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Roper-Suffridge extension operator on a reinhardt domain
Acta Mathematica Scientia 2014,34B(6):1761–1774 http://actams.wipm.ac.cn
ROPER-SUFFRIDGE EXTENSION OPERATOR ON A REINHARDT DOMAIN∗
oõ)
Hongjun LI (
School of Mathematics and Information Science, Henan University, Kaifeng 475004, China E-mail : [email protected]
¾Ô_)
Shuxia FENG (
†
Institute of Contemporary Mathematics, School of Mathematics and Information Science, Henan University, Kaifeng 475004, China E-mail : [email protected] Abstract Let pj ∈ N and pj ≥ 1, j = 2, · · · , k, k ≥ 2 be a fixed positive integer. We introduce a Roper-Suffridge extension operator on the following Reinhardt domain ΩN = {z = (z1 , z2′ , · · · , zk′ )′ ∈ C × Cn2 × · · · × Cnk : |z1 |2 + ||z2 ||p22 + · · · + ||zk ||pkk < 1} given k 1 1 P by F (z) = (f (z1 ) + f ′ (z1 ) Pj (zj ), (f ′ (z1 )) p2 z2′ , · · · , (f ′ (z1 )) pk zk′ )′ , where f is a normalj=2
ized biholomorphic function on the unit disc D, and for 2 ≤ j ≤ k, Pj : Cnj −→ C is a homogeneous polynomial of degree pj and zj = (zj1 , · · · , zjnj )′ ∈ Cnj , nj ≥ 1, pj ≥ 1, nj 1 P ||zj ||j = ( |zjl |pj ) pj . In this paper, some conditions for Pj are found under which the l=1
operator preserves the properties of almost starlikeness of order α, starlikeness of order α and strongly starlikeness of order α on ΩN , respectively. Key words
Reinhardt domain; Roper-Suffridge operator; almost starlike of order α; starlike mapping of order α; strongly starlike mapping of order α
2010 MR Subject Classification
1
32A10
Introduction
In 1995, Roper and Suffridge [1] introduced an extension operator. For a normalized locally biholomorphic function f on the unit disk D in C, the operator is defined by ′ p Φn (f )(z) = f (z1 ), f ′ (z1 )z0′ , (1.1)
n P 12 o n z ∈ Cn : ||z|| = |zj |2 < 1 , and the branch of the square root is j=1 p chosen such that f ′ (0) = 1 . It is well known that the Roper-Suffridge extension operator has the following properties.
where z ∈ B n =
∗ Received
September 6, 2013; received March 6, 2014. Supported by the National Natural Science Foundation of China (11001074, 11061015, 11101124). † Corresponding author: Shuxia FENG.
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In [1], Roper and Suffridge proved that if f is a normalized convex function on D, then Φn (f ) is a normalized convex mapping on B n . In [2, 3], Graham and Kohr proved that the Roper-Suffridge operator preserves the properties of starlikeness and Bloch on B n , respectively. In 2002, Gong and Liu [4, 5] introduced the definition of ε starlike mappings and obtained that the operator Φn,(1/p) (f )(z) = (f (z1 ), (f ′ (z1 ))1/p z0′ )′ , (1.2) maps the ε starlike functions on D to the ε starlike mappings on the Reinhardt domain Ωn,p = {z = (z1 , z0′ )′ ∈ C × Cn−1 : |z1 |2 + kz0 kpp < 1}, (1.3) P n 1/p , p ≥ 1. When ε = 0 and ε = 1, where z0 = (z2 , · · · , zn )′ ∈ Cn−1 , kz0 kp = |zj |p j=2
Φn,(1/p) (f ) maps the starlike function and convex function on D to the starlike mapping and the convex mapping on Ωn,p , respectively. Furthermore, Gong and Liu [6] proved that the operator Φn,(1/p2 ),··· ,(1/pn ) (f )(z) = (f (z1 ), (f ′ (z1 ))1/p2 z2 , · · · , (f ′ (z1 ))1/pn zn )′ ,
(1.4)
maps the ε starlike functions on D to the ε starlike mappings on the Reinhardt domain n X n 2 pj Ωn,p2 ,··· ,pn = z ∈ C : |z1 | + |zj | < 1 , (1.5) j=2
where pj ≥ 1, j = 2, · · · , n. The extension operator (1.4) on the domain Ωn,p2 ,··· ,pn is also studied by Liu and Liu [7], Feng and Liu [8]. In 2005, Muir [9] modified the Roper-Suffridge extension operator as p F (z) = (f (z1 ) + f ′ (z1 )P (z0 ), f ′ (z1 )z0′ )′ , (1.6)
where P (z0 ) is a homogeneous polynomial of degree 2 with respect to z0 , and f , z1 and z0 are defined as above. He proved that this operator preserves starlikeness and convexity if and only if ||P || ≤ 1/4 and ||P || ≤ 1/2, respectively. The modified operator plays an important role in studying the extreme points of convex mappings on B n (see [10, 11]). Recently, the modified Roper-Suffridge extension operator (1.6) on the unit ball is also studied by Wang and Liu [12], Feng and Yu [13]. But the modified Roper-Suffridge extension operator in [12, 13] has a little difference from [9]. In [12, 13], P (z0 ) is a homogeneous polynomial of degree k with respect to z0 , where k ≥ 2. In 2011, Wang and Gao [14] introduced a new Roper-Suffridge extension operator on the following Reinhardt domain Ωn,p2 ,··· ,pn given by F (z) = (f (z1 ) + f ′ (z1 )
k X
p
1
1
aj zj j , (f ′ (z1 )) p2 z2 , · · · , (f ′ (z1 )) pn zn ),
(1.7)
j=2
where f is a normalized locally biholomorphic function on the unit disc D, pj are positive integer, aj are complex constants, j = 2, · · · , n. Some conditions for aj are found under which the operator preserves the properties of almost starlikeness of order α and starlikeness of order α, respectively. In this paper, we introduce a Roper-Suffridge extension operator F (z) = (f (z1 ) + f ′ (z1 )
k X j=2
1
1
Pj (zj ), (f ′ (z1 )) p2 z2′ , · · · , (f ′ (z1 )) pk zk′ )′ ,
(1.8)
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which can preserve the properties of almost starlikeness of order α, starlikeness of order α and strongly starlikeness of order α on the domain ΩN given by different conditions for Pj , j = 2, · · · , k, respectively, where ΩN is definded as n o ΩN = z = (z1 , z2′ , · · · , zk′ )′ ∈ C × Cn2 × · · · × Cnk : |z1 |2 + kz2 kp22 + · · · + kzk kpkk < 1 , (1.9) and for j = 2, · · · , k, nj ≥ 1, pj ≥ 1, zj = (zj1 , · · · , zjnj )′ ∈ Cnj , kzj kj =
|zjl |pj
l=1
N = 1 + n2 + · · · + nk .
Cn
P nj
1/pj
,
n For notations, let D = {z ∈ C : |z| < 1} be the unit disk in C, B n = z = (z1 , · · · , zn )′ ∈ o 21 P n < 1 be the unit ball in Cn . Let Ω be a domain in Cn . Denote : kzk = |zj |2 j=1
H(Ω) be the set of all holomorphic mappings from Ω into Cn . A mapping f ∈ H(Ω) is called normalized if f (0) = 0, Jf (0) = In , where Jf (0) is the complex Jacob matrix of f at the origin and In is the identity matrix on Cn . A mapping f ∈ H(Ω) is said to be locally biholomorphic if det Jf (z) 6= 0 for ∀z ∈ Ω. A normalized mapping f ∈ H(Ω) is said to be convex if there exists a point z ∈ Ω such that f (z) = tf (z1 ) + (1 − t)f (z2 ) for ∀z1 , z2 ∈ Ω and ∀t ∈ [0, 1]. A normalized mapping f ∈ H(Ω) is said to be starlike if tf (z) ∈ H(Ω), for ∀z ∈ Ω and ∀t ∈ [0, 1]. Assume that P : Cn −→ C is a homogeneous polynomial of degree k. Then P satisfies P (λz) = λk P (z) ∂P ∂P for ∀z ∈ Cn and ∀λ ∈ C. It is easy to see that ∇P (z)z = kP (z), where ∇P (z) = ( ∂z , · · · , ∂z ) 1 n n is the gradient of P (z). The norm of P is given by kP k = sup{|P (z)| : z ∈ ∂B }. A domain Ω is said to be a Reinhardt domain if (eiθ1 z1 , · · · , eiθn zn ) ∈ Ω holds for ∀z ∈ Ω and ∀θ1 , · · · , θn ∈ R. A domain Ω is said to be a circular domain if eiθ z ∈ Ω holds for ∀z ∈ Ω and ∀θ ∈ R. Denote ρ(z) = inf{t > 0 : z/t ∈ Ω} to be the Minkowski functional of the domain Ω. If Ω is a bounded circular convex domain, then Ω is a Banach space in Cn with respect to this norm, and Ω = {z ∈ Cn : ρ(z) < 1}. Definition1.1 [15] Suppose Ω is a bounded starlike circular domain in Cn . Its Minkowski functional ρ(z) is C 1 except for a lower-dimensional manifold. Let 0 ≤ α < 1. A normalized locally biholomorphic mapping f ∈ H(Ω) is said to be an almost starlike mapping of order α if Re
2 ∂ρ (z)Jf−1 (z)f (z) ≥ α, ∀z ∈ Ω \ {0}. ρ(z) ∂z
(1.10)
When Ω = B n , the Minkowski functional ρ(z) = kzk, and the above inequality becomes Rez ′ Jf−1 (z)f (z) ≥ α||z||2 , ∀z ∈ B n \ {0}.
