http://scma.maragheh.ac.ir
PARABOLIC STARLIKE MAPPINGS OF THE UNIT BALL Bn
SAMIRA RAHROVI1
Abstract. Letf be a locally univalent function on the unit disk U. We consider the normalized extensions off to the Euclidean unit ballBn⊆Cngiven by
Φn,γ(f)(z) =(
f(z1),
(
f′(z
1)
)γ
ˆ z)
, whereγ∈[0,1/2],z= (z1,z)ˆ ∈Bnand
Ψn,β(f)(z) =
(
f(z1),
(
f(z1) z1
)β
ˆ z
)
,
in whichβ∈[0,1],f(z1)̸= 0 andz= (z1,z)ˆ ∈Bn. In the caseγ= 1/2, the function Φn,γ(f) reduces to the well known Roper-Suffridge extension operator. By using different methods, we prove that iff is parabolic starlike mapping onU then Φn,γ(f) and Ψn,β(f) are parabolic starlike mappings onBn.
1. Introduction
Let Cn be the vector space of n-complex variables z = (z1, . . . , zn) with the Euclidean inner product ⟨z, w⟩ = ∑n
k=1zkw¯k and Euclidean
norm ∥z∥ =⟨z, z⟩1/2. The open ball {z ∈Cn :∥z∥< r} is denoted by
Bnr and the unit ballB1nbyBn. In the case of one complex variable,B1
is denoted by U. It is convenient, if n≥2 to write a vector z∈ Cn as
z= (z1,zˆ), where z1 ∈C and ˆz= (z2, . . . , zn)∈Cn−1.
LetH(Bn,Cn) denotes the topological vector space of all holomorphic mappingsF :Bn→Cn. LetF ∈H(Bn), we say thatF is normalized if
F(0) = 0 andDF(0) =I, whereDF is the Fr´echet differential ofF and
I is the identity operator on Cn. Let S(Bn) be the set of normalized
2010Mathematics Subject Classification. 30C45.
Key words and phrases. Roper-Suffridge extention operator, Biholomorphic map-ping, Parabolic starlike function.
Received: 13 May 2015, Accepted: 4 August 2015.
biholomorphic mappings on Bn, and S1 = S is the classical family of univalent mappings of U.
A mapf ∈S(Bn) is said to be a convex if its image is convex domain in Cn, and starlike if its image is a starlike domain with respect to 0. We denote the classes of normalized convex and starlike mappings on
Bn respectively by K(Bn) and S∗(Bn).
In 1995, Roper and Suffridge [8] introduced an extension operator which gives a way of extending a (locally) univalent function on the unit disk U to a (locally) univalent mapping of Bn into Cn. This operator
is defined for a normalized locally biholomorphic function f in the unit diskU inCby (see [3] and [8])
[Φn(f)] (z) =
(
f(z1),
√
f′(z 1)ˆz
)
,
where z = (z1,zˆ) ∈ Bn and we choose the branch of the square root
such that √
f′(z 1)
z1=0
= 1.
The following results illustrate the importance and usefulness of the Roper-Suffridge extension operator
Φn(K)⊆K(Bn), Φn(S∗)⊆S∗(Bn).
The first was proved by Roper and Suffridge when they introduced their operator [8], while the second result was given by Graham and Kohr [1]. Till now, it has been difficult to make constant the concrete convex mappings and starlike mappings on Bn. By making use of the
Roper-Suffridge extension operator, we may easily give many concrete exam-ples about these mappings. This is one important reason why people are interested in this extension operator. A good treatment of further applications of the Roper-Suffridge extension operator can be found in the recent book by Graham and Kohr [3].
The authors [4] considered the following operator
Φn,γ(f)(z) =(f(z1),(f′(z1))γzˆ), z= (z1,zˆ)∈Bn,
whereγ ∈[0,1/2] andf is a locally univalent function inU, normalized byf(0) =f′(0)−1 = 0. We choose the branch of the power function such that (f′(z
1))γ|z1=0 = 1. Of course whenγ = 1/2, we obtain the
Roper-Suffridge extension operator. In [4], a number of extension results were obtained related to the operator Φn,γ,γ ∈[0,1/2]: iff ∈S, then Φn,γ(f)
can be embedded in a Loewner chain and moreover Φn,γ(f)∈S0(Bn).
In particular, if f ∈S∗, then Φn,γ(f)∈S∗(Bn). It was also proved that
In [2], Graham and Kohr introduced another extension operator for the locally biholomorphic function f on U by
Ψn,β(f)(z) =
(
f(z1),
(
f(z1)
z1
)β
ˆ
z
)
, z= (z1,zˆ)∈Bn,
where β ∈ [0,1] and f(z1) ̸= 0, when z1 ∈ U\{0}, and we choose the
branch of the power function such that (f(z1)
z1 )β
z1=0
= 1. They proved
that the operator Ψn,β(f) maps the normalized starlike function on U
to a normalized starlike mapping on Bn. When β = 1, it was proved
and discussed by Pfaltzgraff and Suffridge [7].
