SECOND ORDER DIFFERENTIAL
SUBORDINATIONS OF HOLOMORPHIC MAPPINGS
ON BOUNDED CONVEX BALANCED
DOMAINS IN C n
2nd Order Differential Subordinations
of Holomorphic Mappings
Yu-Can Zhu and Ming-Sheng Liu
vol. 8, iss. 4, art. 104, 2007
YU-CAN ZHU
MING-SHENG LIU
Title Page
Department of Mathematics
Fuzhou University,
Fuzhou, 350002 Fujian, P. R. China
EMail: zhuyucan@fzu.edu.cn
Department of Mathematics
South China Normal University,
Guangzhou, 510631 Guangdong, P. R. China
EMail: liumsh@scnu.edu.cn
Contents
JJ
II
J
I
Received:
09 November, 2006
Accepted:
04 November, 2007
Communicated by:
G. Kohr
2000 AMS Sub. Class.:
32H02, 30C45.
Key words:
Differential subordination, biholomorphic convex mapping, convex balanced domain, Minkowski functional.
Full Screen
Abstract:
In this paper, we obtain some second order differential subordinations of holomorphic mappings on a bounded convex balanced domain Ω in Cn . These results
imply some first order differential subordinations of holomorphic mappings on
a bounded convex balanced domain Ω in Cn . When Ω is the unit disc in the
complex plane C, these results are just ones of Miller and Mocanu et al. about
differential subordinations of analytic functions on the unit disc in the complex
plane C.
Close
Page 1 of 30
Go Back
Acknowledgements:
The authors thank the referee for his helpful comments and suggestions to improve our manuscript.
This research is partly supported by the National Natural Science Foundation of
China(No.10471048), the Doctoral Foundation of the Education Committee of
China(No.20050574002), the Natural Science Foundation of Fujian Province,
China (No.Z0511013) and the Education Commission Foundation of Fujian
Province, China (No.JB04038).
2nd Order Differential Subordinations
of Holomorphic Mappings
Yu-Can Zhu and Ming-Sheng Liu
vol. 8, iss. 4, art. 104, 2007
Title Page
Contents
JJ
II
J
I
Page 2 of 30
Go Back
Full Screen
Close
Contents
1
Introduction
4
2
Main Results and Their Proofs
7
2nd Order Differential Subordinations
of Holomorphic Mappings
Yu-Can Zhu and Ming-Sheng Liu
vol. 8, iss. 4, art. 104, 2007
Title Page
Contents
JJ
II
J
I
Page 3 of 30
Go Back
Full Screen
Close
1.
Introduction
Let Cn be the space of n complex variables z = (z1 , z2 , . .p
. , zn ) with the Euclidian
Pn
inner product hz, wi = j=1 zj wj and the norm kzk = hz, zi. A domain Ω is
called a balanced domain in Cn if λz ∈ Ω for all z ∈ Ω and λ ∈ C with |λ| ≤ 1.
The Minkowski functional of the balanced domain Ω is
o
n
z
ρ(z) = inf t > 0, ∈ Ω , z ∈ Cn .
t
Suppose that Ω is a bounded convex balanced domain in Cn , and ρ(z) is the
Minkowski functional of Ω. Then ρ(·) is a norm of Cn such that
Ω = {z ∈ Cn : ρ(z) < 1},
ρ(λz) = |λ|ρ(z)
for λ ∈ C, z ∈ Cn (see [20]).
Let pj > 1 (j = 1, 2, . . . , n). Then
(
Dp =
n
(z1 , z2 , . . . , zn ) ∈ C :
n
X
)
pj
|zj |
<1
is a bounded convex balanced domain, and the Minkowski functional ρ(z) of Dp
satisfies
n X
zj pj
(1.1)
ρ(z) = 1.
j=1
ρ(z) =
n
j=1
|zj |p
1/p
is the Minkowski functional of domain
(
)
n
X
Bp = z ∈ Cn :
|zj |p < 1 , where p > 1.
j=1
Yu-Can Zhu and Ming-Sheng Liu
vol. 8, iss. 4, art. 104, 2007
Title Page
Contents
j=1
P
2nd Order Differential Subordinations
of Holomorphic Mappings
JJ
II
J
I
Page 4 of 30
Go Back
Full Screen
Close
Let Df (z) and D2 f (z)(·, ·) denote the first Fréchet derivative and the second
Fréchet derivative for a holomorphic mapping f : Ω → Cn respectively. Then they
have the matrix representation
!
n
2
X
∂
f
(z)
∂fj (z)
j
Df (z) =
, D2 f (z)(b, ·) =
bl
,
∂zk
∂z
k ∂zl
1≤j, k≤n
l=1
1≤j, k≤n
n
n
where b = (b1 , b2 , . . . , bn ) ∈ C . The mapping f : Ω → C is called locally
biholomorphic if the matrix Df (z) is nonsingular at each point z in Ω.