(1.11)
In particular, when α = 0, f is a starlike mapping on Ω. Definition 1.2 [16] Suppose Ω is a bounded starlike circular domain in Cn , and its Minkowski functional ρ(z) is C 1 except for a lower-dimensional manifold. Let 0 < α < 1. A normalized locally biholomorphic mapping f ∈ H(Ω) is said to be a starlike mapping of order α if 2 ∂ρ 1 1 −1 (1.12) ρ(z) ∂z (z)Jf (z)f (z) − 2α < 2α , ∀z ∈ Ω \ {0}. When Ω = B n , the above inequality reduces to 1 ′ −1 1 1 z J (z)f (z) − < , ∀z ∈ B n \ {0}. ||z|| f 2α 2α
(1.13)
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Definition 1.3 [17] Let 0 < α ≤ 1. Suppose Ω is a bounded starlike circular domain in Cn , and its Minkowski functional ρ(z) is C 1 except for a lower-dimensional manifold. A normalized locally biholomorphic mapping f (z) ∈ H(Ω) is said to be a strongly starlike mapping of order α on Ω if π 2 ∂ρ −1 arg (z)Jf (z)f (z) < α, ∀z ∈ Ω \ {0}. (1.14) ρ(z) ∂z 2 When Ω = B n , the inequality (1.14) becomes π arghJf−1 (z)f (z), zi < α, ∀z ∈ B n \ {0}. 2 When n = 1, the inequality (1.15) becomes ′ arg zf (z) < π α, f (z) 2
(1.15)
(1.16)
it is the definition of strongly starlike function of order α on D. From Definition 1.3, we know that if f is a strongly starlike mapping of order α on Ω, then f is a starlike mapping on Ω, furthermore f is a strongly starlike mapping of order β on Ω, where 0 < β < α. ′ If f (z) ∈ H(Ω) is a normalized locally biholomorphic mapping, A is a normal matrix(AA = ′ A A), then f is said to be spirallike mapping relative to A, if e−At Ω and n f (z) ∈ f (Ω), for ∀z ∈ o −1 2 ∂ρ ∀t ≥ 0. f is a spirallike mapping relative to A if and only if Re ρ(z) ∂z (z)Jf (z)Af (z) ≥ 0,
∀z ∈ Ω \ {0} (see [16]). When A = e−iβ I, f is said to be spirallike mapping of type β (β ∈ R, |β| < π2 ). When n = 1, f is a spirallike function of type β on D if and only if ([18]) zf ′ (z) Re eiβ > 0, ∀z ∈ D. (1.17) f (z) In one complex variable, Stankiewicz [18] has given an interesting relationship between strongly starlike function of order α and spirallike function of type β. If we denote S ∗ (α) to be the family of strongly starlike functions of order α, where 0 < α ≤ 1, and SP (β) to be the family of spirallike functions of type β, where β ∈ R, |β| < π2 , then \ S ∗ (α) = SP (β). (1.18) |β|≤ π 2 (1−α)
2
Some Lemmas
In order to prove the main results, we need the following lemmas. Lemma 2.1 [19] Suppose Ω ⊆ Cn is a bounded starlike circular domain in Cn , and its Minkowski function ρ(z) is C 1 on Ω except for a lower-dimensional manifold, then ∂ρ (z)z = ρ(z), ∀z ∈ Cn ; ∂z ∂ρ 2Re (z0 )z0 = 1, ∀z0 ∈ ∂Ω; ∂z ∂ρ ∂ρ (λz) = (z), ∀λ ∈ [0, +∞); ∂z ∂z ∂ρ iθ ∂ρ (e z) = e−iθ (z), ∀θ ∈ R. ∂z ∂z 2Re
(2.1)
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Lemma 2.2 [20] Let h(z) be a holomorphic function on D. If Reh(z) > 0 and h(0) > 0, then 2Reh(z) |h′ (z)| ≤ . (2.2) 1 − |z|2 Lemma 2.3 [21] Let f be a normalized biholomorphic function on D, then ′′ (1 − |z|2 ) f (z) − 2z ≤ 4, ∀z ∈ D. ′ f (z)
(2.3)
Lemma 2.4 [22] If ρ(z) is a Minkowski functional of the Reinhardt domain n X Ωp1 ,p2 ,··· ,pn = z ∈ Cn : |zj |pj < 1 , pj ≥ 1, j = 1, · · · , n, j=1
then ∂ρ (zj ) = ∂z
zj pj −2 pj zj ρ(z) n P zj pj , pj ρ(z) 2ρ(z)
j = 1, · · · , n.
(2.4)
j=1
Lemma 2.5 [23] Suppose f ∈ H(D). Then Ref (z) > 0 (∀z ∈ D) if and only if there is an increasing function µ on [0, 2π], which satisfies µ(2π) − µ(0) = Ref (0), such that Z 2π 1 + ze−iθ f (z) = dµ(θ) + iImf (0), z ∈ D. (2.5) 1 − ze−iθ 0 Lemma 2.6 [8] Suppose w ∈ C. Then (i) Re[(1 − w2 )(1 − w)2 ] = (1 − |w|2 )|1 − w|2 ; (ii) Re[(1 + 2w − w2 )(1 − w)2 ] = (1 − |w|2 )2 − 2|w|2 |1 − w|2 . Lemma 2.7 [17] Suppose 0 < α 6 1, f ∈ Ω is a normalized locally biholomorphic mapping, then f (z) is a strongly starlike mapping of order of α if only and if f is a spirallike mapping with respect to eiβ I, where β ∈ R, |β| ≤ π2 (1 − α).
3
Main Results
Theorem 3.1 Let 0 ≤ α < 1 and let f be an almost starlike function of order α on the unit disc D. For j = 2, · · · , k, Pj : Cnj −→ C is a homogeneous polynomial of zj with degree pj ≥ 1. If ||Pj || ≤ 1−α 4 , j = 2, · · · , k, then F (z) = (f (z1 ) + f ′ (z1 )
k X
1
1
Pj (zj ), (f ′ (z1 )) p2 z2′ , · · · , (f ′ (z1 )) pk zk′ )′
j=2
is an almost starlike mapping of order α on the domain ΩN , where the branches are chosen 1 such that (f ′ (0)) pj = 1. Proof By the definition of almost starlike mapping of order α, we only need to prove that the following inequality Re holds for any z ∈ ΩN and z 6= 0.
2 ∂ρ (z)JF−1 (z)F (z) ≥ α ρ(z) ∂z
(3.1)
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If (z2 , · · · , zk ) = 0, then F (z) = (f (z1 ), 0, · · · , 0)′ . Therefore, we can get the result by the definition of function f . Note that the mapping F is holomorphic at every point z = (z1 , z2′ , · · · , zk′ )′ ∈ ΩN . Let z = ζu = eiθ |ζ|u, u ∈ ∂ΩN , and ζ ∈ D \ {0}, then we have 2 ∂ρ (z)JF−1 (z)F (z) ≥ α Re ρ(z) ∂z 2 ∂ρ iθ −1 iθ iθ ⇐⇒ Re (e |ζ|u)J (e |ζ|u)F (e |ζ|u) ≥α F ρ(eiθ |ζ|u) ∂z 2 e−iθ ∂ρ iθ ⇐⇒ Re (e |ζ|u)JF−1 (eiθ |ζ|u)F (eiθ |ζ|u) ≥ α |ζ| ∂z JF−1 (ζu)F (ζu) 2∂ρ ⇐⇒ Re (u) ≥ α. (3.2) ∂z ζ The expression
J −1 (ζu)F (ζu) 2∂ρ Re (u) F ∂z ζ
is the real part of a holomorphic function with respect to ζ, and thus, it is a harmonic function. By the minimum principle for harmonic functions, we know that it attains its minimum on |ζ| = 1. Therefore, we only need to prove that inequality (3.1) holds for (z2′ , · · · , zk′ )′ 6= 0 and z ∈ ∂ΩN , i.e., ρ(z) = 1. It means that we need to prove 2∂ρ −1 Re (z)JF (z)F (z) ≥ α, ∀z ∈ ∂ΩN , (z2 , · · · , zk ) 6= 0. (3.3) ∂z Since
hence
where
k X
f (z1 ) + f ′ (z1 ) Pj (zj ) j=2 1 (f ′ (z1 )) p2 z2 F (z) = , .. . 1 (f ′ (z1 )) pk zk
λ1
α2
···
αk
β2 λ2 In2 · · · 0 JF (z) = .. .. , .. . . . βk 0 · · · λk Ink λ1 = f ′ (z1 ) + f ′′ (z1 )
k X
(3.4)
1
Pj (zj ), λj = (f ′ (z1 )) pj , j = 2, · · · , k,
j=2
αj = f ′ (z1 )∇Pj (zj ), βj =
1 1 ′ −1 (f (z1 )) pj f ′′ (z1 )zj , j = 2, · · · , k. pj
We denote X = JF−1 (z)F (z), where X = (x1 , x2 , · · · , xk )′ , xj ∈ Cnj for j = 2, · · · , k, then
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F (z) = JF (z)X. So
′
k X
f (z1 ) + f (z1 ) Pj (zj ) λ α2 · · · αk x 1 1 j=2 β2 λ2 In2 · · · 0 x2 1 ′ p2 = z (f (z )) . 2 1 .. .. .. .. . . . . . .. βk 0 · · · λk Ink xk 1 (f ′ (z1 )) pk zk
(3.5)
Consequently k k k X X X (f ′ (z1 ) + f ′′ (z1 ) Pj (zj ))x1 + f ′ (z1 ) ∇Pj (zj )xj = f (z1 ) + f ′ (z1 ) Pj (zj ), j=2 j=2 j=2 1 ′ 1 1 1 −1 (f (z1 )) p2 f ′′ (z1 )z2 x1 + (f ′ (z1 )) p2 x2 = (f ′ (z1 )) p2 z2 , p2 · ·· 1 1 1 1 (f ′ (z1 )) pk −1 f ′′ (z1 )zk x1 + (f ′ (z1 )) pk xk = (f ′ (z1 )) pk zk . pk Note that ∇Pj (zj ) = pj Pj (zj ), j = 2, · · · , k, by easy computation we can get k f (z1 ) X − (pj − 1)Pj (zj ), x = 1 f ′ (z1 ) j=2 k ′′ ′′ X x2 = 1 − f (z1 )f (z1 ) + f (z1 ) (p − 1)P (z ) j j j z2 , p2 (f ′ (z1 ))2 p2 f ′ (z1 ) j=2 ··· k f (z1 )f ′′ (z1 ) f ′′ (z1 ) X (pj − 1)Pj (zj ) zk . xk = 1 − pk (f ′ (z1 ))2 + pk f ′ (z1 ) j=2
(3.6)
Hence, by (2.4) we have
2
∂ρ (z)JF−1 (z)F (z) = ∂z
G(z) , k P pj 2 2|z1 | + pj ||zj ||j
(3.7)
j=2
where
k f (z1 ) X G(z) = 2z1 ′ − (pj − 1)Pj (zj ) f (z1 ) j=2 k k X f (z1 )f ′′ (z1 ) f ′′ (z1 ) X pj + pj ||zj ||j 1 − + (pl − 1)Pl (zl ) . pj (f ′ (z1 ))2 pj f ′ (z1 ) j=2
(3.8)
l=2
f (z1 ) z1 f ′ (z1 ) −α, then 1) ′ other hand, ff′(z (z1 )
Let h(z1 ) =
h(0) = 1−α > 0, and by lemma 2.2 we know that |h′ (z)| <
On the
= [z1 (h(z1 ) + α)]′ , i.e.,
f (z1 )f ′′ (z1 ) = 1 − α − h(z1 ) − z1 h′ (z1 ). (f ′ (z1 ))2
Substituting (3.9) into (3.8), we obtain that k X f (z1 )f ′′ (z1 ) p G(z) = 2|z1 |2 (h(z1 ) + α) + 1− pj ||zj ||j j ′ (z ))2 p (f j 1 j=2
2Reh(z) 1−|z|2 .
