not reproducible]. We use (26) and [absolute value of h([z.sub.1])] < 1 to see that

[mathematical expression not reproducible]. (35)

Note that [mathematical expression not reproducible] implies that

[mathematical expression not reproducible], (36)

lead to

[mathematical expression not reproducible]. (37)

Therefore

[mathematical expression not reproducible] (38)

provided that q [less than or equal to] [delta]/(n + 1).

Lemma 7 and (38) eventually lead us to the following inequality string:

[mathematical expression not reproducible], (39)

where [P.sub.j] = 0 for j < 2/(1 - 4[gamma]) and

[mathematical expression not reproducible]. (40)

This completes the proof.

Setting [w.sub.(1)] = ... = [w.sub.(s)] = 0 in Theorem 11, the operator (4) reduces to the generalized Roper-Suffridge operator (6) and the Bergman-Hartogs domain [OMEGA] reduces to the unit ball [B.sup.n]; thus, we draw the following conclusion on [B.sup.n] in [C.sup.n].

Corollary 12. Let f([z.sub.1]) be a strong and almost spiral-like function of type [beta] and order [alpha] on D with [alpha] [member of] [0,1) and [beta] [member of] (-[pi]/2, [pi]/2), c [member of] (0,1). Let F(z) be the mapping denoted by (6) with [mathematical expression not reproducible]. Then F(z) is a strong and almost spiral-like mapping of type [beta] and order [alpha] on [B.sup.n] provided that [P.sub.j] = 0 for j < 2/(1 - 4[gamma]) and

[mathematical expression not reproducible]. (41)

Remark 13. Setting [alpha] = 0 and [beta] = 0 in Theorem 11 and Corollary 12, respectively, we get the corresponding results for strong and almost star-like mappings of order [alpha] and strong spiral-like mapping of type [beta].

If we have the precise growth result of strong and almost spiral-like mapping of type [beta] and order [alpha] and apply the growth result in Theorem 11, we can get more precise conclusion. But, up to now we only have the growth theorem of strong spiral-like function of type [beta] [22]. In the following, we apply it in the process of discussing the invariance of strong spiral-like function of type [beta].

Theorem 14. Let f([z.sub.1]) be a strong spiral-like function of type [beta] on D with [beta] [member of] (-[pi]/2, [pi]/2) and c [member of] (0,1). Let F(w,z) be the mapping denoted by (4) with [mathematical expression not reproducible]. Then F(w,z) is a strong spiral-like mapping of type [beta] and order [alpha] on [OMEGA] provided that [P.sub.j] = 0 for j < 2/(1 - 4[gamma]) and

[mathematical expression not reproducible], (42)

where [delta] = max{[p.sub.1] [[delta].sub.1], ..., [p.sub.s] [[delta].sub.s]}.

Proof. Setting [alpha] = 0 in (39) and applying Lemma 8 we obtain the following inequality string:

[mathematical expression not reproducible], (43)

where [P.sub.j] = 0 for j < 2/(1 - 4[gamma]) and

[mathematical expression not reproducible]. (44)

Hence the assertion follows.

4. The Invariance of [S.sup.*.sub.[OMEGA]]([beta], A, B)

As above, write [mathematical expression not reproducible]. In this study the perturbed Extension Operator (4) preserving the geometric characteristics of [S.sup.*.sub.[OMEGA]] ([beta], A, B) on [OMEGA] and then generalize the conclusion on [B.sup.n] and get the invariance of some subclasses.

Theorem 15. Let f([z.sub.1]) [member of] [S.sup.*.sub.D]([beta], A, B) with -1 [less than or equal to] A < B < (A + 1)/2 < 1 and [beta] [member of] (-[pi]/2, [pi]/2). Let F(w, z) be the mapping denoted by (4) with [mathematical expression not reproducible]. Then F(w, z) [member of] [S.sup.*.sub.[OMEGA]]([beta], A, B) provided that p, = 0 for j <2/(1 - 4[gamma]) and

[mathematical expression not reproducible], (45)

where [delta] = max{[p.sub.1] [[delta].sub.1], ..., [p.sub.s] [[delta].sub.s]}.

Proof. By Definition 2, we need to prove

[mathematical expression not reproducible]. (46)

Similar to Theorem 11, we need only to prove that (46) holds for (w, z) [member of] [partial derivative][OMEGA] which implies [rho](w, z) = 1.