The class of all holomorphic mappings f : Ω → Cn is denoted by H(Ω, Cn ).
Assume f, g ∈ H(Ω, Cn ). Then we say that the mapping f is subordinate to g,
written f ≺ g or f (z) ≺ g(z), if there exists a holomorphic mapping w : Ω → Ω
with w(0) = 0 such that f (z) ≡ g(w(z)) for all z ∈ Ω. If g is a biholomorphic
mapping, then f (z) ≺ g(z) if and only if f (Ω) ⊂ g(Ω) and f (0) = g(0).
In classical results of geometric function theory, differential subordinations provide some simple proofs. They play a key role in the study of some integral operators, differential equations, and properties of subclasses of univalent functions,
etc. S.S. Miller and P.T. Mocanu et al. have obtained some deep results for differential subordinations [10, 11, 12, 13, 16, 14]. There is a excellent text Differential
Subordinations Theory and Applications, by S.S. Miller and P.T. Mocanu [15].
The geometric function theory of several complex variables has been studied by
many authors. Many important results for biholomorphic convex or starlike mappings in Cn have been obtained (see [2, 3]). Some differential subordinations of
analytic functions in the complex plane are also extended to Cn [4, 6, 8, 15, 22]. But
there are very few results on second order differential subordinations of holomorphic
mappings in Cn .
In this paper, we obtain some second order differential subordinations of holomorphic mappings on a bounded convex balanced domain Ω in Cn . These results
2nd Order Differential Subordinations
of Holomorphic Mappings
Yu-Can Zhu and Ming-Sheng Liu
vol. 8, iss. 4, art. 104, 2007
Title Page
Contents
JJ
II
J
I
Page 5 of 30
Go Back
Full Screen
Close
imply some first order differential subordinations of holomorphic mappings on a
bounded convex balanced domain Ω in Cn . When Ω is the unit disc in the complex
plane C, these results are just those of Miller and Mocanu et al. about differential
subordinations of analytic functions on the unit disc in the complex plane C.
2nd Order Differential Subordinations
of Holomorphic Mappings
Yu-Can Zhu and Ming-Sheng Liu
vol. 8, iss. 4, art. 104, 2007
Title Page
Contents
JJ
II
J
I
Page 6 of 30
Go Back
Full Screen
Close
2.
Main Results and Their Proofs
In the following, we always assume that the domain Ω is a bounded convex balanced
domain in Cn and ρ(z) is the Minkowski functional of Ω. Then ρ(·) is a norm of Cn
such that
Ω = {z ∈ Cn : ρ(z) < 1}, ρ(λz) = |λ|ρ(z)
for λ ∈ C, z ∈ Cn .
In order to derive our main results, we need the following lemmas.
Lemma 2.1. Suppose that ρ(z) is twice differentiable in Ω − {0}, and let w ∈
H(Ω, Cn ) with w(z) 6≡ 0 and w(0) = 0. If z0 ∈ Ω − {0} satisfies
ρ(w(z0 )) =
max ρ(w(z)),
2nd Order Differential Subordinations
of Holomorphic Mappings
Yu-Can Zhu and Ming-Sheng Liu
vol. 8, iss. 4, art. 104, 2007
Title Page
ρ(z)≤ρ(z0 )
Contents
then there exists a real number t ≥ 1/2 such that
∂ρ
(2.1)
Dw(z0 )(z0 ), (w0 ) = tρ(w0 ),
∂z
and
(2.2)
JJ
II
J
I
Page 7 of 30
Go Back
∂ρ
2
Re D w(z0 )(z0 , z0 ), (w0 )
∂z
( n
)
n
X ∂2ρ
X
∂2ρ
≥ Re
(w0 )bj bl −
(w0 )bj bl − tρ(w0 ),
∂zj ∂z l
∂zj ∂zl
j,l=1
j,l=1
where w0 = w(z0 ), Dw(z0 )(z0 ) = (b1 , b2 , . . . , bn ).
Full Screen
Close
Proof. Since ρ(w0 ) =
max ρ(w(z)), then we have w0 6= 0. Otherwise, there is
ρ(z)≤ρ(z0 )
w(z) ≡ 0, which contradicts the hypothesis of Lemma 2.1.
Let w(z) = (w1 (z), w2 (z), . . . , wn (z)), γ(t) = w(eit z0 ) = (γ1 (t), γ2 (t), . . . , γn (t)).