(3.9)
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X k k f ′′ (z1 ) X pj ||z || − 2z (pl − 1)Pl (zl ) j j 1 f ′ (z1 ) j=2 l=2
= 2|z1 |2 h(z1 ) + 2α|z1 |2 +
k X j=2
k X
f ′′ (z1 ) + ′ f (z1 ) = (2|z1 |2 +
p ||zj ||j j
− 2z1
j=2
k X
p pj − (1 − α − h(z1 ) − z1 h′ (z1 )) ||zj ||j j
X k
(pl − 1)Pl (zl )
l=2
p
||zj ||j j )h(z1 ) + 2α|z1 |2 +
j=2
+z1 h′ (z1 )
p
||zj ||j j +
j=2
Since z ∈ ∂ΩN , i.e., |z1 |2 +
j=2
p
(pj − 1 + α)||zj ||j j
j=2
k X k P
k X
X k k f (z1 ) X pj ||z || (pl − 1)Pl (zl ). − 2z j 1 j f ′ (z1 ) j=2 ′′
l=2
p
||zj ||j j = 1, so
G(z) = (1 + |z1 |2 )h(z1 ) + 2α|z1 |2 +
k X
p
(pj − 1 + α)||zj ||j j + z1 h′ (z1 )(1 − |z1 |2 )
j=2
+
f ′′ (z1 ) (1 − |z1 |2 ) − 2z1 f ′ (z1 )
By (2.2) and (2.3), we get
X k
(pl − 1)Pl (zl ).
l=2
ReG(z) ≥ (1 + |z1 |2 )Reh(z1 ) + 2α|z1 |2 +
k X
p
(pj − 1 + α)||zj ||j j
j=2
−(1 − |z1 |2 )|z1 |
2Reh(z1 ) − 1 − |z1 |2
k X l=2
≥ (1 + |z1 |2 )Reh(z1 ) + 2α|z1 |2 +
′′ f (z1 ) (pl − 1)|Pl (zl )| ′ (1 − |z1 |2 ) − 2z1 f (z1 )
k X
p
(pj − 1 + α)||zj ||j j − 2|z1 |Reh(z1 )
j=2
−4
k X
(pl − 1)|Pl (zl )|.
l=2
p
Since ||Pj (zj )|| ≤ ||Pj || · ||zj ||j j ≤
pj 1−α 4 ||zj ||j ,
j = 2, · · · , k, therefor
ReG(z) ≥ (1 − |z1 |)2 Reh(z1 ) + 2α|z1 |2 +
k X j=2
≥ (1 − |z1 |)2 Reh(z1 ) + 2α|z1 |2 +
k X
p (pj − 1 + α) − 4(pj − 1)||Pj || ||zj ||j j (pj − 1 + α) − 4(pj − 1)
j=2
≥ 2α|z1 |2 + α
k X
p
pj ||zj ||j j .
j=2
So Re
2∂ρ (z)JF−1 (z)F (z) = ∂z
ReG(z) ≥α k P pj 2 2|z1 | + pj ||zj ||j j=2
1 − α p ||zj ||j j 4
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holds for ∀z ∈ ∂ΩN , (z2 , · · · , zk ) 6= 0. Consequently, inequality (3.1) holds for z ∈ ΩN and z 6= 0.
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Theorem 3.2 Let 0 < α < 1 and let f be a starlike function of order α on the unit disc D. For j = 2, · · · , k, Pj : Cnj −→ C is a homogeneous polynomial of zj with degree pj ≥ 1. If kPj k ≤ 1−|2α−1| , j = 2, · · · , k, then 8α F (z) = (f (z1 ) + f ′ (z1 )
k X
1
1
Pj (zj ), (f ′ (z)) p2 z2′ , · · · , (f ′ (z)) pk zk′ )′ ,
j=2
is a starlike mapping of order α on the domain ΩN , where the branches are chosen such that 1 (f ′ (0)) pj = 1. Proof By the definition of starlike mapping of order α, we only need to prove that the following inequality 2 ∂ρ 1 1 −1 (3.10) ρ(z) ∂z (z)JF (z)F (z) − 2α < 2α holds for z ∈ ΩN \ {0}. If (z2′ , · · · , zk′ )′ = 0, then F (z) = (f (z1 ), 0, · · · , 0)′ . Therefore, we can get the result by the definition of function f . Note that the mapping F is holomorphic at every point z = (z1 , z2′ , · · · , zk′ )′ ∈ ΩN . Let z = ζu = eiθ |ζ|u, u ∈ ∂ΩN , and ζ ∈ D \ {0}, then we have 2 ∂ρ 1 1 −1 < (z)J (z)F (z) − F ρ(z) ∂z 2α 2α 1 ∂ρ iθ 1 2 −1 iθ iθ < ⇐⇒ iθ (e |ζ|u)JF (e |ζ|u)F (e |ζ|u) − ρ(e |ζ|u) ∂z 2α 2α −iθ 2 e ∂ρ iθ 1 1 < ⇐⇒ (e |ζ|u)JF−1 (eiθ |ζ|u)F (eiθ |ζ|u) − |ζ| ∂z 2α 2α −1 2∂ρ 1 J (ζu)F (ζu) 1 ⇐⇒ < (u) F − . (3.11) ∂z ζ 2α 2α The expression J −1 (ζu)F (ζu) 2∂ρ 1 (u) F − ∂z ζ 2α is holomorphic with respect to ζ. Thus, the maximum modules principle for holomorphic functions yields that it attains its maximum on |ζ| = 1. Therefore, we only need to prove that inequality (3.10) holds for z ∈ ∂ΩN and (z2′ , · · · , zk′ )′ 6= 0. From the proof of Theorem 3.1, we can get ∂ρ G(z) 2 (z)JF−1 (z)F (z) = , k P ∂z p 2|z1 |2 + pj ||zj ||j j j=2
where G(z) is defined by equality (3.8). Hence 2
∂ρ 1 (z)JF−1 (z)F (z) − = ∂z 2α
H(z) , k P p 2α 2|z1 |2 + pj ||zj ||j j j=2
where H(z) = 4α|z1 |2
k X f (z1 ) f (z1 )f ′′ (z1 ) pj + 2α p ||z || 1 − j j j z1 f ′ (z1 ) pj (f ′ (z1 ))2 j=2
(3.12)
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′′ k k X f (z1 ) X p ||zl ||pl l − 2z1 − 2|z1 |2 + (pj − 1)Pj (zj ) ′ pj ||zj ||j j f (z1 ) j=2 j=2
k X
l=2
k X f (z1 )f ′′ (z1 ) f (z1 ) 1 pj 2 − 1 + 2α pj ||zj ||j 1 − − = 2|z1 | 2α z1 f ′ (z1 ) 2α pj (f ′ (z1 ))2 j=2 ′′ f (z1 ) 2 (1 − |z1 | ) − 2z1 . +2α (pj − 1)Pj (zj ) ′ f (z1 ) j=2 k X
1) − 1. Then |h(z1 )| < 1 by the definition of f (z1 ). According to Let h(z1 ) = 2α z1ff(z′ (z 1) Schwarz-Pick lemma, we obtain that
1 − |h(z1 )|2 . 1 − |z1 |2
|h′ (z1 )| ≤
(3.13)
On the other hand, we can get 1 + h(z1 ) + z1 h′ (z1 ) f (z1 )f ′′ (z1 ) = 1 − , (f ′ (z1 ))2 2α
(3.14)
hence 2
H(z) = 2|z1 | h(z1 ) + 2α
k X
p pj ||zj ||j j
j=2
1 1 1 + h(z1 ) + z1 h′ (z1 ) 1− − 1− 2α pj 2α
′′ f (z1 ) 2 +2α (pj − 1)Pj (zj ) ′ (1 − |z1 | ) − 2z1 f (z1 ) j=2 k X
k k k X X X pj p p 2 = 2|z1 | + ||zj ||j h(z1 ) + ||zj ||j j z1 h′ (z1 ) + (2α − 1) pj ||zj ||j j j=2
j=2
j=2
′′ f (z1 ) 2 +2α (pj − 1)Pj (zj ) ′ (1 − |z1 | ) − 2z1 , f (z1 ) j=2 k X
and
|H(z)| ≤ (2|z1 |2 +
k X
p
||zj ||j j )|h(z1 )| +
j=2
+|2α − 1| ·
k X j=2
k X
(pj −
p 1)||zj ||j j
j=2
′′ f (z1 ) 2 + 2α (pj − 1)|Pj (zj )| ′ (1 − |z1 | ) − 2z1 f (z1 ) j=2
≤ (1 + |z1 |2 )|h(z1 )| + (1 − |z1 |2 )|z1 |
+2α
k X
p
||zj ||j j |z1 | · |h′ (z1 )| k X
k X 1 − |h(z1 )|2 p + |2α − 1| (pj − 1)||zj ||j j 1 − |z1 |2 j=2
(pj − 1)|Pj (zj )| · 4
j=2
≤ (1 + |z1 |2 )(|h(z1 )| − 1) + (1 + |z1 |2 ) + 2|z1 |(1 − |h(z1 )|) +|2α − 1|
k X j=2
p
(pj − 1)||zj ||j j + 8α
k X
(pj − 1)|Pj (zj )|
j=2
≤ (1 + |z1 |2 ) + (|h(z1 )| − 1)(1 − |z1 |)2 + |2α − 1|
k X j=2
p
(pj − 1)||zj ||j j
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k X
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p
(pj − 1)||Pj || · ||zj ||j j
j=2
2
≤ (1 + |z1 | ) + |2α − 1|
k X
(pj −
p 1)||zj ||j j
+ 8α
j=2
= (1 + |z1 |2 ) +
k X
k X
(pj − 1)
j=2
1 − |2α − 1| p ||zj ||j j 8α
p
(pj − 1)||zj ||j j
j=2
k k X X pj pj 2 = 1− ||zj ||j + |z1 | + pj ||zj ||j j=2
= 2|z1 |2 +
j=2
k X
p
pj ||zj ||j j .
j=2
Hence
2∂ρ 1 1 −1 ∂z (z)JF (z)F (z) − 2α < 2α
holds for z ∈ ∂ΩN \ {0}, and (z2′ , · · · , zk′ )′ 6= 0.
