CHIRILA, TEODORA

English

Research Publication Journals Repository

ADV  MATH
SCI  JOURNAL
Advances in Mathematics: Scientic Journal 1 (2012), no.2, 139145
ISSN 1857-8365, UDC: 517.546
AN EXTENSION OPERATOR AND LOEWNER CHAINS ON SOME
REINHARDT DOMAINS IN Cn
TEODORA CHIRIL
Presented at the 8th International Symposium GEOMETRIC FUNCTION THEORY AND
APPLICATIONS, 27-31 August 2012, Ohrid, Republic of Macedonia.
Abstract. In this paper we are concerned with an extension operator n;p;,   0,
p  1, that provides a way of extending a locally biholomorphic mapping f 2 H(Bn)
to a locally biholomorphic mapping F 2 H(
n;p), where 
n;p = fz = (z0; zn+1) 2
Cn+1 : kz0k2+jzn+1jp < 1g. By using the method of Loewner chains, we prove that if
f 2 S0(Bn), then n;p;(f) 2 S0(
n;p), for p  maxf2n; 1g and  2 [0; 1=(n+ 1)].
In particular, if f 2 S(Bn), then n;p;(f) 2 S(
n;p) and if f is spirallike of type
 2 ( =2; =2) on Bn, then n;p;(f) is also spirallike of type  on 
n;p. Finally,
we consider the preservation of "-starlikeness under the operator n;p;.
1. Introduction and preliminaries
Let Cn denote the space of n complex variables z = (z1; : : : ; zn) with the Euclidean
inner product hz;wi =
Pn
j=1 zjwj and the Euclidean norm kzk = hz; zi
1=2. The open
ball fz 2 Cn : kzk < rg is denoted by Bnr and the unit ball B
n
1 is denoted by B
n. In the
case of one complex variable, B1 is denoted by U .
Let L(Cn;Cm) denote the space of linear continuous operators from Cn into Cm with
the standard operator norm, and let In be the identity of L(C
n;Cn). If 
 is a domain in
Cn, we denote by H(
) the set of holomorphic mappings from 
 into Cn. If 0 2 
, such
a mapping f is said to be normalized if f(0) = 0 and Df(0) = In.
From now on, we assume that 
 is a domain in Cn that contains the origin. We say
that f 2 H(
) is locally biholomorphic on 
 if the complex Jacobian matrix Df(z) is
nonsingular at each z 2 
. Let Jf (z) = detDf(z). Let LSn(
) be the set of normalized
locally biholomorphic mappings on 
 and let S(
) be the set of normalized biholomorphic
mappings on 
. A map f 2 S(
) is said to be convex if its image is a convex domain in
Cn, and starlike if the image is a starlike domain with respect to the origin. We denote the
classes of normalized convex and starlike mappings on 
 respectively by K(
) and S(
).
2010 Mathematics Subject Classication. 32H02, 30C45.
Key words and phrases. Loewner chain, parametric representation, Pfaltzgra-Suridge extension
operator, Roper-Suridge extension operator, spirallike mapping of type , starlike mapping, "-starlike
mapping.
139
140 T. CHIRIL
In one variable we write LS1(B
1) = LS, S(B1) = S, K(B1) = K and S(B1) = S. A
mapping f 2 S(
) is spirallike of type  2 ( =2; =2) if the spiral exp( e i )f(z)
(  0) is contained in f(
) for any z 2 
. We denote by S^(
) the class of normalized
spirallike mappings of type  on 
.
We next present the notion of "-starlikeness due to Gong and Liu (see [3]). This notion
interpolates between starlikeness and convexity as " ranges from 0 to 1.
Denition 1.1. Let f : 
! Cn be a biholomorphic mapping such that f(0) = 0. We
say that f is "-starlike, 0  "  1, if f(
) is starlike with respect to each point in
"f(
), i.e.
(1  )f(z) + "f(w) 2 f(
);  2 [0; 1]; z; w 2 
:
When " = 0 we obtain the family of starlike mappings on 
, and when " = 1 we obtain
the family of convex mappings on 
. The analytical characterization of "-starlikeness was
given in [4].
For n  1, set z0 = (z1; : : : ; zn) 2 C
n and z = (z0; zn+1) 2 C
n+1. Let p  1 and