Then we have γj (t) = wj (eit z0 ) (j = 1, 2, . . . , n), γ(0) = w(z0 ) = w0 and
!
n
n
it
it z )
X
X
(t)
∂w
(e
∂w
(e
z
)
dγ
dγj (t)
j
0
j
j
0
= −ie−it
zk0 ,
= ieit
zk0 ,
dt
∂z
dt
∂z
k
k
k=1
k=1
where z0 = (z10 , z20 , . . . , zn0 ). Set L(t) = ρ(γ(t)) (−π ≤ t ≤ π). Some straightforward calculations yield
n
n
X
dγj (t) X ∂ρ
dγj (t)
∂ρ
0
(γ(t)) ·
+
(γ(t)) ·
L (t) =
∂zj
dt
∂z j
dt
j=1
j=1
"
#
n
X
∂ρ
∂wj (eit z0 ) 0
it
= −2 Im e
(γ(t)) ·
zk
∂zj
∂zk
j,k=1
∂ρ
it
it
= −2 Im Dw(e z0 )(e z0 ), (γ(t)) ,
∂z
"
n
n
n
X
X
X
∂wj
∂ρ
∂ 2 wj 0 0
∂ρ
(γ(t)) ·
+ ie2it
(γ(t))
zk zl
L00 (t) = −2 Im ieit
∂z
∂z
∂z
∂z
j
k
j
k ∂zl
j,k=1
j,k=1
l=1
" n
#
n
X X ∂2ρ
∂wl
it 0 ∂wj
it 0
− 2 Im i
(γ(t)) ·
· (e zm )
· (e zk )
∂zj ∂zl
∂zm
∂zk
j,k=1 l,m=1
" n
#
n
X X
∂2ρ
∂wl
∂w
j
0 )
(γ(t)) ·
· (eit zm
· (eit zk0 ) .
+ 2 Im i
∂z
∂z
∂z
∂z
j
l
m
k
j,k=1 l,m=1
#
2nd Order Differential Subordinations
of Holomorphic Mappings
Yu-Can Zhu and Ming-Sheng Liu
vol. 8, iss. 4, art. 104, 2007
Title Page
Contents
JJ
II
J
I
Page 8 of 30
Go Back
Full Screen
Close
Noting L(0) = max L(t) , we have L0 (0) = 0 and L00 (0) ≤ 0. It follows that
−π≤t≤π
∂ρ
Im Dw(z0 )(z0 ), (w0 ) = 0,
∂z
(2.3)
and
(2.4)
∂ρ
∂ρ
2
Re D w(z0 )(z0 , z0 ), (w0 ) + Re Dw(z0 )(z0 ), (w0 )
∂z
∂z
#
" n
X ∂2ρ
∂2ρ
+ Re
(w0 )bj bl −
(w0 )bj bl ≥ 0.
∂zj ∂zl
∂zj ∂zl
j,l=1
2nd Order Differential Subordinations
of Holomorphic Mappings
Yu-Can Zhu and Ming-Sheng Liu
vol. 8, iss. 4, art. 104, 2007
Title Page
n
On the other hand, by Schwarz’s Lemma in C [19], we have
ρ(w(z))
ρ(w0 )
≤
ρ(z)
ρ(z0 )
Let
ϕ(r) =
for 0 < ρ(z) ≤ ρ(z0 ).
ρ(w(rz0 ))
ρ(w(rz0 ))
=
.
ρ(rz0 )
rρ(z0 )
Then ϕ(1) = max ϕ(r) . It follows that
0<r≤1
ϕ0 (1) = lim−
r→1
ϕ(r) − ϕ(1)
≥ 0.
r−1
By a simple calculation, we obtain
ρ(w0 )
2
∂ρ
ϕ (1) = −
+
Re Dw(z0 )(z0 ), (w0 ) ≥ 0.
ρ(z0 )
ρ(z0 )
∂z
0
Contents
JJ
II
J
I
Page 9 of 30
Go Back
Full Screen
Close
If we let
1
∂ρ
t=
Re Dw(z0 )(z0 ), (w0 ) ,
ρ(w0 )
∂z
then we have t ≥ 1/2, therefore (2.1) of Lemma 2.1 holds, and (2.2) follows from
(2.3) and (2.4). This completes the proof.
Remark 1. Since ρ(tz) = tρ(z) for t > 0, then for z ∈ Cn − {0}, we have
n
n
X
X
dρ(tz) ∂ρ
∂ρ
∂ρ
(2.5)
ρ(z) =
=
zj +
z j = 2 Re z, (z) .
dt t=1 j=1 ∂zj
∂z
∂z
j
j=1
z
For any z ∈ Cn − {0}, we have ρ( ρ(z)
) = 1. Letting w(z) ≡ z in (2.1), we obtain
1
that there exists a real number t ≥ 2 such that
∂ρ
z, (z) = tρ(z) ≥ 0, z ∈ Cn − {0}.