Notation When n2 = · · · = nk = 1, (1.8) reduces to (1.7). Theorems 3.1 and 3.2 in this paper are Theorems 3.1 and 3.2 in [14]. Theorem 3.3 Let 0 < α ≤ 1 and let f be a strongly starlike function of order α on the unit disc D. For j = 2, · · · , k, Pj : Cnj −→ C is a homogeneous polynomial of zj with degree pj ≥ 1. If ||Pj || ≤ 14 cos π2 (1 − α) , j = 2, · · · , k, then F (z) = (f (z1 ) + f ′ (z1 )
k X
1
1
Pj (zj ), (f ′ (z1 )) p2 z2′ , · · · , (f ′ (z1 )) pk zk′ )′
j=2
is a strongly starlike mapping of order α on the domain ΩN , where the branches are chosen 1 such that (f ′ (0)) pj = 1. Proof By Lemma 2.7, we only need to prove F (z) is spirallike mapping of type β, for all |β| ≤ π2 (1 − α). From the definition of spirallike mapping ([16]), we need to prove −1 −iβ 2 ∂ρ Re e (z)Jf (z)f (z) ≥ 0, ∀z ∈ ΩN \ {0}. (3.15) ρ(z) ∂z Similarly to the inequality (3.1), we only need to prove inequality (3.13) holds for (z2′ , · · · , zk′ )′ 6= 0 and z ∈ ∂ΩN , i.e. ρ(z) = 1. It means that we need to prove ∂ρ Re 2e−iβ (z)JF−1 (z)F (z) ≥ 0, ∀z ∈ ∂ΩN , (z2′ , · · · , zk′ )′ 6= 0. (3.16) ∂z From equality (3.7), we have 2e−iβ
∂ρ (z)JF−1 (z)F (z) = ∂z
e−iβ G(z) , k P p 2|z1 |2 + pj ||zj ||j j
(3.17)
j=2
where G(z) is defined by (3.8). From (1.18), we know f is spirallike function of type β for |β| ≤ π2 (1 − α). Let h(z1 ) = −iβ f (z1 ) e z1 f ′ (z1 ) , z1 ∈ D. By (1.17), we know that Reh(z1 ) > 0; and by Lemma 2.5, there is an
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increasing function µ on [0, 2π], which satisfies µ(2π) − µ(0) = Reh(0) = cos β, and Z 2π 1 + z1 e−iθ h(z1 ) = dµ(θ) − i sin β, z1 ∈ D. 1 − z1 e−iθ 0 By simple calculation we can get Z 2π 2z1 e−iθ z1 h′ (z1 ) = dµ(θ), (1 − z1 e−iθ )2 0 f (z1 )f ′′ (z1 ) = 1 − eiβ (z1 h′ (z1 ) + h(z1 )), (f ′ (z1 ))2
then Re{e−iβ G(z)} k X f (z1 ) −iβ − e 2z = Re 2|z1 |2 e−iβ (pj − 1)Pj (zj ) 1 z1 f ′ (z1 ) j=2 +e−iβ
k f (z1 )f ′′ (z1 ) f ′′ (z1 ) X p + (p − 1)P (z ) ||zj ||j j pj − l l l (f ′ (z1 ))2 f ′ (z1 ) j=2
k X
l=2
k X p = Re 2|z1 |2 h(z1 ) + e−iβ ||zj ||j j pj − 1 + eiβ (z1 h′ (z1 ) + h(z1 )) j=2
−e
−iβ
2z1
k X
(pj − 1)Pj (zj ) + e
−iβ
j=2
k X
p ||zj ||j j
j=2
k f ′′ (z1 ) X (pl − 1)Pl (zl ) f ′ (z1 ) l=2
k k X X p = Re 2|z1 |2 h(z1 ) − e−iβ 2z1 (pj − 1)Pj (zj ) + e−iβ (pj − 1)||zj ||j j j=2
+
k X
p ||zj ||j j (z1 h′ (z1 )
+ h(z1 )) + e
j=2
−iβ
j=2
k X
p ||zj ||j j
j=2
k f (z1 ) X (pl − 1)Pl (zl ) f ′ (z1 ) ′′
l=2
k n ′′ X 2 −iβ 2 f (z1 ) = Re 2|z1 | h(z1 ) + e (pj − 1)Pj (zj ) (1 − |z1 | ) ′ − 2z1 f (z1 ) j=2 +e
−iβ
k X
(pj −
j=2
p 1)||zj ||j j
+ (1 − |z1 | )(z1 h (z1 ) + h(z1 )) 2
′
k X p = Re (1 + |z1 |2 )h(z1 ) + (1 − |z1 |2 )z1 h′ (z1 ) + cos β (pj − 1)||zj ||j j j=2
k X f ′′ (z1 ) +Re e−iβ (pj − 1)Pj (zj ) (1 − |z1 |2 ) ′ − 2z1 f (z1 ) j=2
k n o X p > Re (1 + |z1 |2 )h(z1 ) + (1 − |z1 |2 )z1 h′ (z1 ) + cos β (pj − 1)||zj ||j j j=2
−
k X j=2
f ′′ (z1 ) − 2z1 . (pj − 1)|Pj (zj )| (1 − |z1 |2 ) ′ f (z1 )
By Lemma 2.3, we have
′′ (1 − |z|2 ) f (z) − 2z ≤ 4. ′ f (z)
(3.18)
(3.19) (3.20)
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p − α) , and |Pj (zj )| ≤ ||Pj || · ||zj ||j j , we get k k iX hπ ′′ X p 2 f (z1 ) (pj − 1)|Pj (zj )| (1 − |z1 | ) ′ − 2z1 ≤ cos (1 − α) (pj − 1)||zj ||j j . f (z ) 2 1 j=2 j=2 π π Note |β| ≤ 2 (1 − α), and 0 < α ≤ 1, so cos β ≥ cos 2 (1 − α) . Hence k k X X f ′′ (z1 ) p cos β (pj − 1)||zj ||j j − (pj − 1)|Pj (zj )| (1 − |z1 |2 ) ′ − 2z1 ≥ 0. (3.21) f (z1 ) j=2 j=2 Since ||Pj || ≤
1 4
cos
π
2 (1
Next, we just prove
Re{(1 + |z1 |2 )h(z1 ) + (1 − |z1 |2 )z1 h′ (z1 )} ≥ 0.
(3.22)
Substituting (3.18) and (3.19) into the left of (3.22), we have n o Re (1 + |z1 |2 )h(z1 ) + (1 − |z1 |2 )z1 h′ (z1 ) Z 2π n o 1 (1 + |z1 |2 )[1 − (z1 e−iθ )2 ] + 2z1 e−iθ (1 − |z1 |2 ) dµ(θ). = Re −iθ 2 (1 − z1 e ) 0
Let w = z1 e−iθ , then w ∈ D. Since µ is increasing, we just prove 1 2 2 2 Re [(1 + |w| )(1 − w ) + 2w(1 − |w| )] ≥ 0. (1 − w)2
By lemma 2.6, we obtain 1 2 2 2 Re [(1 + |w| )(1 − w ) + 2w(1 − |w| )] (1 − w)2 1 = Re[(1 + |w|2 )(1 − w2 )(1 − w)2 + 2w(1 − |w|2 )(1 − w)2 ] |1 − w|4 1 = Re[(1 + |w|2 )(1 − |w|2 )|1 − w|2 + 2w(1 − |w|2 )(1 − w)2 ] |1 − w|4 1 − |w|2 = Re[(1 + |w|2 )|1 − w|2 − (1 − w2 )(1 − w)2 + (1 − w2 )(1 − w)2 + 2w(1 − w)2 ] |1 − w|4 1 − |w|2 = Re[(1 + |w|2 )|1 − w|2 − |1 − w|2 (1 − |w|2 ) + (1 + 2w − w2 )(1 − w)2 ] |1 − w|4 1 − |w|2 = [2|w|2 |1 − w|2 + (1 − |w|2 )2 − 2|w|2 |1 − w|2 ] |1 − w|4 (1 − |w|2 )3 = |1 − w|4 ≥ 0. Theorem 3.3 is completely proved.