Remark 1.1. Let g : U → C be a holomorphic univalent function such that g(0) = 1, g(¯η) = g(η) for η ∈ U (so, g has real coefficients in its power series expansion), Reg(η) > 0 on U and assume g satisfies the following
min
|η|=rReg(η) = min{g(r), g(−r)},
max
|η|=rReg(η) = max{g(r), g(−r)},
(1.1)
forr ∈(0,1).
For example, the condition (1.1) is satisfied by all functions which are convex in the direction of the imaginary axis and symmetric about the real axis (see [6]).
Let
Mg =
{
h∈H(Bn) :h(0) = 0, Dh(0) =In,
⟨
h(z),∥zz∥2 ⟩
∈g(U), z∈Bn\{0}
}
.
For g(ξ) = 1+1−ξξ, ξ ∈ U, we obtain the well known set Mg = M of
mapping with ”positive real part on Bn”, i.e.
Mg =
{
h∈H(Bn) :h(0) = 0, Dh(0) =In, Re
⟨
h(z),∥zz∥2 ⟩
>0, z∈Bn\{0}
}
.
Now, we give the definition of parabolic starlike mappings on Bn (see [5]). Let
q(η) = 1 + 4
π2
(
log1 +
√η
1−√η
)2
Then q is a biholomorphic mapping from U onto domain Ω, where
Ω ={
w=u+iv:v2 <4u}
={w:|w−1|<1 + Rew}.
We note that Ω is a parabolic region in the right half- plane.
Definition 1.2. Letf be a normalized locally biholomorphic mapping on Bn, we say thatf is a parabolic starlike mapping if
⟨
[Df(z)]−1f(z), z ∥z∥2
⟩
∈g(U), z∈Bn\{0},
where g= 1q.
Letf be a parabolic starlike mapping onBn. Since
Re⟨
[Df(z)]−1f(z), z⟩
>0,
parabolic starlike mappings are starlike mappings by Suffridge [9]. In order to prove the main results, we need the following lemma:
Lemma 1.3. [5]. Let g= 1q. Then, g(U) is starlike with respect to 1.
2. Main Results
Theorem 2.1. Let f : U → C be a normalized locally univalent func-tion, which satisfies the condition
z1f′(z1)
f(z1) − 1
<1, z1∈U. (2.1)
Also, let F = Φn,γ(f), then
∥z∥2
⟨DF−1(z)F(z), z⟩ −1
<1, z∈Bn\{0},
and hence F is a parabolic starlike mapping on Bn.
Proof. Without loss of generality, we may assume thatf is holomorphic on the closed unit disk U, since otherwise we use the function fr(z1) =
f(rz1)/rforr ∈(0,1), which is holomorphic onU. Taking into account the minimum principle for harmonic functions, we have to prove that
1
⟨DF−1(z)F(z), z⟩ −1
<1, z= (z1,zˆ)∈Bn, ∥z∥= 1,
i.e.
Re⟨
DF−1(z)F(z), z⟩
> 1
2, z∈∂B
n.
We know that the inequality (2.1) is equivalent to
Re
{
f(z1)
z1f′(z1)
}
> 1
Sincef is parabolic starlike,F = Φn,γ(f) is starlike onBn, and hence
biholomorphic. A short computation yields that
DF(z) =
f′(z1) 0
γ(f′(z1))γ−1f′′(z1)ˆz (f′(z1))γ
,
therefore,
DF−1(z)F(z) =
f(z1)
f′(z1)
−γf′′(z1)f(z1)
(f′(z1))2 zˆ+ ˆz
.
Then we have
⟨
DF(z)−1F(z), z⟩
= f(z1)
f′(z 1)
¯
z1+∥zˆ∥2
(
1−γf(z1)f
′′(z 1) (f′(z
1))2
)
=|z1|2
f(z1)
z1f′(z1)
+∥zˆ∥2− ∥zˆ∥2γf(z1)f
′′(z 1) (f′(z
1))2
,
forz= (z1,zˆ)∈Bn. We must show that
Re⟨DF(z)−1F(z), z⟩> 1
2, z∈∂B
n.
(2.2)
Therefore, by making use of the equality |z1|2+
n
∑
j=2|
zj|2 = 1, we must
show that
|z1|2Re
{
f(z1)
z1f′(z1)
}
+(
1− |z1|2)−γ(1− |z1|2)Re
{
f(z1)f′′(z1) (f′(z1))2
}
> 1
2. If z= (z1,0)∈Bn, then
Re⟨DF(z)−1F(z), z⟩>|z1|2Re
{
f(z1)
z1f′(z1)
}
> ∥z∥
2
2 ,
and hence we may assume ˆz̸= 0. ThenF is holomorphic in a neighbor-hood of each z∈B¯n for each ˆz̸= 0. In view of the minimum principle for harmonic functions, it suffices to prove that
Re⟨DF(z)−1F(z), z⟩ ≥ 1
2, ∥z∥= 1.
Letp(z1) = z1ff(z′(1z)1) for|z1|<1. Sincef is parabolic starlike onU, we
obtain that Rep(z1)> 12. Also letq(z1) = 2p(z1)−1. Then Req(z1)>0 for|z1|<1, and thus (see e.g. [3], Theorem 2.1.3)
Re{
z1q′(z1)}≥ −2|z1| 1− |z1|2
Therefore, we deduce that
Re{
z1p′(z1)}≥ − 2|z1| 1− |z1|2
Rep(z1) + |
z1| 1− |z1|2
.