∂z
Hence it follows from (2.5) that
∂ρ
ρ(z) = 2 z, (z) ,
∂z
z ∈ Cn − {0}.
|ξ|≤r0
2
ξ0 g 0 (ξ0 ) ≤ −
| a − g(ξ0 ) |
,
2 Re(a − g(ξ0 ))
and
(2.7)
Yu-Can Zhu and Ming-Sheng Liu
vol. 8, iss. 4, art. 104, 2007
Title Page
Contents
JJ
II
J
I
Page 10 of 30
Go Back
Lemma 2.2 ([10]). Let g(ξ) = a + b1 ξ + b2 ξ 2 + · · · be analytic in |ξ| < 1 with
g(ξ) 6≡ 0. If ξ0 = r0 eiθ0 (0 < r0 < 1) and Re g(ξ0 ) = min Re g(ξ), then
(2.6)
2nd Order Differential Subordinations
of Holomorphic Mappings
Re{ξ02 g 00 (ξ0 ) + ξ0 g 0 (ξ0 )} ≤ 0.
Full Screen
Close
Lemma 2.3. Suppose that ρ(z) is differentiable in Ω − {0}. Let h : Ω → Cn be a
biholomorphic convex mapping with h(0) = 0. Then for every z ∈ Ω − {0}, we have
2 Dh(z)−1 h(z), ∂ρ (z) − ρ(z) ≤ ρ(z).
∂z
Proof. For each z ∈ Ω − {0}, we let g(ξ) = hDh(z)−1 (h(z) − h(ξz)), ∂ρ
(z)i for
∂z
|ξ| ≤ 1. Then g(ξ) is analytic in |ξ| ≤ 1 and
∂ρ
−1
g(ξ) = Dh(z) h(z), (z) + b1 ξ + · · · .
∂z
From the result in [3, 7], we have Re g(ξ) > 0 for all |ξ| < 1. Hence we obtain
2nd Order Differential Subordinations
of Holomorphic Mappings
Yu-Can Zhu and Ming-Sheng Liu
vol. 8, iss. 4, art. 104, 2007
Title Page
Contents
0 = Re g(1) = min Re g(ξ).
|ξ|≤1
By a simple calculation, we may obtain
∂ρ
∂ρ
ρ(z)
0
−1
g (1) = − Dh(z) Dh(z)(z), (z) = − z, (z) = −
.
∂z
∂z
2
JJ
II
J
I
Page 11 of 30
Go Back
By (2.6), we have
−ρ(z) Re a + |a|2 ≤ 0,
D
E
∂ρ
−1
where a = (Dh(z) h(z), ∂z (z) . It follows that
∂ρ
2 Dh(z)−1 h(z), (z) − ρ(z) ≤ ρ(z).
∂z
Full Screen
Close
Lemma 2.4 ([23]). Suppose that ρ(z) is twice differentiable in Ω − {0}. If f : Ω →
Cn is a biholomorphic convex mapping, then we have
( n
X ∂2ρ
(2.8) Re
bl bm
∂z
l ∂zm
l,m=1
)
n
X
∂2ρ
∂ρ
+
bl bm − Df (z)−1 D2 f (z)(b, b),
≥0
∂z
∂z
l ∂z m
l,m=1
=
for every z = (z1 , z2 , . . . , zn ) ∈ Ω−{0}, b = (b1 , b2 , . . . , bn ) ∈ Cn with Re b, ∂ρ
∂z
0.
Lemma 2.5. Assume that ρ(z) is differentiable in Ω − {0}. Then
∂ρ
(2.9)
ρ(z) = 2 z, (z) , z ∈ Cn − {0},
∂z
(2.10)
z ∈ Cn − {0}, w ∈ Cn .
II
J
I
Go Back
Full Screen
is the normalD vector of ∂ΩE
z at z. For
(z)
every z, w ∈ Cn with ρ(z) = 1, ρ(w) = 1, we have Re z − w, ∂ρ
∂z
follows that
∂ρ
∂ρ
(2.11)
2 Re w, (z) ≤ 2 Re z, (z) = ρ(z) = 1.
∂z
∂z
JJ
Page 12 of 30
Ωz = {w ∈ Cn : ρ(w) < ρ(z)}.
∂ρ
(z)
∂z
vol. 8, iss. 4, art. 104, 2007
Contents
Proof. From Remark 1, we only need to prove (2.10). Let z ∈ Cn − {0} and
Then Ωz is a convex domain in Cn , and
Yu-Can Zhu and Ming-Sheng Liu
Title Page
and
2 w, ∂ρ (z) ≤ ρ(w),
∂z
2nd Order Differential Subordinations
of Holomorphic Mappings
≥ 0. It
Close
D
E
∂ρ
(z)
∂z
When w,
= 0, it is obvious that (2.10) holds.
D
E
z
w
(z)
6= 0, then ρ(w) 6= 0. Using ρ(z)
to substitute for z and ρ(w)
e−iθ
When w, ∂ρ
∂z
to substitute for w in (2.11), we obtain
∂ρ
2 Re w, (z) ≤ ρ(w), z ∈ Cn − {0}, w ∈ Cn ,
∂z
E
D
(z)
and ∂ρ
(λz) = ∂ρ
(z) for all λ ∈ (0, +∞) and z ∈
where θ = arg w, ∂ρ
∂z
∂z
∂z
Ω − {0}. This completes the proof.