References
[1] Roper K, Suffridge T. Convex mappings on the unit ball of Cn . J d’Analyse Math, 1995, 65: 333–347 [2] Graham I, Kohr G. Univalent mappings associated with the Roper-Suffridge extension operator. J d’Analyse Math, 2000, 81: 331–342 [3] Graham I, Kohr G. Loewner chains and the Roper-Suffridge extension operator. J Math Anal Appl, 2000, 247: 448–465 [4] Gong S, Liu T S. On the Roper-Suffridge extension operator. J d’Analyse Math, 2002, 88: 397–404
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[5] Liu T S, Gong S. The family of ε starlike mappings (I). Chin Ann Math, 2002, 23A(3): 273–282 [6] Gong S, Liu T S. The generalized Roper-Suffridge extension operator. J Math Anal Appl, 2003, 284(2): 425–434 [7] Liu X S, Liu T S. The generalized Roper-Suffridge extension operator on a Reinhardt domain and the unit ball in a complex Hilbert space. Chin Ann Math, 2005, 26A(5): 721–730 [8] Feng S X, Liu T S. The generalized Roper-Suffridge extension operator. Acta Math Sci, 2008, 28B(1): 63–80 [9] Muir J. A modification of the Roper-Suffridge extension operator. Comput Methods and Funct Theory, 2005, 5(1): 237–251 [10] Muir J, Suffridge T. A generalization of half-plane mappings to the ball in C n . Trans Amer Math Society, 2007, 359(4): 1485–1498 [11] Muir J, Suffridge T. Extreme points for convex mappings of B n . J d’Analyse Math, 2006, 98: 169–182 [12] Wang J F, Liu T S. A modified Roper-Suffridge extension operator for some holomorphic mappings. Chin Ann Math, 2010, 31A(4): 487–496 [13] Feng S X, Yu L. Modified Roper-Suffridge operator for some holomorphic mappings. Front Math China, 2011, 6(3): 411–426 [14] Wang J F, Gao C L. A new Roper-Suffridge extension operator on a Reinhardt domain. Abst Appl Anal, 2011, 2011: Artile ID 865496 [15] Feng S X, Lu K P. The growth theorem for almost starlike mappings of order on bounded starlike circular domains. Chin Quart J Math, 2000, 15(2): 50–56 [16] Liu H. Class of Starlike Mappings, its Extensions and Subclasses in Several Complex Variables [D]. Hefei: University of Science and Technology of China, 1999 (In Chinese) [17] Liu H, Li X S. The growth theorem for strongly starlike mappings of order α on bounded starlike circular domains. Chin Quart J Math, 2000, 15(3): 28–33 [18] Stankiewicz J. Queleques problemes extremaux dans les classes α-angulairment etoiles. Ann Universitaties, Mariae Curie-Sklodowska, 1966, 20: 59–75 [19] Liu T S, Ren G B. The growth theorem for starlike mappings on bounded starlike circular domains. Chin Ann Math, 1998, 19B(4): 401–408 [20] Graham I, Kohr G. Geometric function theory in one and higher dimensions. New York: Marcel Dekker, 2003 [21] Muir J. A class of Loewner chain preserving extension operators. J Math Anal Appl, 2008, 337(2): 862–879 [22] Zhang W J, Liu T S. On decomposition theorem of normalized biholomorphic convex mappings in Reinhardt domains. Sci in China, 2003, 46A(1): 94–106 [23] Duren P. Univalent Functions. New York: Springer-Verlag, 1983
ROPER-SUFFRIDGE EXTENSION OPERATOR ON A REINHARDT DOMAIN∗
oõ)
Hongjun LI (
School of Mathematics and Information Science, Henan University, Kaifeng 475004, China E-mail : [email protected]
¾Ô_)
Shuxia FENG (
†
Institute of Contemporary Mathematics, School of Mathematics and Information Science, Henan University, Kaifeng 475004, China E-mail : [email protected] Abstract Let pj ∈ N and pj ≥ 1, j = 2, · · · , k, k ≥ 2 be a fixed positive integer. We introduce a Roper-Suffridge extension operator on the following Reinhardt domain ΩN = {z = (z1 , z2′ , · · · , zk′ )′ ∈ C × Cn2 × · · · × Cnk : |z1 |2 + ||z2 ||p22 + · · · + ||zk ||pkk < 1} given k 1 1 P by F (z) = (f (z1 ) + f ′ (z1 ) Pj (zj ), (f ′ (z1 )) p2 z2′ , · · · , (f ′ (z1 )) pk zk′ )′ , where f is a normalj=2
ized biholomorphic function on the unit disc D, and for 2 ≤ j ≤ k, Pj : Cnj −→ C is a homogeneous polynomial of degree pj and zj = (zj1 , · · · , zjnj )′ ∈ Cnj , nj ≥ 1, pj ≥ 1, nj 1 P ||zj ||j = ( |zjl |pj ) pj . In this paper, some conditions for Pj are found under which the l=1
operator preserves the properties of almost starlikeness of order α, starlikeness of order α and strongly starlikeness of order α on ΩN , respectively. Key words
Reinhardt domain; Roper-Suffridge operator; almost starlike of order α; starlike mapping of order α; strongly starlike mapping of order α
2010 MR Subject Classification
1
32A10
Introduction
In 1995, Roper and Suffridge [1] introduced an extension operator. For a normalized locally biholomorphic function f on the unit disk D in C, the operator is defined by ′ p Φn (f )(z) = f (z1 ), f ′ (z1 )z0′ , (1.1)
n P 12 o n z ∈ Cn : ||z|| = |zj |2 < 1 , and the branch of the square root is j=1 p chosen such that f ′ (0) = 1 . It is well known that the Roper-Suffridge extension operator has the following properties.
where z ∈ B n =
∗ Received
September 6, 2013; received March 6, 2014. Supported by the National Natural Science Foundation of China (11001074, 11061015, 11101124). † Corresponding author: Shuxia FENG.
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In [1], Roper and Suffridge proved that if f is a normalized convex function on D, then Φn (f ) is a normalized convex mapping on B n . In [2, 3], Graham and Kohr proved that the Roper-Suffridge operator preserves the properties of starlikeness and Bloch on B n , respectively. In 2002, Gong and Liu [4, 5] introduced the definition of ε starlike mappings and obtained that the operator Φn,(1/p) (f )(z) = (f (z1 ), (f ′ (z1 ))1/p z0′ )′ , (1.2) maps the ε starlike functions on D to the ε starlike mappings on the Reinhardt domain Ωn,p = {z = (z1 , z0′ )′ ∈ C × Cn−1 : |z1 |2 + kz0 kpp < 1}, (1.3) P n 1/p , p ≥ 1. When ε = 0 and ε = 1, where z0 = (z2 , · · · , zn )′ ∈ Cn−1 , kz0 kp = |zj |p j=2
Φn,(1/p) (f ) maps the starlike function and convex function on D to the starlike mapping and the convex mapping on Ωn,p , respectively. Furthermore, Gong and Liu [6] proved that the operator Φn,(1/p2 ),··· ,(1/pn ) (f )(z) = (f (z1 ), (f ′ (z1 ))1/p2 z2 , · · · , (f ′ (z1 ))1/pn zn )′ ,
(1.4)
maps the ε starlike functions on D to the ε starlike mappings on the Reinhardt domain n X n 2 pj Ωn,p2 ,··· ,pn = z ∈ C : |z1 | + |zj | < 1 , (1.5) j=2
where pj ≥ 1, j = 2, · · · , n. The extension operator (1.4) on the domain Ωn,p2 ,··· ,pn is also studied by Liu and Liu [7], Feng and Liu [8]. In 2005, Muir [9] modified the Roper-Suffridge extension operator as p F (z) = (f (z1 ) + f ′ (z1 )P (z0 ), f ′ (z1 )z0′ )′ , (1.6)
where P (z0 ) is a homogeneous polynomial of degree 2 with respect to z0 , and f , z1 and z0 are defined as above. He proved that this operator preserves starlikeness and convexity if and only if ||P || ≤ 1/4 and ||P || ≤ 1/2, respectively. The modified operator plays an important role in studying the extreme points of convex mappings on B n (see [10, 11]). Recently, the modified Roper-Suffridge extension operator (1.6) on the unit ball is also studied by Wang and Liu [12], Feng and Yu [13]. But the modified Roper-Suffridge extension operator in [12, 13] has a little difference from [9]. In [12, 13], P (z0 ) is a homogeneous polynomial of degree k with respect to z0 , where k ≥ 2. In 2011, Wang and Gao [14] introduced a new Roper-Suffridge extension operator on the following Reinhardt domain Ωn,p2 ,··· ,pn given by F (z) = (f (z1 ) + f ′ (z1 )
k X
p
1
1
aj zj j , (f ′ (z1 )) p2 z2 , · · · , (f ′ (z1 )) pn zn ),
(1.7)
j=2
where f is a normalized locally biholomorphic function on the unit disc D, pj are positive integer, aj are complex constants, j = 2, · · · , n. Some conditions for aj are found under which the operator preserves the properties of almost starlikeness of order α and starlikeness of order α, respectively. In this paper, we introduce a Roper-Suffridge extension operator F (z) = (f (z1 ) + f ′ (z1 )
k X j=2
1
1
Pj (zj ), (f ′ (z1 )) p2 z2′ , · · · , (f ′ (z1 )) pk zk′ )′ ,
(1.8)
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which can preserve the properties of almost starlikeness of order α, starlikeness of order α and strongly starlikeness of order α on the domain ΩN given by different conditions for Pj , j = 2, · · · , k, respectively, where ΩN is definded as n o ΩN = z = (z1 , z2′ , · · · , zk′ )′ ∈ C × Cn2 × · · · × Cnk : |z1 |2 + kz2 kp22 + · · · + kzk kpkk < 1 , (1.9) and for j = 2, · · · , k, nj ≥ 1, pj ≥ 1, zj = (zj1 , · · · , zjnj )′ ∈ Cnj , kzj kj =
|zjl |pj
l=1
N = 1 + n2 + · · · + nk .
Cn
P nj
1/pj
,
n For notations, let D = {z ∈ C : |z| < 1} be the unit disk in C, B n = z = (z1 , · · · , zn )′ ∈ o 21 P n < 1 be the unit ball in Cn . Let Ω be a domain in Cn . Denote : kzk = |zj |2 j=1
H(Ω) be the set of all holomorphic mappings from Ω into Cn . A mapping f ∈ H(Ω) is called normalized if f (0) = 0, Jf (0) = In , where Jf (0) is the complex Jacob matrix of f at the origin and In is the identity matrix on Cn . A mapping f ∈ H(Ω) is said to be locally biholomorphic if det Jf (z) 6= 0 for ∀z ∈ Ω. A normalized mapping f ∈ H(Ω) is said to be convex if there exists a point z ∈ Ω such that f (z) = tf (z1 ) + (1 − t)f (z2 ) for ∀z1 , z2 ∈ Ω and ∀t ∈ [0, 1]. A normalized mapping f ∈ H(Ω) is said to be starlike if tf (z) ∈ H(Ω), for ∀z ∈ Ω and ∀t ∈ [0, 1]. Assume that P : Cn −→ C is a homogeneous polynomial of degree k. Then P satisfies P (λz) = λk P (z) ∂P ∂P for ∀z ∈ Cn and ∀λ ∈ C. It is easy to see that ∇P (z)z = kP (z), where ∇P (z) = ( ∂z , · · · , ∂z ) 1 n n is the gradient of P (z). The norm of P is given by kP k = sup{|P (z)| : z ∈ ∂B }. A domain Ω is said to be a Reinhardt domain if (eiθ1 z1 , · · · , eiθn zn ) ∈ Ω holds for ∀z ∈ Ω and ∀θ1 , · · · , θn ∈ R. A domain Ω is said to be a circular domain if eiθ z ∈ Ω holds for ∀z ∈ Ω and ∀θ ∈ R. Denote ρ(z) = inf{t > 0 : z/t ∈ Ω} to be the Minkowski functional of the domain Ω. If Ω is a bounded circular convex domain, then Ω is a Banach space in Cn with respect to this norm, and Ω = {z ∈ Cn : ρ(z) < 1}. Definition1.1 [15] Suppose Ω is a bounded starlike circular domain in Cn . Its Minkowski functional ρ(z) is C 1 except for a lower-dimensional manifold. Let 0 ≤ α < 1. A normalized locally biholomorphic mapping f ∈ H(Ω) is said to be an almost starlike mapping of order α if Re
2 ∂ρ (z)Jf−1 (z)f (z) ≥ α, ∀z ∈ Ω \ {0}. ρ(z) ∂z
(1.10)
When Ω = B n , the Minkowski functional ρ(z) = kzk, and the above inequality becomes Rez ′ Jf−1 (z)f (z) ≥ α||z||2 , ∀z ∈ B n \ {0}.