n;p = fz = (z
0; zn+1) 2 C
n+1 : kz0k2 + jzn+1j
p < 1g:
The complex ellipsoid 
n;p is a balanced convex domain in C
n+1 (see e.g. [10]) and
Bn  f0g  
n;p  B
n  U .
We next refer to the notions of subordination and Loewner chains.
Let f; g 2 H(
n;p). We say that f is subordinate to g (and write f  g) if there
is a Schwarz mapping v (i.e. v 2 H(
n;p) and kv(z)k  kzk, z 2 
n;p) such that
f(z) = g(v(z)), z 2 
n;p.
Denition 1.2. A mapping f : 
n;p [0;1)! C
n is called a Loewner chain if f(; t)
is biholomorphic on 
n;p, f(0; t) = 0, Df(0; t) = e
tIn for t  0, and f(z; s)  f(z; t)
whenever 0  s  t <1 and z 2 
n;p.
The requirement f(z; s)  f(z; t) is equivalent to the condition that there is a
unique biholomorphic Schwarz mapping v = v(z; s; t) called the transition mapping
associated to f(z; t) such that
f(z; s) = f(v(z; s; t); t); z 2 
n;p; t  s  0:
Various results concerning Loewner chains can be found in [6].
Remark 1.1. Certain subclasses of S(
n;p) can be characterized in terms of Loewner
chains. In particular, f is starlike if and only if f(z; t) = etf(z) is a Loewner chain.
Also, f is spirallike of type  if and only if f(z; t) = e(1 ia)tf(eiatz) is a Loewner
chain, where a = tan (see e.g. [9]).
The notion of parametric representation is related to that of a Loewner chain (see e.g.
[9]).
Denition 1.3. A normalized mapping f 2 H(
n;p) has parametric representation
if there exists a Loewner chain f(z; t) such that fe tf(; t)gt0 is a normal family on

n;p and f(z) = f(z; 0), z 2 
n;p.
Let S0(
n;p) be the set of mappings which have parametric representation on 
n;p.
A key role in our discussion is played by the following Schwarz-type lemma for the
Jacobian determinant of a holomorphic mapping from Bn into Bn (see [13]):
AN EXTENSION OPERATOR ON SOME REINHARDT DOMAINS IN CN 141
Lemma 1.1. Let  2 H(Bn) be such that  (Bn)  Bn. Then
(1.1) jJ (z)j 

1  k (z)k2
1  kzk2
n+1
2
; z 2 Bn:
Denition 1.4. Let   0. The extension operator n;p; : LSn(B
n)! LSn+1(
n;p)
is dened by
n;p;(f)(z) = (f(z
0); zn+1[Jf (z
0)]
2
p ); z = (z0; zn+1) 2 
n;p:
We choose the branch of the power function such that [Jf (z
0)]
2
p

z0=0
= 1:
If  = 1=(n + 1), the operator n;p;1=(n+1) was studied in [9]. In the case p = 2,

n;2 = B
n+1 and the operator n;2; was recently studied in [1]. If p = 2 and  =
1=(n+1), the operator n;2;1=(n+1) reduces to the Pfaltzgra-Suridge extension operator
(see [11]). If n = 1, p = 2 and  = 1=2, then 1;2;1=2 reduces to the well-known
Roper-Suridge extension operator. For n  2, the Roper-Suridge extension operator
	n : LS ! LSn(B
n) is dened by (see [12])
	n(f)(z) = (f(z1); ~z
p
f 0(z1)); z = (z1; ~z) 2 B
n:
We choose the branch of the power function such that
p
f 0(z1)