Lemma 2.6. Suppose that ρ(z) is differentiable in Ω − {0}, and let h : Ω → Cn be
a biholomorphic convex mapping with h(0) = 0. Then for every z ∈ Ω − {0} and
vector ξ ∈ Cn , the inequality
2 Dh(z)−1 (ξ), ∂ρ (z) ≤ (1 + ρ(z))2 ρ(Dh(0)−1 (ξ))
∂z
holds.
2nd Order Differential Subordinations
of Holomorphic Mappings
Yu-Can Zhu and Ming-Sheng Liu
vol. 8, iss. 4, art. 104, 2007
Title Page
Contents
JJ
II
J
I
Page 13 of 30
Proof. Without loss of generality, we may assume that h is a biholomorphic convex
mapping on Ω. If not, then we can replace h(z) by hr (z) = h(rz), where 0 < r < 1.
For any fixed z ∈ Ω − {0}, from the proof of Theorem 2.1 in [5, 9], there exist
ze ∈ ∂Ω and µ ∈ (0, 1) such that h(z) = µh(e
z ) and
1−µ≥
1 − ρ(z)
.
1 + ρ(z)
Let g(w) = h−1 [(1 − µ)h(w) + µh(e
z )]. Since h is a biholomorphic convex mapping
on Ω, then g ∈ H(Ω, Cn ) with g(Ω) ⊂ Ω and g(0) = z. For every ξ ∈ Cn − {0},
Go Back
Full Screen
Close
we set
ξ
∂ρ
ψ(λ) = 2 g λ
, (z) ,
ρ(ξ)
∂z
then ψ(λ) is an analytic function in |λ| < 1. By Lemma 2.5, we obtain
ξ
|ψ(λ)| ≤ ρ g λ
<1
ρ(ξ)
for all |λ| < 1, and
2nd Order Differential Subordinations
of Holomorphic Mappings
Yu-Can Zhu and Ming-Sheng Liu
vol. 8, iss. 4, art. 104, 2007
∂ρ
ξ
, (z) λ + · · · .
ψ(λ) = ρ(z) + 2 Dg(0)
∂z
ρ(ξ)
0
Title Page
2
From the classical result in [1], we have |ψ (0)| ≤ 1 − |ψ(0)| . It follows that
2 Dg(0)(ξ), ∂ρ (z) ≤ (1 − ρ(z)2 )ρ(ξ).
∂z
Since Dh(z)Dg(0) = (1 − µ)Dh(0), then
2 Dh(z)−1 Dh(0)(ξ), ∂ρ (z) ≤ 1 (1 − ρ(z)2 )ρ(ξ) ≤ (1 + ρ(z))2 ρ(ξ)
1−µ
∂z
Contents
JJ
II
J
I
Page 14 of 30
Go Back
Full Screen
n
for all ξ ∈ C , z ∈ Ω − {0}.
Set ζ = Dh(0)(ξ), then ξ = Dh(0)−1 ζ and
2 Dh(z)−1 (ζ), ∂ρ (z) ≤ (1 + ρ(z))2 ρ(Dh(0)−1 (ζ)),
∂z
which completes the proof.
Close
Theorem 2.7. Let f, g ∈ H(Ω, Cn ) with f (0) = g(0), and let g be biholomorphic
convex on Ω. Suppose that ρ(z) is twice differentiable in Ω − {0}. If f is not
subordinate to g, then there exist points z0 ∈ Ω − {0}, w0 ∈ ∂Ω with 0 < ρ(z0 ) <
1, ρ(w0 ) = 1 and there is a real number t ≥ 1/2 such that
1. f (z0 ) = g(w0 ),
D
E
2. Dg(w0 )−1 Df (z0 )(z0 ), ∂ρ
(w
)
= t, and
0
∂z
D
E
3. Re Dg(w0 )−1 D2 f (z0 )(z0 , z0 ), ∂ρ
(w
)
≥ −t.
0
∂z
Proof. If f is not subordinate to g, then there exist points z0 ∈ Ω − {0}, w0 ∈ ∂Ω
with 0 < ρ(z0 ) < 1, ρ(w0 ) = 1 such that f (z0 ) = g(w0 ) and f (Dr ) ⊂ g(Ω), where
Dr = {z ∈ Cn : ρ(z) < r} and r = ρ(z0 ).
Let w(z) = g −1 (f (z)). Then w : Dr → Ω is a holomorphic mapping with
w(z) 6≡ 0 and w(0) = 0 satisfying f (z) = g(w(z)) for z ∈ Dr . Hence
1 = ρ(w0 ) =
max ρ(w(z)).