(1.11)
In particular, when α = 0, f is a starlike mapping on Ω. Definition 1.2 [16] Suppose Ω is a bounded starlike circular domain in Cn , and its Minkowski functional ρ(z) is C 1 except for a lower-dimensional manifold. Let 0 < α < 1. A normalized locally biholomorphic mapping f ∈ H(Ω) is said to be a starlike mapping of order α if 2 ∂ρ 1 1 −1 (1.12) ρ(z) ∂z (z)Jf (z)f (z) − 2α < 2α , ∀z ∈ Ω \ {0}. When Ω = B n , the above inequality reduces to 1 ′ −1 1 1 z J (z)f (z) − < , ∀z ∈ B n \ {0}. ||z|| f 2α 2α
(1.13)
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Definition 1.3 [17] Let 0 < α ≤ 1. Suppose Ω is a bounded starlike circular domain in Cn , and its Minkowski functional ρ(z) is C 1 except for a lower-dimensional manifold. A normalized locally biholomorphic mapping f (z) ∈ H(Ω) is said to be a strongly starlike mapping of order α on Ω if π 2 ∂ρ −1 arg (z)Jf (z)f (z) < α, ∀z ∈ Ω \ {0}. (1.14) ρ(z) ∂z 2 When Ω = B n , the inequality (1.14) becomes π arghJf−1 (z)f (z), zi < α, ∀z ∈ B n \ {0}. 2 When n = 1, the inequality (1.15) becomes ′ arg zf (z) < π α, f (z) 2
(1.15)
(1.16)
it is the definition of strongly starlike function of order α on D. From Definition 1.3, we know that if f is a strongly starlike mapping of order α on Ω, then f is a starlike mapping on Ω, furthermore f is a strongly starlike mapping of order β on Ω, where 0 < β < α. ′ If f (z) ∈ H(Ω) is a normalized locally biholomorphic mapping, A is a normal matrix(AA = ′ A A), then f is said to be spirallike mapping relative to A, if e−At Ω and n f (z) ∈ f (Ω), for ∀z ∈ o −1 2 ∂ρ ∀t ≥ 0. f is a spirallike mapping relative to A if and only if Re ρ(z) ∂z (z)Jf (z)Af (z) ≥ 0,
∀z ∈ Ω \ {0} (see [16]). When A = e−iβ I, f is said to be spirallike mapping of type β (β ∈ R, |β| < π2 ). When n = 1, f is a spirallike function of type β on D if and only if ([18]) zf ′ (z) Re eiβ > 0, ∀z ∈ D. (1.17) f (z) In one complex variable, Stankiewicz [18] has given an interesting relationship between strongly starlike function of order α and spirallike function of type β. If we denote S ∗ (α) to be the family of strongly starlike functions of order α, where 0 < α ≤ 1, and SP (β) to be the family of spirallike functions of type β, where β ∈ R, |β| < π2 , then \ S ∗ (α) = SP (β). (1.18) |β|≤ π 2 (1−α)
2
Some Lemmas
In order to prove the main results, we need the following lemmas. Lemma 2.1 [19] Suppose Ω ⊆ Cn is a bounded starlike circular domain in Cn , and its Minkowski function ρ(z) is C 1 on Ω except for a lower-dimensional manifold, then ∂ρ (z)z = ρ(z), ∀z ∈ Cn ; ∂z ∂ρ 2Re (z0 )z0 = 1, ∀z0 ∈ ∂Ω; ∂z ∂ρ ∂ρ (λz) = (z), ∀λ ∈ [0, +∞); ∂z ∂z ∂ρ iθ ∂ρ (e z) = e−iθ (z), ∀θ ∈ R. ∂z ∂z 2Re
(2.1)
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Lemma 2.2 [20] Let h(z) be a holomorphic function on D. If Reh(z) > 0 and h(0) > 0, then 2Reh(z) |h′ (z)| ≤ . (2.2) 1 − |z|2 Lemma 2.3 [21] Let f be a normalized biholomorphic function on D, then ′′ (1 − |z|2 ) f (z) − 2z ≤ 4, ∀z ∈ D. ′ f (z)
(2.3)
Lemma 2.4 [22] If ρ(z) is a Minkowski functional of the Reinhardt domain n X Ωp1 ,p2 ,··· ,pn = z ∈ Cn : |zj |pj < 1 , pj ≥ 1, j = 1, · · · , n, j=1
then ∂ρ (zj ) = ∂z
zj pj −2 pj zj ρ(z) n P zj pj , pj ρ(z) 2ρ(z)
j = 1, · · · , n.
(2.4)
j=1
Lemma 2.5 [23] Suppose f ∈ H(D). Then Ref (z) > 0 (∀z ∈ D) if and only if there is an increasing function µ on [0, 2π], which satisfies µ(2π) − µ(0) = Ref (0), such that Z 2π 1 + ze−iθ f (z) = dµ(θ) + iImf (0), z ∈ D. (2.5) 1 − ze−iθ 0 Lemma 2.6 [8] Suppose w ∈ C. Then (i) Re[(1 − w2 )(1 − w)2 ] = (1 − |w|2 )|1 − w|2 ; (ii) Re[(1 + 2w − w2 )(1 − w)2 ] = (1 − |w|2 )2 − 2|w|2 |1 − w|2 . Lemma 2.7 [17] Suppose 0 < α 6 1, f ∈ Ω is a normalized locally biholomorphic mapping, then f (z) is a strongly starlike mapping of order of α if only and if f is a spirallike mapping with respect to eiβ I, where β ∈ R, |β| ≤ π2 (1 − α).
3
Main Results
Theorem 3.1 Let 0 ≤ α < 1 and let f be an almost starlike function of order α on the unit disc D. For j = 2, · · · , k, Pj : Cnj −→ C is a homogeneous polynomial of zj with degree pj ≥ 1. If ||Pj || ≤ 1−α 4 , j = 2, · · · , k, then F (z) = (f (z1 ) + f ′ (z1 )
k X
1
1
Pj (zj ), (f ′ (z1 )) p2 z2′ , · · · , (f ′ (z1 )) pk zk′ )′
j=2
is an almost starlike mapping of order α on the domain ΩN , where the branches are chosen 1 such that (f ′ (0)) pj = 1. Proof By the definition of almost starlike mapping of order α, we only need to prove that the following inequality Re holds for any z ∈ ΩN and z 6= 0.
2 ∂ρ (z)JF−1 (z)F (z) ≥ α ρ(z) ∂z
(3.1)
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If (z2 , · · · , zk ) = 0, then F (z) = (f (z1 ), 0, · · · , 0)′ . Therefore, we can get the result by the definition of function f . Note that the mapping F is holomorphic at every point z = (z1 , z2′ , · · · , zk′ )′ ∈ ΩN . Let z = ζu = eiθ |ζ|u, u ∈ ∂ΩN , and ζ ∈ D \ {0}, then we have 2 ∂ρ (z)JF−1 (z)F (z) ≥ α Re ρ(z) ∂z 2 ∂ρ iθ −1 iθ iθ ⇐⇒ Re (e |ζ|u)J (e |ζ|u)F (e |ζ|u) ≥α F ρ(eiθ |ζ|u) ∂z 2 e−iθ ∂ρ iθ ⇐⇒ Re (e |ζ|u)JF−1 (eiθ |ζ|u)F (eiθ |ζ|u) ≥ α |ζ| ∂z JF−1 (ζu)F (ζu) 2∂ρ ⇐⇒ Re (u) ≥ α. (3.2) ∂z ζ The expression
J −1 (ζu)F (ζu) 2∂ρ Re (u) F ∂z ζ
is the real part of a holomorphic function with respect to ζ, and thus, it is a harmonic function. By the minimum principle for harmonic functions, we know that it attains its minimum on |ζ| = 1. Therefore, we only need to prove that inequality (3.1) holds for (z2′ , · · · , zk′ )′ 6= 0 and z ∈ ∂ΩN , i.e., ρ(z) = 1. It means that we need to prove 2∂ρ −1 Re (z)JF (z)F (z) ≥ α, ∀z ∈ ∂ΩN , (z2 , · · · , zk ) 6= 0. (3.3) ∂z Since
hence
where
k X
f (z1 ) + f ′ (z1 ) Pj (zj ) j=2 1 (f ′ (z1 )) p2 z2 F (z) = , .. . 1 (f ′ (z1 )) pk zk
λ1
α2
···
αk
β2 λ2 In2 · · · 0 JF (z) = .. .. , .. . . . βk 0 · · · λk Ink λ1 = f ′ (z1 ) + f ′′ (z1 )
k X
(3.4)
1
Pj (zj ), λj = (f ′ (z1 )) pj , j = 2, · · · , k,
j=2
αj = f ′ (z1 )∇Pj (zj ), βj =
1 1 ′ −1 (f (z1 )) pj f ′′ (z1 )zj , j = 2, · · · , k. pj
We denote X = JF−1 (z)F (z), where X = (x1 , x2 , · · · , xk )′ , xj ∈ Cnj for j = 2, · · · , k, then
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F (z) = JF (z)X. So
′
k X
f (z1 ) + f (z1 ) Pj (zj ) λ α2 · · · αk x 1 1 j=2 β2 λ2 In2 · · · 0 x2 1 ′ p2 = z (f (z )) . 2 1 .. .. .. .. . . . . . .. βk 0 · · · λk Ink xk 1 (f ′ (z1 )) pk zk
(3.5)
Consequently k k k X X X (f ′ (z1 ) + f ′′ (z1 ) Pj (zj ))x1 + f ′ (z1 ) ∇Pj (zj )xj = f (z1 ) + f ′ (z1 ) Pj (zj ), j=2 j=2 j=2 1 ′ 1 1 1 −1 (f (z1 )) p2 f ′′ (z1 )z2 x1 + (f ′ (z1 )) p2 x2 = (f ′ (z1 )) p2 z2 , p2 · ·· 1 1 1 1 (f ′ (z1 )) pk −1 f ′′ (z1 )zk x1 + (f ′ (z1 )) pk xk = (f ′ (z1 )) pk zk . pk Note that ∇Pj (zj ) = pj Pj (zj ), j = 2, · · · , k, by easy computation we can get k f (z1 ) X − (pj − 1)Pj (zj ), x = 1 f ′ (z1 ) j=2 k ′′ ′′ X x2 = 1 − f (z1 )f (z1 ) + f (z1 ) (p − 1)P (z ) j j j z2 , p2 (f ′ (z1 ))2 p2 f ′ (z1 ) j=2 ··· k f (z1 )f ′′ (z1 ) f ′′ (z1 ) X (pj − 1)Pj (zj ) zk . xk = 1 − pk (f ′ (z1 ))2 + pk f ′ (z1 ) j=2
(3.6)
Hence, by (2.4) we have
2
∂ρ (z)JF−1 (z)F (z) = ∂z
G(z) , k P pj 2 2|z1 | + pj ||zj ||j
(3.7)
j=2
where
k f (z1 ) X G(z) = 2z1 ′ − (pj − 1)Pj (zj ) f (z1 ) j=2 k k X f (z1 )f ′′ (z1 ) f ′′ (z1 ) X pj + pj ||zj ||j 1 − + (pl − 1)Pl (zl ) . pj (f ′ (z1 ))2 pj f ′ (z1 ) j=2
(3.8)
l=2
f (z1 ) z1 f ′ (z1 ) −α, then 1) ′ other hand, ff′(z (z1 )
Let h(z1 ) =
h(0) = 1−α > 0, and by lemma 2.2 we know that |h′ (z)| <
On the
= [z1 (h(z1 ) + α)]′ , i.e.,
f (z1 )f ′′ (z1 ) = 1 − α − h(z1 ) − z1 h′ (z1 ). (f ′ (z1 ))2
Substituting (3.9) into (3.8), we obtain that k X f (z1 )f ′′ (z1 ) p G(z) = 2|z1 |2 (h(z1 ) + α) + 1− pj ||zj ||j j ′ (z ))2 p (f j 1 j=2
2Reh(z) 1−|z|2 .