Since f([z.sub.1]) [member of] [S.sup.*.sub.D] ([beta], A, B), by Definition 2, we have

[mathematical expression not reproducible]. (47)

Let

[mathematical expression not reproducible]. (48)

Then [absolute value of h([z.sub.1])] < 1 and h(0) = -B. Letting g([z.sub.1]) = (B - A)/(1 - [B.sup.2])(h([z.sub.1]) + B), we get g([z.sub.1]) [member of] H(D), g(0) = 0 and [absolute value of h([z.sub.1])] = [absolute value of ((1 - [B.sup.2])/(B - A))g([z.sub.1]) - B] < 1. In view of ((1 - [B.sup.2])/(B - A))g([z.sub.1]) - [absolute value of B] [less than or equal to] [absolute value of ((1 - [B.sup.2])/(B - A)) g ([z.sub.1]) - B], we obtain [absolute value of g([z.sub.1])] < (1 + [absolute value of B]) (B - A)/ (1 - [B.sup.2]) [less than or equal to] 1 where - 1 [less than or equal to] A < B < (A + 1)/2 < 1. By Schwarz Lemma, we get [absolute value of h([z.sub.1])] = [absolute value of g([z.sub.1])] [less than or equal to] [absolute value of [z.sub.1]] which implies that

[absolute value of (h([z.sub.1]) + B)] [less than or equal to] [1 - [B.sup.2]/B - A] [absolute value of [z.sub.1]]. (49)

Let

[mathematical expression not reproducible]. (50)

Then, (31) and (48) lead to the following equation string:

[mathematical expression not reproducible]. (51)

By (49), (38), and Lemma 7 there comes the following inequality string:

[mathematical expression not reproducible], (52)

where [P.sub.j] = 0 for j < 2/(1 - 4[gamma]) and

[mathematical expression not reproducible]. (53)

Therefore (46) holds, which follows F(w, z) [member of] [S.sup.*.sub.[OMEGA]] ([beta], A, B).

Setting [w.sub.(1)] = ... = [w.sub.(s)] = 0 in Theorem 15, we draw the following invariance of the mappings [mathematical expression not reproducible] ([beta], A, B) under the generalized Roper-Suffridge operator (6) on the unit ball [B.sup.n] in [C.sup.n].

Corollary 16. Let f([z.sub.1]) [member of] [S.sup.*.sub.D]([beta], A, B) with -1 [less than or equal to] A < B < (A + 1)/2 < 1 and [beta] [member of] (-[pi]/2, [pi]/2). Let F(z) be the mapping denoted by (6) with [mathematical expression not reproducible]. Then [mathematical expression not reproducible] provided that [P.sub.j] = 0 for j < 2/(1 - 4[gamma]) and

[mathematical expression not reproducible]. (54)

In the process of discussing the invariance of [S.sup.*.sub.[OMEGA]](A, B), if we apply the growth theorem of [S.sup.*.sub.[OMEGA]](A, B) in [23], we can get the following more precise conclusion.

Theorem 17. Let f([z.sub.1]) [member of] [S.sup.*.sub.D] (A,B) with -1 [less than or equal to] A < B < (A + 1)/2 < 1. Let F(w,z) be the mapping denoted by (4) with [mathematical expression not reproducible]. Let q [greater than or equal to] [delta]/(n + 1) where [delta] = max{[p.sub.1] [[delta].sub.1], ..., [p.sub.s] [[delta].sub.s]}. Then we have the following conclusions:

(1) For A [not equal to] 0, F (w, z) [member of] [S.sup.*.sub.D] (A,B) provided that

[mathematical expression not reproducible]. (55)

(2) For A = 0 and B >0, F(w, z) [member of] [S.sup.*.sub.[OMEGA]] (A, B) provided that

[mathematical expression not reproducible]. (56)

(3) For A = 0 and B <0, F(w, z) [member of] [S.sup.*.sub.[OMEGA]] (A,B) provided that

[[infinity].summation over (j=2)] (j - 1) [parallel] [P.sub.j] [parallel] [less than or equal to] (1 - [gamma]) (B - A)/2 (1 + [absolute value of B]). (57)

Proof. Setting [beta] = 0 in the proof of Theorem 15, by using (52) and Lemma 9 we obtain the following conclusions:

(1) For the case that A [not equal to] 0 we get

[mathematical expression not reproducible], (58)

where

[mathematical expression not reproducible]. (59)

(2) For the case that A = 0 and B > 0 we have

[mathematical expression not reproducible], (60)

where

[mathematical expression not reproducible]. (61)

(3) For the case that A = 0 and B < 0, similar to (2), we get

[mathematical expression not reproducible], (62)

where

[[infinity].summation over (j=2)] (j - 1) [parallel] [P.sub.j] [parallel] [less than or equal to] (1 - [gamma]) (B - A)/2 (1 + [absolute value of B]). (63)

From (1) to (3) we get the desired conclusion.

Remark 18. Setting A =-1 = -B - 2[alpha] and A = -B = -[alpha] in Theorems 15 and 17 and Corollary 16, respectively, we get the corresponding results for spiral-like mappings of type [beta] and order [alpha] and strong spiral-like mappings of type [beta] and order [alpha].

5. The Invariance of Almost Spiral-Like Mappings of Type [beta] and Order [alpha]

In the following, we mainly discuss the perturbed Extension Operator (4) preserving the geometric characteristics of almost spiral-like mappings of type [beta] and order [alpha] on [OMEGA], and thus we get the conclusion on [B.sup.n] as well as the results about some subclasses.