ρ(z)≤ρ(z0 )
By a simple calculation, we have
2nd Order Differential Subordinations
of Holomorphic Mappings
Yu-Can Zhu and Ming-Sheng Liu
vol. 8, iss. 4, art. 104, 2007
Title Page
Contents
JJ
II
J
I
Page 15 of 30
Go Back
−1
Dw(z0 )(z0 ) = Dg(w0 ) Df (z0 )(z0 ),
−1
Dg(w0 ) D f (z0 )(z0 , z0 ) = Dg(w0 )−1 D2 g(w0 )(Dw(z0 )(z0 ), Dw(z0 )(z0 ))
Full Screen
2
+ D2 w(z0 )(z0 , z0 ).
From (2.1), there is a real number t ≥ 1/2 such that
∂ρ
Dw(z0 )(z0 ), (w0 ) = tρ(w0 ) = t.
∂z
Close
So we obtain
∂ρ
−1
Dg(w0 ) Df (z0 )(z0 ), (w0 ) = t,
∂z
and
∂ρ
−1 2
Re Dg(w0 ) D f (z0 )(z0 , z0 ), (w0 )
∂z
∂ρ
−1 2
= Re Dg(w0 ) D g(w0 )(Dw(z0 )(z0 ), Dw(z0 )(z0 )), (w0 )
∂z
∂ρ
+ Re D2 w(z0 )(z0 , z0 ), (w0 )
∂z
∂ρ
−1 2
≥ Re Dg(w0 ) D g(w0 )(a, a), (w0 )
∂z
)
( n
n
X
X ∂2ρ
∂2ρ
(w0 )aj al − t,
(w0 )aj al −
+ Re
∂z
∂z
j ∂zl
j ∂z l
j,l=1
j,l=1
where a = Dw(z0 )(z0 ) = (a1 , a2 , . . . , an ). If we let b = (b1 , b2 , . . . , bn ) with
bj = iaj , then we have
∂ρ
∂ρ
Re b, (w0 ) = Re i Dw(z0 )(z0 ), (w0 )
= Re{it} = 0.
∂z
∂z
From Lemma 2.4, we obtain
∂ρ
−1 2
Re Dg(w0 ) D f (z0 )(z0 , z0 ), (w0 )
∂z
2nd Order Differential Subordinations
of Holomorphic Mappings
Yu-Can Zhu and Ming-Sheng Liu
vol. 8, iss. 4, art. 104, 2007
Title Page
Contents
JJ
II
J
I
Page 16 of 30
Go Back
Full Screen
Close
∂ρ
−1 2
≥ − Re Dg(w0 ) D g(w0 )(b, b), (w0 )
∂z
( n
)
n
X ∂2ρ
X
∂2ρ
+ Re
(w0 )bj bl +
(w0 )bj bl − t
∂zj ∂z l
∂zj ∂zl
j,l=1
j,l=1
≥ −t.
2nd Order Differential Subordinations
of Holomorphic Mappings
This completes the proof.
Yu-Can Zhu and Ming-Sheng Liu
Remark 2. When n = 1, Ω is the unit disc in the complex plane C and ρ(z) = |z|(z ∈
C), we may obtain Lemma 1 in [14] from Theorem 2.7. Theorem 2.7 will play a
key role in studying some second order differential subordinations of holomorphic
mappings on a bounded convex balanced domain Ω in Cn .
vol. 8, iss. 4, art. 104, 2007
Let Ω1 be a set of Cn , and let h be a biholomorphic convex mapping on Ω. Suppose that ρ(z) is twice differentiable in Ω − {0}. We define Ψ(Ω1 , h) to be the class
of maps ψ : Cn × Cn × Cn × Ω → Cn that satisfy the following conditions:
Contents
1. ψ(h(0), 0, 0, 0) ∈ Ω1 , and
−1
II
J
I
E
∂ρ
(w)
∂z
2. ψ(α, β, γ, z) ∈
/ Ω1 for α = h(w),
Dh(w) (β),
= t,
D
E
Re Dh(w)−1 (γ), ∂ρ
(w) ≥ −t, and z ∈ Ω, where ρ(w) = 1 and t ≥ 1/2.
∂z
Theorem 2.8. Let ψ ∈ Ψ(Ω1 , h). If f ∈ H(Ω, Cn ) with f (0) = h(0) satisfies
ψ(f (z), Df (z)(z), D2 f (z)(z, z), z) ∈ Ω1
for all z ∈ Ω, then f (z) ≺ h(z).
JJ
Page 17 of 30
D
(2.12)
Title Page
Go Back
Full Screen
Close
Proof. If f is not subordinate to h, then by Theorem 2.7, there exist points z0 ∈
Ω − {0}, w0 ∈ ∂Ω with 0 < ρ(z0 ) < 1, ρ(w0 ) = 1 and there is a real number
t ≥ 1/2 such that
∂ρ
−1
f (z0 ) = h(w0 ),
Dh(w0 ) Df (z0 )(z0 ), (w0 ) = t,
∂z
and
∂ρ
−1 2
Re Dh(w0 ) D f (z0 )(z0 , z0 ), (w0 ) ≥ −t.