(3.9)
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X k k f ′′ (z1 ) X pj ||z || − 2z (pl − 1)Pl (zl ) j j 1 f ′ (z1 ) j=2 l=2
= 2|z1 |2 h(z1 ) + 2α|z1 |2 +
k X j=2
k X
f ′′ (z1 ) + ′ f (z1 ) = (2|z1 |2 +
p ||zj ||j j
− 2z1
j=2
k X
p pj − (1 − α − h(z1 ) − z1 h′ (z1 )) ||zj ||j j
X k
(pl − 1)Pl (zl )
l=2
p
||zj ||j j )h(z1 ) + 2α|z1 |2 +
j=2
+z1 h′ (z1 )
p
||zj ||j j +
j=2
Since z ∈ ∂ΩN , i.e., |z1 |2 +
j=2
p
(pj − 1 + α)||zj ||j j
j=2
k X k P
k X
X k k f (z1 ) X pj ||z || (pl − 1)Pl (zl ). − 2z j 1 j f ′ (z1 ) j=2 ′′
l=2
p
||zj ||j j = 1, so
G(z) = (1 + |z1 |2 )h(z1 ) + 2α|z1 |2 +
k X
p
(pj − 1 + α)||zj ||j j + z1 h′ (z1 )(1 − |z1 |2 )
j=2
+
f ′′ (z1 ) (1 − |z1 |2 ) − 2z1 f ′ (z1 )
By (2.2) and (2.3), we get
X k
(pl − 1)Pl (zl ).
l=2
ReG(z) ≥ (1 + |z1 |2 )Reh(z1 ) + 2α|z1 |2 +
k X
p
(pj − 1 + α)||zj ||j j
j=2
−(1 − |z1 |2 )|z1 |
2Reh(z1 ) − 1 − |z1 |2
k X l=2
≥ (1 + |z1 |2 )Reh(z1 ) + 2α|z1 |2 +
′′ f (z1 ) (pl − 1)|Pl (zl )| ′ (1 − |z1 |2 ) − 2z1 f (z1 )
k X
p
(pj − 1 + α)||zj ||j j − 2|z1 |Reh(z1 )
j=2
−4
k X
(pl − 1)|Pl (zl )|.
l=2
p
Since ||Pj (zj )|| ≤ ||Pj || · ||zj ||j j ≤
pj 1−α 4 ||zj ||j ,
j = 2, · · · , k, therefor
ReG(z) ≥ (1 − |z1 |)2 Reh(z1 ) + 2α|z1 |2 +
k X j=2
≥ (1 − |z1 |)2 Reh(z1 ) + 2α|z1 |2 +
k X
p (pj − 1 + α) − 4(pj − 1)||Pj || ||zj ||j j (pj − 1 + α) − 4(pj − 1)
j=2
≥ 2α|z1 |2 + α
k X
p
pj ||zj ||j j .
j=2
So Re
2∂ρ (z)JF−1 (z)F (z) = ∂z
ReG(z) ≥α k P pj 2 2|z1 | + pj ||zj ||j j=2
1 − α p ||zj ||j j 4
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holds for ∀z ∈ ∂ΩN , (z2 , · · · , zk ) 6= 0. Consequently, inequality (3.1) holds for z ∈ ΩN and z 6= 0.
1769
Theorem 3.2 Let 0 < α < 1 and let f be a starlike function of order α on the unit disc D. For j = 2, · · · , k, Pj : Cnj −→ C is a homogeneous polynomial of zj with degree pj ≥ 1. If kPj k ≤ 1−|2α−1| , j = 2, · · · , k, then 8α F (z) = (f (z1 ) + f ′ (z1 )
k X
1
1
Pj (zj ), (f ′ (z)) p2 z2′ , · · · , (f ′ (z)) pk zk′ )′ ,
j=2
is a starlike mapping of order α on the domain ΩN , where the branches are chosen such that 1 (f ′ (0)) pj = 1. Proof By the definition of starlike mapping of order α, we only need to prove that the following inequality 2 ∂ρ 1 1 −1 (3.10) ρ(z) ∂z (z)JF (z)F (z) − 2α < 2α holds for z ∈ ΩN \ {0}. If (z2′ , · · · , zk′ )′ = 0, then F (z) = (f (z1 ), 0, · · · , 0)′ . Therefore, we can get the result by the definition of function f . Note that the mapping F is holomorphic at every point z = (z1 , z2′ , · · · , zk′ )′ ∈ ΩN . Let z = ζu = eiθ |ζ|u, u ∈ ∂ΩN , and ζ ∈ D \ {0}, then we have 2 ∂ρ 1 1 −1 < (z)J (z)F (z) − F ρ(z) ∂z 2α 2α 1 ∂ρ iθ 1 2 −1 iθ iθ < ⇐⇒ iθ (e |ζ|u)JF (e |ζ|u)F (e |ζ|u) − ρ(e |ζ|u) ∂z 2α 2α −iθ 2 e ∂ρ iθ 1 1 < ⇐⇒ (e |ζ|u)JF−1 (eiθ |ζ|u)F (eiθ |ζ|u) − |ζ| ∂z 2α 2α −1 2∂ρ 1 J (ζu)F (ζu) 1 ⇐⇒ < (u) F − . (3.11) ∂z ζ 2α 2α The expression J −1 (ζu)F (ζu) 2∂ρ 1 (u) F − ∂z ζ 2α is holomorphic with respect to ζ. Thus, the maximum modules principle for holomorphic functions yields that it attains its maximum on |ζ| = 1. Therefore, we only need to prove that inequality (3.10) holds for z ∈ ∂ΩN and (z2′ , · · · , zk′ )′ 6= 0. From the proof of Theorem 3.1, we can get ∂ρ G(z) 2 (z)JF−1 (z)F (z) = , k P ∂z p 2|z1 |2 + pj ||zj ||j j j=2
where G(z) is defined by equality (3.8). Hence 2
∂ρ 1 (z)JF−1 (z)F (z) − = ∂z 2α
H(z) , k P p 2α 2|z1 |2 + pj ||zj ||j j j=2
where H(z) = 4α|z1 |2
k X f (z1 ) f (z1 )f ′′ (z1 ) pj + 2α p ||z || 1 − j j j z1 f ′ (z1 ) pj (f ′ (z1 ))2 j=2
(3.12)
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′′ k k X f (z1 ) X p ||zl ||pl l − 2z1 − 2|z1 |2 + (pj − 1)Pj (zj ) ′ pj ||zj ||j j f (z1 ) j=2 j=2
k X
l=2
k X f (z1 )f ′′ (z1 ) f (z1 ) 1 pj 2 − 1 + 2α pj ||zj ||j 1 − − = 2|z1 | 2α z1 f ′ (z1 ) 2α pj (f ′ (z1 ))2 j=2 ′′ f (z1 ) 2 (1 − |z1 | ) − 2z1 . +2α (pj − 1)Pj (zj ) ′ f (z1 ) j=2 k X
1) − 1. Then |h(z1 )| < 1 by the definition of f (z1 ). According to Let h(z1 ) = 2α z1ff(z′ (z 1) Schwarz-Pick lemma, we obtain that
1 − |h(z1 )|2 . 1 − |z1 |2
|h′ (z1 )| ≤
(3.13)
On the other hand, we can get 1 + h(z1 ) + z1 h′ (z1 ) f (z1 )f ′′ (z1 ) = 1 − , (f ′ (z1 ))2 2α
(3.14)
hence 2
H(z) = 2|z1 | h(z1 ) + 2α
k X
p pj ||zj ||j j
j=2
1 1 1 + h(z1 ) + z1 h′ (z1 ) 1− − 1− 2α pj 2α
′′ f (z1 ) 2 +2α (pj − 1)Pj (zj ) ′ (1 − |z1 | ) − 2z1 f (z1 ) j=2 k X
k k k X X X pj p p 2 = 2|z1 | + ||zj ||j h(z1 ) + ||zj ||j j z1 h′ (z1 ) + (2α − 1) pj ||zj ||j j j=2
j=2
j=2
′′ f (z1 ) 2 +2α (pj − 1)Pj (zj ) ′ (1 − |z1 | ) − 2z1 , f (z1 ) j=2 k X
and
|H(z)| ≤ (2|z1 |2 +
k X
p
||zj ||j j )|h(z1 )| +
j=2
+|2α − 1| ·
k X j=2
k X
(pj −
p 1)||zj ||j j
j=2
′′ f (z1 ) 2 + 2α (pj − 1)|Pj (zj )| ′ (1 − |z1 | ) − 2z1 f (z1 ) j=2
≤ (1 + |z1 |2 )|h(z1 )| + (1 − |z1 |2 )|z1 |
+2α
k X
p
||zj ||j j |z1 | · |h′ (z1 )| k X
k X 1 − |h(z1 )|2 p + |2α − 1| (pj − 1)||zj ||j j 1 − |z1 |2 j=2
(pj − 1)|Pj (zj )| · 4
j=2
≤ (1 + |z1 |2 )(|h(z1 )| − 1) + (1 + |z1 |2 ) + 2|z1 |(1 − |h(z1 )|) +|2α − 1|
k X j=2
p
(pj − 1)||zj ||j j + 8α
k X
(pj − 1)|Pj (zj )|
j=2
≤ (1 + |z1 |2 ) + (|h(z1 )| − 1)(1 − |z1 |)2 + |2α − 1|
k X j=2
p
(pj − 1)||zj ||j j
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+8α
k X
1771
p
(pj − 1)||Pj || · ||zj ||j j
j=2
2
≤ (1 + |z1 | ) + |2α − 1|
k X
(pj −
p 1)||zj ||j j
+ 8α
j=2
= (1 + |z1 |2 ) +
k X
k X
(pj − 1)
j=2
1 − |2α − 1| p ||zj ||j j 8α
p
(pj − 1)||zj ||j j
j=2
k k X X pj pj 2 = 1− ||zj ||j + |z1 | + pj ||zj ||j j=2
= 2|z1 |2 +
j=2
k X
p
pj ||zj ||j j .