z1=0
= 1:
Roper and Suridge proved that if f is convex on U then 	n(f) is also convex on B
n.
Graham and Kohr proved that if f is starlike on U then so is 	n(f) on B
n. Graham,
Kohr and Kohr [7] proved that if f has parametric representation on the unit disc, then
	n(f) has the same property on B
n. A general class of extension operators was studied
in [2], [5]. For other extension operators, see [6] and the references therein.
In this paper we prove that if f 2 S0(Bn), then n;p;(f) 2 S
0(
n;p), for p 
maxf2n; 1g and  2 [0; 1=(n+1)]. In particular, we obtain various consequences related
to the preservation of starlikeness and spirallikeness of type  under n;p;. Finally, we
consider the preservation of "-starlikeness under the operator n;p;.
2. Parametric representation and the operator n;p;
We begin this section with the following main result. If  = 1=(n + 1), see [9] and if
p = 2, see [1] (see also [8] for  = 1=(n+ 1) and p = 2).
Theorem 2.1. Assume f 2 S0(Bn), p  maxf2n; 1g and  2 [0; 1=(n + 1)]. Then
n;p;(f) 2 S
0(
n;p).
Proof. Since f 2 S0(Bn) there exists a Loewner chain f(z0; t) such that fe tf(z0; t)gt0
is a normal family on Bn and f(z0) = f(z0; 0) for z0 2 Bn. Let F = n;p;(f). Then it is
easy to see that F 2 S(
n;p). Let v = v(z
0; s; t) be the transition mapping associated to
f(z0; t). Then
(2.1) f(z0; s) = f(v(z0; s; t); t); z0 2 Bn; 0  s  t <1:
Let ft(z
0) = f(z0; t) for z0 2 Bn and t  0, and let vs;t(z
0) = v(z0; s; t), z0 2 Bn, t  s  0.
Also let F : 
n;p  [0;1)! C
n+1 be given by
(2.2) F (z; t) = (f(z0; t); zn+1e
t(1  2np )[Jft(z
0)]
2
p )
142 T. CHIRIL
for z = (z0; zn+1) 2 
n;p and t  0. We choose the branch of the power function such
that [Jft(z
0)]
2
p

z0=0
= e
2nt
p :
Let us prove that F (z; t) is a Loewner chain. Indeed, since f(; t) is biholomorphic on
Bn, f(0; t) = 0 and Df(0; t) = etIn, it is not dicult to see that F (; t) is biholomorphic
on 
n;p, F (0; t) = 0 and DF (0; t) = e
tIn+1.
Let Vs;t : 
n;p ! C
n+1 be given by Vs;t(z) = V (z; s; t) where
V (z; s; t) = (v(z0; s; t); zn+1e
(s t)(1  2np )[Jvs;t(z
0)]
2
p )
for z = (z0; zn+1) 2 
n;p and t  s  0. We choose the branch of the power function
such that [Jvs;t(z
0)]
2
p

z0=0
= e
2n(s t)
p . Then Vs;t is biholomorphic on 
n;p, Vs;t(0) = 0,
DVs;t(0) = e
s tIn+1 and Vs;t(
n;p)  
n;p. Indeed, x z 2 
n;p and let w = Vs;t(z). We
have to prove that kw0k2+jwn+1j
p < 1. By Lemma 1.1 and the fact that p  maxf2n; 1g,
 2 [0; 1=(n+ 1)], we obtain
kw0k2 + jwn+1j
p = kvs;t(z
0)k2 + e(s t)(p 2n)jzn+1j
pjJvs;t(z
0)j2
 kvs;t(z
0)k2 + jzn+1j
p