∂z
2nd Order Differential Subordinations
of Holomorphic Mappings
Yu-Can Zhu and Ming-Sheng Liu
vol. 8, iss. 4, art. 104, 2007
2
Set α = f (z0 ), β = Df (z0 )(z0 ), γ = D f (z0 )(z0 , z0 ), then according to the definition of Ψ(Ω1 , h), we have
Title Page
2
ψ(f (z0 ), Df (z0 )(z0 ), D f (z0 )(z0 , z0 ), z0 ) ∈
/ Ω1
Contents
which contradicts (2.12). Hence f (z) ≺ h(z), and the proof of Theorem 2.8 is
complete.
JJ
II
Theorem 2.9. Let A ≥ 0, h ∈ H(Ω, Cn ) be biholomorphic convex with h(0) = 0,
and let ψ(z) ∈ H(Ω, Cn ) with ψ(0) = 0. Suppose that ρ(z) is twice differentiable
in Ω − {0}, k > 4kDh(0)−1 k, and ϕ, φ : Ω1 × Ω → C are holomorphic such that
J
I
Page 18 of 30
Re φ(α, z) ≥ A + |ϕ(α, z) − 1| − Re[ϕ(α, z) − 1] + kρ(ψ(z))
Full Screen
for all (α, z) ∈ h(Ω) × Ω, where Ω1 is a domain of Cn with h(Ω) ⊂ Ω1 and
kDh(0)−1 k = sup ρ(Dh(0)−1 (ξ)). If f ∈ H(Ω, Cn ) with f (0) = 0 satisfies
Close
ρ(ξ)≤1
AD2 f (z)(z, z) + φ(f (z), z)Df (z)(z) + ϕ(f (z), z)f (z) + ψ(z) ≺ h(z),
then f (z) ≺ h(z).
Go Back
Proof. Without loss of generality, we may assume that f and g satisfy the conditions
of Theorem 2.9 on Ω. If not, then we can replace f (z) by fr (z) = f (rz), ψ(z) by
ψr (z) = ψ(rz), and h(z) by hr (z) = h(rz), where 0 < r < 1. We would then prove
fr (z) ≺ hr (z) for all 0 < r < 1. By letting r → 1− , we obtain f (z) ≺ h(z).
Let
ψ(α, β, γ, z) = Aγ + φ(α, z)β + ϕ(α, z)α + ψ(z),
E
D
E
D
∂ρ
−1
(w)
≥ −t,
and let α = h(w), Dh(w)−1 (β), ∂ρ
(w)
=
t,
Re
Dh(w)
(γ),
∂z
∂z
where ρ(w) = 1, t ≥ 1/2. If we set
2nd Order Differential Subordinations
of Holomorphic Mappings
Yu-Can Zhu and Ming-Sheng Liu
vol. 8, iss. 4, art. 104, 2007
ψ(α, β, γ, z) = h(w) + λDh(w)(w),
Title Page
then we have
Contents
λw = ADh(w)−1 (γ) + φ(α, z)Dh(w)−1 (β)
+ [ϕ(α, z) − 1]Dh(w)−1 h(w) + Dh(w)−1 (ψ(z)).
E
D
(w)
= ρ(w) = 1 from Remark 1, we obtain
Since 2 w, ∂ρ
∂z
∂ρ
∂ρ
−1
(2.13) λ = 2A Dh(w) (γ), (w) + 2φ(α, z) Dh(w) (β), (w)
∂z
∂z
∂ρ
∂ρ
−1
−1
+ 2[ϕ(α, z) − 1] Dh(w) h(w), (w) + 2 Dh(w) (ψ(z)), (w) .
∂z
∂z
−1
By Lemma 2.3, we have
2 Dh(w)−1 h(w), ∂ρ (w) − 1 ≤ 1.
∂z
JJ
II
J
I
Page 19 of 30
Go Back
Full Screen
Close
By Lemma 2.6, we obtain
Re λ ≥ −2At + 2t Re φ(α, z) + Re[ϕ(α, z) − 1]
− |ϕ(α, z) − 1| − 4kDh(0)−1 kρ(ψ(z))
≥ (2t − 1){|ϕ(α, z) − 1| − Re[ϕ(α, z) − 1]}
+ (k − 4kDh(0)−1 k)ρ(ψ(z)) ≥ 0.