Theorem 19. Let f([z.sub.1]) be an almost spiral-like function of type [beta] and order [alpha] on D with [alpha] [member of] [0,1/2) [union] (1/2,1) and [beta] [member of] (-[pi]/2, [pi]/2). Let F(w, z) be the mapping denoted by (4) with [mathematical expression not reproducible]. Let [P.sub.j] = 0 (j = 2,3) and q [greater than or equal to] [delta]/(n + 1) where [delta] = max{[p.sub.1] [[delta].sub.1], ..., [p.sub.s] [[delta].sub.s]}. Then F(w, z) is an almost spiral-like mapping of type [beta] and order [alpha] on [OMEGA] provided that

[mathematical expression not reproducible]. (64)

Proof. By Definition 3, we need to prove

[mathematical expression not reproducible], (65)

which is equivalent to

[mathematical expression not reproducible]. (66)

Similar to Theorem 11, the left side of (66) is the real part of a holomorphic mapping and thus is a harmonic function. Due to the minimum principle of harmonic functions, we need only to prove that (66) holds for z [member of] [partial derivative][OMEGA] which implies that [rho](w, z) = 1.

Let

(1 - i tan [beta]) [f/[z.sub.1] f'] - [alpha] = h([z.sub.1]). (67)

Since f([z.sub.1]) is an almost spiral-like function of type [beta] and order [alpha] on D, then h([z.sub.1]) [member of] H(D), Rh([z.sub.1]) > 0, h(0) = 1 - i tan [beta] - [alpha]. Therefore [absolute value of (h([z.sub.1]) -(1 - [alpha] - i tan [beta]))/(h([z.sub.1]) + (1 - [alpha] + i tan [beta]))] < 1. Let

g([z.sub.1]) = (1 - h([z.sub.1]) - (1 - [alpha] - i tan [beta])/h([z.sub.1]) + (1 - [alpha] + i tan [beta]). (68)

Then [absolute value of (g([z.sub.1]) < 1, g(0) = 0. Moreover Rh([z.sub.1]) > 0 implies h([z.sub.1]) [not equal to] -(1 - [alpha] + i tan [beta]); thus g([z.sub.1]) [member of] H(D). Applying Schwarz Lemma we obtain [absolute value of g([z.sub.1])] [less than or equal to] [absolute value of [z.sub.1]]; in other words,

[absolute value of (1 + 2(1 - [alpha])/h([z.sub.1]) = -(1 - [alpha] + i tan [beta]))] [greater than or equal to] 1/[absolute value of [z.sub.1]], (69)

which leads to

[absolute value of (h([z.sub.1]) -(1 - [alpha] + i tan [beta]))] [greater than or equal to] 2(1 - [alpha]) [absolute value of [z.sub.1]]/1 - [absolute value of [z.sub.1]]. (70)

Let

[mathematical expression not reproducible]. (71)

We then get the following equation string by (31) and (67):

[mathematical expression not reproducible]. (72)

Therefore, by (70), (38), and Lemma 10 there comes the following inequality string:

[mathematical expression not reproducible], (73)

where [P.sub.j] = 0 (j = 2, 3) and

[mathematical expression not reproducible]. (74)

This is equal to (66) and hence completes the proof.

Setting [w.sub.(1)] = ... = [w.sub.(s)] = 0 in Theorem 19, we draw the following invariance of almost spiral-like mappings of type [beta] and order [alpha] under the generalized Roper-Suffridge operator (6) on the unit ball [B.sup.n] in [C.sup.n].

Corollary 20. Let f([z.sub.1]) be an almost spiral-like function of type [beta] and order [alpha] on D with [alpha] [member of] [0,1/2) [union] (1/2,1) and [beta] [member of] (-[pi]/2, [pi]/2). Let F(z) be the mapping denoted by (6) with [mathematical expression not reproducible]. Then F(z) is an almost spiral-like mapping of type [beta] and order [alpha] on [B.sup.n] provided that

[mathematical expression not reproducible]. (75)

Remark 21. Setting [alpha] = 0 and [beta] = 0 in Theorem 19 and Corollary 20, respectively, we get the corresponding results for spiral-like mappings of type [beta] and almost star-like mappings of order [alpha].

https://doi.org/10.1155/2017/3512326

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by NSF of China (no. 11271359) and Science and Technology Research Projects of Henan Provincial Education Department (no. 17A110041).

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Chaojun Wang, (1) Yanyan Cui, (1, 2) and Hao Liu (3)

(1) College of Mathematics and Statistics, Zhoukou Normal University, Zhoukou, Henan 466001, China

(2) College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, Hebei 050016, China

(3) Institute of Contemporary Mathematics, Henan University, Kaifeng, Henan 475001, China

Correspondence should be addressed to Yanyan Cui; cui9907081@163.com

Received 3 March 2017; Accepted 8 May 2017; Published 15 June 2017

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Title Annotation:Research Article
Author:Wang, Chaojun; Cui, Yanyan; Liu, Hao
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Date:Jan 1, 2017
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