1  kvs;t(z
0)k2
1  kz0k2
(n+1)
 kvs;t(z
0)k2 +
jzn+1j
p
1  kz0k2
(1  kvs;t(z
0)k2)
< kvs;t(z
0)k2 + 1  kvs;t(z
0)k2 = 1;
for z = (z0; zn+1) 2 
n;p and t  s  0. Hence w = (w
0; wn+1) 2 
n;p and thus
Vs;t(
n;p)  
n;p, as claimed.
Further, taking into account (2.1), we easily deduce that F (z; s) = F (V (z; s; t); t) for
z 2 
n;p and t  s  0. Indeed,
F (V (z; s; t); t) = (f(v(z0; s; t); t); zn+1e
(s t)(1  2np )et(1 
2n
p )[Jvs;t(z
0)]
2
p [Jft(vs;t(z
0))]
2
p )
= (f(z0; s); zn+1e
s(1  2np )[Jfs(z
0)]
2
p ) = F (z; s);
for all z 2 
n;p and t  s  0. Here we have used (2.1) and the fact that
Jfs(z
0) = Jft(vs;t(z
0))Jvs;t(z
0); z0 2 Bn; t  s  0:
Therefore we have proved that F (z; t) is a Loewner chain.
It remains to prove that fe tF (z; t)gt0 is a normal family. Since fe
 tf(z0; t)gt0 is
a normal family on Bn, there exists a sequence ftkgk2N such that 0 < tk ! 1 and
e tkf(z0; tk) ! g(z
0) locally uniformly on Bn as k ! 1, where g is a holomorphic
mapping on Bn. Since g(0) = 0 and Dg(0) = In, we deduce in view of Hurwitz's theorem
for holomorphic mappings that g 2 S(Bn). Further, applying Vitali's theorem in several
complex variables, we deduce that n;p;(e
 tkf(z; tk)) = e
 tkF (z; tk)! n;p;(g) locally
uniformly on 
n;p as k!1. This completes the proof. 
Taking into account Theorem 2.1 and Remark 1.1, we may prove that the operator
n;p; preserves the notions of starlikeness and spirallikeness of type . We omit the
proofs of the next corollaries (see [1] in the case p = 2). Corollary 2.1 was obtained in [9]
in the case  = 1=(n+ 1) (see also [8] for  = 1=(n+ 1) and p = 2).
AN EXTENSION OPERATOR ON SOME REINHARDT DOMAINS IN CN 143
Corollary 2.1. Assume f 2 S(Bn), p  maxf2n; 1g,  2 [0; 1=(n + 1)]. Then
F = n;p;(f) 2 S
(
n;p).
Corollary 2.2. Assume f 2 S^(B
n), where  2 ( =2; =2). Then F = n;p;(f) 2
S^(
n;p) for p  maxf2n; 1g and  2 [0; 1=(n+ 1)].
3. "-starlikeness and the operator n;p;
We next discuss the case of "-starlike mappings associated with the operator n;p;,
for  2
h
1
n+1 ;
p
2n
i
. To this end, for a 2 (0; 1], let

a;n;p; = fz = (z
0; zn+1) 2 C
n+1 : jzn+1j
p < a2n(1  kz0k2)(n+1)g:
Then Bn  f0g  
a;n;p;  
n;p. For a = 1 and  =
1
n+1 , we obtain 
1;n;p; 1n+1 = 
n;p.
We have obtained the following result regarding "-starlikeness, which when " = 1 gives
a partial answer to the question of whether n;p; preserves convexity. If p = 2, see [1].
Theorem 3.1. Let " 2 [0; 1] and f : Bn ! Cn be a normalized "-starlike mapping.
Also let F = n;p;(f), where  2
h
1
n+1 ;
p
2n
i
and let a1; a2 > 0 be such that a1+a2  1.
Then
(1  )F (z) + "F (w) 2 F (
a1+a2;n;p;); z 2 
a1;n;p;; w 2 
a2;n;p;;  2 [0; 1]:
Proof. Since f is biholomorphic on Bn, it follows that F = n;p;(f) is also biholomor-
phic on 
n;p. Fix  2 [0; 1] and let z 2 
a1;n;p;, w 2 
a2;n;p;. We want to nd a point
u = (u0; un+1) 2 
a1+a2;n;p; such that
(1  )F (z) + "F (w) = F (u);
i.e. f(u0) = (1  )f(z0) + "f(w0) and
un+1[Jf (u
0)]
2
p = (1  )zn+1[Jf (z
0)]
2
p + "wn+1[Jf (w
0)]
2
p :
If  = 0, let u = z. If  = 1, then using the fact that f is "-starlike and the equality
"F (w) = F (u), we easily deduce that u = (u0; un+1) 2 
a2;n;p;  
a1+a2;n;p;. Hence,
it suces to assume that  2 (0; 1). Since f is "-starlike, we obtain that
u0 = f 1((1  )f(z0) + "f(w0)):
Then u0 = u0(z0; w0) can be viewed as a mapping from Bn Bn into Bn. Let
un+1 = (1  )zn+1