j=2
Hence
2∂ρ 1 1 −1 ∂z (z)JF (z)F (z) − 2α < 2α
holds for z ∈ ∂ΩN \ {0}, and (z2′ , · · · , zk′ )′ 6= 0.
Notation When n2 = · · · = nk = 1, (1.8) reduces to (1.7). Theorems 3.1 and 3.2 in this paper are Theorems 3.1 and 3.2 in [14]. Theorem 3.3 Let 0 < α ≤ 1 and let f be a strongly starlike function of order α on the unit disc D. For j = 2, · · · , k, Pj : Cnj −→ C is a homogeneous polynomial of zj with degree pj ≥ 1. If ||Pj || ≤ 14 cos π2 (1 − α) , j = 2, · · · , k, then F (z) = (f (z1 ) + f ′ (z1 )
k X
1
1
Pj (zj ), (f ′ (z1 )) p2 z2′ , · · · , (f ′ (z1 )) pk zk′ )′
j=2
is a strongly starlike mapping of order α on the domain ΩN , where the branches are chosen 1 such that (f ′ (0)) pj = 1. Proof By Lemma 2.7, we only need to prove F (z) is spirallike mapping of type β, for all |β| ≤ π2 (1 − α). From the definition of spirallike mapping ([16]), we need to prove −1 −iβ 2 ∂ρ Re e (z)Jf (z)f (z) ≥ 0, ∀z ∈ ΩN \ {0}. (3.15) ρ(z) ∂z Similarly to the inequality (3.1), we only need to prove inequality (3.13) holds for (z2′ , · · · , zk′ )′ 6= 0 and z ∈ ∂ΩN , i.e. ρ(z) = 1. It means that we need to prove ∂ρ Re 2e−iβ (z)JF−1 (z)F (z) ≥ 0, ∀z ∈ ∂ΩN , (z2′ , · · · , zk′ )′ 6= 0. (3.16) ∂z From equality (3.7), we have 2e−iβ
∂ρ (z)JF−1 (z)F (z) = ∂z
e−iβ G(z) , k P p 2|z1 |2 + pj ||zj ||j j
(3.17)
j=2
where G(z) is defined by (3.8). From (1.18), we know f is spirallike function of type β for |β| ≤ π2 (1 − α). Let h(z1 ) = −iβ f (z1 ) e z1 f ′ (z1 ) , z1 ∈ D. By (1.17), we know that Reh(z1 ) > 0; and by Lemma 2.5, there is an
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increasing function µ on [0, 2π], which satisfies µ(2π) − µ(0) = Reh(0) = cos β, and Z 2π 1 + z1 e−iθ h(z1 ) = dµ(θ) − i sin β, z1 ∈ D. 1 − z1 e−iθ 0 By simple calculation we can get Z 2π 2z1 e−iθ z1 h′ (z1 ) = dµ(θ), (1 − z1 e−iθ )2 0 f (z1 )f ′′ (z1 ) = 1 − eiβ (z1 h′ (z1 ) + h(z1 )), (f ′ (z1 ))2
then Re{e−iβ G(z)} k X f (z1 ) −iβ − e 2z = Re 2|z1 |2 e−iβ (pj − 1)Pj (zj ) 1 z1 f ′ (z1 ) j=2 +e−iβ
k f (z1 )f ′′ (z1 ) f ′′ (z1 ) X p + (p − 1)P (z ) ||zj ||j j pj − l l l (f ′ (z1 ))2 f ′ (z1 ) j=2
k X
l=2
k X p = Re 2|z1 |2 h(z1 ) + e−iβ ||zj ||j j pj − 1 + eiβ (z1 h′ (z1 ) + h(z1 )) j=2
−e
−iβ
2z1
k X
(pj − 1)Pj (zj ) + e
−iβ
j=2
k X
p ||zj ||j j
j=2
k f ′′ (z1 ) X (pl − 1)Pl (zl ) f ′ (z1 ) l=2
k k X X p = Re 2|z1 |2 h(z1 ) − e−iβ 2z1 (pj − 1)Pj (zj ) + e−iβ (pj − 1)||zj ||j j j=2
+
k X
p ||zj ||j j (z1 h′ (z1 )
+ h(z1 )) + e
j=2
−iβ
j=2
k X
p ||zj ||j j
j=2
k f (z1 ) X (pl − 1)Pl (zl ) f ′ (z1 ) ′′
l=2
k n ′′ X 2 −iβ 2 f (z1 ) = Re 2|z1 | h(z1 ) + e (pj − 1)Pj (zj ) (1 − |z1 | ) ′ − 2z1 f (z1 ) j=2 +e
−iβ
k X
(pj −
j=2
p 1)||zj ||j j
+ (1 − |z1 | )(z1 h (z1 ) + h(z1 )) 2
′
k X p = Re (1 + |z1 |2 )h(z1 ) + (1 − |z1 |2 )z1 h′ (z1 ) + cos β (pj − 1)||zj ||j j j=2
k X f ′′ (z1 ) +Re e−iβ (pj − 1)Pj (zj ) (1 − |z1 |2 ) ′ − 2z1 f (z1 ) j=2
k n o X p > Re (1 + |z1 |2 )h(z1 ) + (1 − |z1 |2 )z1 h′ (z1 ) + cos β (pj − 1)||zj ||j j j=2
−
k X j=2
f ′′ (z1 ) − 2z1 . (pj − 1)|Pj (zj )| (1 − |z1 |2 ) ′ f (z1 )
By Lemma 2.3, we have
′′ (1 − |z|2 ) f (z) − 2z ≤ 4. ′ f (z)
(3.18)
(3.19) (3.20)
No.6
H.J. Li & S.X. Feng: ROPER-SUFFRIDGE EXTENSION OPERATOR
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p − α) , and |Pj (zj )| ≤ ||Pj || · ||zj ||j j , we get k k iX hπ ′′ X p 2 f (z1 ) (pj − 1)|Pj (zj )| (1 − |z1 | ) ′ − 2z1 ≤ cos (1 − α) (pj − 1)||zj ||j j . f (z ) 2 1 j=2 j=2 π π Note |β| ≤ 2 (1 − α), and 0 < α ≤ 1, so cos β ≥ cos 2 (1 − α) . Hence k k X X f ′′ (z1 ) p cos β (pj − 1)||zj ||j j − (pj − 1)|Pj (zj )| (1 − |z1 |2 ) ′ − 2z1 ≥ 0. (3.21) f (z1 ) j=2 j=2 Since ||Pj || ≤
1 4
cos
π
2 (1
Next, we just prove
Re{(1 + |z1 |2 )h(z1 ) + (1 − |z1 |2 )z1 h′ (z1 )} ≥ 0.
(3.22)
Substituting (3.18) and (3.19) into the left of (3.22), we have n o Re (1 + |z1 |2 )h(z1 ) + (1 − |z1 |2 )z1 h′ (z1 ) Z 2π n o 1 (1 + |z1 |2 )[1 − (z1 e−iθ )2 ] + 2z1 e−iθ (1 − |z1 |2 ) dµ(θ). = Re −iθ 2 (1 − z1 e ) 0
Let w = z1 e−iθ , then w ∈ D. Since µ is increasing, we just prove 1 2 2 2 Re [(1 + |w| )(1 − w ) + 2w(1 − |w| )] ≥ 0. (1 − w)2
By lemma 2.6, we obtain 1 2 2 2 Re [(1 + |w| )(1 − w ) + 2w(1 − |w| )] (1 − w)2 1 = Re[(1 + |w|2 )(1 − w2 )(1 − w)2 + 2w(1 − |w|2 )(1 − w)2 ] |1 − w|4 1 = Re[(1 + |w|2 )(1 − |w|2 )|1 − w|2 + 2w(1 − |w|2 )(1 − w)2 ] |1 − w|4 1 − |w|2 = Re[(1 + |w|2 )|1 − w|2 − (1 − w2 )(1 − w)2 + (1 − w2 )(1 − w)2 + 2w(1 − w)2 ] |1 − w|4 1 − |w|2 = Re[(1 + |w|2 )|1 − w|2 − |1 − w|2 (1 − |w|2 ) + (1 + 2w − w2 )(1 − w)2 ] |1 − w|4 1 − |w|2 = [2|w|2 |1 − w|2 + (1 − |w|2 )2 − 2|w|2 |1 − w|2 ] |1 − w|4 (1 − |w|2 )3 = |1 − w|4 ≥ 0. Theorem 3.3 is completely proved.
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