(2.3)
On the other hand, since
f′′(z 1)f(z1)
(f′(z1))2 = 1−z1p ′(z
1)−p(z1),
we deduce from (2.3) that
|z1|2Re
{
f(z1)
z1f′(z1)
}
+(
1− |z1|2)−γ(1− |z1|2)Re
{
f(z1)f′′(z1) (f′(z
1))2
}
=|z1|2Rep(z1) +(1− |z1|2)−γ(1− |z1|2)Re{1−z1p′(z1)−p(z1)}
=(
|z1|2+γ
(
1− |z1|2
))
Rep(z1) +γ
(
1− |z1|2
)
Re{
z1p′(z1)
}
+(
1− |z1|2)(1−γ)
≥(
|z1|2+γ
(
1− |z1|2
))
Rep(z1) +
(
1− |z1|2
)
(1−γ)
+γ(
1− |z1|2)
(
−2|z1| 1− |z1|2
Rep(z1) + |
z1| 1− |z1|2
)
=(
|z1|2+γ
(
1− |z1|2
))
Rep(z1)−2γ|z1|Rep(z1) +γ|z1|
+(
1− |z1|2)(1−γ)
≥ 12(
γ+ (1−γ)|z1|2)+(1− |z1|2)(1−γ)
= 1
2γ+ 1−γ
2 |z1| 2+(
1− |z1|2
)
(1−γ)
= 1
2γ+ (1−γ)
(
1−1 2|z1|
2
)
≥ 12γ+1−γ
2 =
1 2.
Hence the relation (2.2) holds, as desired. This completes the proof. □
In view of Theorem 2.1, we obtain the following particular cases. This result, was obtained in [5], in the caseγ = 12.
Corollary 2.2. Let f :U → C be a normalized locally univalent func-tion, which satisfies the condition
z1f′(z1)
f(z1) −1
<1, z1∈U.
Also, let F = Φn(f) where
Φn(f)(z) = (f(z1),
√
f′(z
then
∥z∥2
⟨DF−1(z)F(z), z⟩ −1
<1, z∈Bn\{0},
and hence F is a parabolic starlike mapping on Bn.
In the next Theorem, through a different method, we show that Ψn,β(f) is also a parabolic starlike mapping onBn.
Theorem 2.3. Assume that f be a parabolic starlike function on U. For any β ∈[0,1], letF = Ψn,β(f). Then Ψn,β(f) is a parabolic starlike mapping on Bn.
Proof. Letf ∈S be a parabolic starlike mapping onU, since parabolic starlike mappings are starlike mappings with respect to 1 (Lemma 1.3) then we must show that
F(z) = Φn,β(f)(z) =
[
f(z1)
(f(z 1)
z1 )β
ˆ
z
]
,
is a parabolic starlike map with respect to 1 on Bn. For this purpose,
through simple calculations we have
DF(z) =
[
f′(z1) 0
β(f(z1)
z1
)β−1(
f′(z
1)
z1 −
f(z1)
z2 1
)
ˆ
z (f(z1)
z1 )β
]
,
and
DF−1(z)F(z) =
f(z1)
f′(z1) β( f(z1)
z1f′(z1) −1 )
ˆ
z+ ˆz
,
therefore
⟨DF(z)−1F(z), z⟩= f(z1)
f′(z1)z¯1+β∥zˆ∥ 2
(
f(z1)
z1f′(z1) − 1
)
+∥zˆ∥2.
Now we will show that
1
∥z∥2⟨DF(z)
−1F(z), z⟩ −1
<1.
For this purpose we have
1
∥z∥2
{
|z1|2f(z1)
z1f′(z1)
+β∥zˆ∥2
(
f(z1)
z1f′(z1) − 1
)
+∥zˆ∥2
}
−1
=
1
∥z∥2
{
(|z1|2+β∥zˆ∥2)
f(z1)
z1f′(z1)−
β∥zˆ∥2+∥zˆ∥2
}
−1
=
|z1|2+β∥zˆ∥2
∥z∥2
(
f(z1)
z1f′(z1) − 1
)
+|z1|
2+β∥zˆ∥2
∥z∥2 +
(1−β)∥zˆ∥2
∥z∥2 −1
=
|z1|2+β∥zˆ∥2
∥z∥2
(
f(z1)
z1f′(z1) − 1
)
+(1−β)∥zˆ∥ 2
∥z∥2 −
(1−β)∥zˆ∥2
∥z∥2
= |z1|
2+β∥zˆ∥2
∥z∥2
f(z1)
z1f′(z1) − 1
≤ |z1|
2+β∥zˆ∥2
∥z∥2 ≤1,
where 0≤β≤1. This completes the proof. □
Acknowledgment. The authors are thankful to the referees and the editors for their valuable comments and suggestions that improved the presentation of this paper.
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1 Department of Mathematics, Faculty of Basic Science, University of