Jf (z
0)
Jf (u0)
 2
p
+ "wn+1

Jf (w
0)
Jf (u0)
 2
p
:
We prove that u = (u0; un+1) 2 
a1+a2;n;p;. It is obvious that
@u0
@z0
= (1  )[Df(u0)] 1Df(z0) and
@u0
@w0
= "[Df(u0)] 1Df(w0):
Hence
un+1 = (1  )
p 2n
p zn+1[Ju0
z0
]
2
p + (")
p 2n
p wn+1[Ju0
w0
]
2
p :
144 T. CHIRIL
Next, let 1=~p = 2n=p. Since p  2n, it follows that ~p  1. Also let 1=~q = 1   1=~p.
Using Lemma 1.1 in the previous equation, we obtain
jun+1j
 (1  )1=~qjzn+1j

1  ku0(z0; w0)k2
1  kz0k2
 (n+1)
p
+ (")1=~qjwn+1j

1  ku0(z0; w0)k2
1  kw0k2
 (n+1)
p
= (1  ku0k2)
(n+1)
p

(1  )1=~q

jzn+1j
p
(n+1)
1  kz0k2
 (n+1)
p
+ (")1=~q

jwn+1j
p
(n+1)
1  kw0k2
 (n+1)
p

:
We have two cases:
First case (compare with Corollary 2.1). If " = 0 (i.e. f is starlike), then we obtain
that
jun+1j  (1  ku
0k2)
(n+1)
p (1  )1=~q
jzn+1j
(1  kz0k2)
(n+1)
p
< a
2n
p
1 (1  ku
0k2)
(n+1)
p :
Here we have used the fact that z = (z0; zn+1) 2 
a1;n;p;. Hence jun+1j
p < a2n1 (1  
ku0k2)(n+1), i.e. u = (u0; un+1) 2 
a1;n;p;. On the other hand, since 
a1;n;p; 

a1+a2;n;p;, we deduce that u = (u
0; un+1) 2 
a1+a2;n;p;, as desired.
Second case. For " 2 (0; 1], using Hölder's inequality we obtain
jun+1j
 (1  ku0k2)
(n+1)
p (1  + ")1=~q

jzn+1j
p
(n+1)
1  kz0k2
n+1
2n
+

jwn+1j
p
(n+1)
1  kw0k2
n+1
2n
1=~p
< (1  ku0k2)
(n+1)
p (a1 + a2)
1=~p:
Therefore, we have proved that jun+1j
p < (a1 + a2)
2n(1   ku0k2)(n+1), i.e. u =
(u0; un+1) 2 
a1+a2;n;p;. This completes the proof. 
Taking " = 1 in Theorem 3.1, we obtain the following convexity result for the operator
n;p;. If p = 2, see [1].
Corollary 3.1. If f 2 K(Bn) and F = n;p;(f), where  2
h
1
n+1 ;
p
2n
i
, then (1  
)F (z) + F (w) 2 F (
a1+a2;n;p;), z 2 
a1;n;p;, w 2 
a2;n;p;,  2 [0; 1], where
a1; a2 > 0, a1 + a2  1.
Taking  = 1n+1 and a1 = 1  a2 in Corollary 3.1 and using the fact that 
1;n;p; 1n+1 =

n;p, we obtain the following corollary. In the case " = 1, see [9] and when p = 2, see [1].
Corollary 3.2. If f is a normalized "-starlike mapping on Bn, " 2 [0; 1], a 2 (0; 1)
and F = n;p; 1
n+1
(f), p  2n=(n+ 1), then
(1  )F (z) + "F (w) 2 F (
n;p); z 2 
a;n;p; 1
n+1
; w 2 
1 a;n;p; 1
n+1
;  2 [0; 1]:
Acknowledgment
This work was supported by a grant of the Romanian National Authority for Scientic
Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0899. The author is
indebted to Gabriela Kohr for valuable suggestions during the preparation of this paper.
AN EXTENSION OPERATOR ON SOME REINHARDT DOMAINS IN CN 145
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Faculty of Mathematics and Computer Science
Babe³-Bolyai University
Str. M. Kog lniceanu no. 1, 400084 Cluj-Napoca, Romania
E-mail address: teodora.andrica@ubbcluj.ro


AN EXTENSION OPERATOR AND LOEWNER CHAINS ON SOME REINHARDT DOMAINS IN C^n

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AN EXTENSION OPERATOR AND LOEWNER CHAINS ON SOME REINHARDT DOMAINS IN C^n

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