(2.14)
Now we verify that ψ(α, β, γ, z) ∈
/ h(Ω). Suppose not, then there exists w1 ∈ Ω
such that ψ(α, β, γ, z) = h(w1 ). From the result in [3, 7, 18, 19], we have
∂ρ
−1
− Re λ = 2 Re Dh(w) (h(w) − h(w1 )), (w) > 0,
∂z
which contradicts (2.14), hence ψ(α, β, γ, z) ∈
/ h(Ω). By Theorem 2.8, we obtain
f (z) ≺ h(z), and the proof is complete.
2nd Order Differential Subordinations
of Holomorphic Mappings
Yu-Can Zhu and Ming-Sheng Liu
vol. 8, iss. 4, art. 104, 2007
Title Page
Contents
JJ
II
J
I
Corollary 2.10. Let A ≥ 0, h ∈ H(Ω, Cn ) be biholomorphic convex with h(0) = 0,
and let ψ(z) ∈ H(Ω, Cn ) with ψ(0) = 0. Suppose that k > 4kDh(0)−1 k, ρ(z) is
twice differentiable in Ω − {0}, and B(z), C(z) ∈ H(Ω, C) satisfy
Page 20 of 30
Re B(z) ≥ A + |C(z) − 1| − Re[C(z) − 1] + kρ(ψ(z))
Go Back
for all z ∈ Ω, where kDh(0)−1 k = sup ρ(Dh(0)−1 (ξ)). If f ∈ H(Ω, Cn ) with
Full Screen
ρ(ξ)≤1
f (0) = 0 satisfies
AD2 f (z)(z, z) + B(z)Df (z)(z) + C(z)f (z) + ψ(z) ≺ h(z),
then f (z) ≺ h(z).
Close
Corollary 2.11. Let A ≥ 0, h ∈ H(Ω, Cn ) be biholomorphic convex. Suppose that
ρ(z) is twice differentiable in Ω − {0}, and B(z) ∈ H(Ω, C) with Re B(z) ≥ A for
all z ∈ Ω. If f ∈ H(Ω, Cn ) with f (0) = h(0) satisfies
AD2 f (z)(z, z) + B(z)Df (z)(z) + f (z) ≺ h(z),
then f (z) ≺ h(z).
Corollary 2.12. Let h ∈ H(Ω, Cn ) be biholomorphic convex with h(0) = 0. Suppose that ρ(z) is twice differentiable in Ω − {0} and φ : Ω1 → C is holomorphic
such that Re φ(h(z)) ≥ 0 for all z ∈ Ω. If f ∈ H(Ω, Cn ) with f (0) = 0 satisfies
2nd Order Differential Subordinations
of Holomorphic Mappings
Yu-Can Zhu and Ming-Sheng Liu
vol. 8, iss. 4, art. 104, 2007
f (z) + φ(f (z))Df (z)(z) ≺ h(z),
then f (z) ≺ h(z).
Remark 3. When n = 1, we have Df (z)(z) = zf 0 (z) and D2 f (z)(z, z) = z 2 f 00 (z).
From Corollary 2.10, we may obtain Theorem 2 in [13], Theorem 3.1a in [15], Theorem 1 for case 1 in [12] and Theorem 1 in [14]. From Corollary 2.11, we may
obtain Corollary 2.1 in [13].
Example 2.1. Let β > 0 and γ ∈ C with 2 Re γ ≥ β. The unit ball in Cn is denoted
z
by B = {z ∈ Cn : kzk < 1}. If u ∈ Cn with kuk = 1, then h(z) = 1−hz,ui
is a
biholomorphic convex mapping on B (see [17]). By a simple calculation, we have
z
(2.15)
Re β
,u + γ
1 − hz, ui
Re γ|1 − hz, ui|2 + β[Rehz, ui − |hz, ui|2 ]
=
|1 − hz, ui|2
β|1 − hz, ui|2 + 2β[Rehz, ui − |hz, ui|2 ]
≥
2|1 − hz, ui|2
Title Page
Contents
JJ
II
J
I
Page 21 of 30
Go Back
Full Screen
Close
β(1 − |hz, ui|2 )
=
>0
2|1 − hz, ui|2
for all z ∈ B. If f ∈ H(B, Cn ) with f (0) = 0, then by Corollary 2.12, we have
f (z) +
Df (z)(z)
z
z
≺
=⇒ f (z) ≺
.
βhf (z), ui + γ
1 − hz, ui
1 − hz, ui
Example 2.2. Let A ≥ 0, β ≥ 0 and γ ∈ C with Re γ ≥ β/2 + A, u ∈ Cn with
kuk = 1. If f ∈ H(B, Cn ) with f (0) = 0, then by Theorem 2.9, Corollary 2.11 and
(2.15), we have
hz, ui
2
AD f (z)(z, z) + β
+ γ Df (z)(z) + f (z)
1 − hz, ui
z
z
≺
=⇒ f (z) ≺
,
1 − hz, ui
1 − hz, ui
and
AD2 f (z)(z, z)+[βhf (z), ui+γ]Df (z)(z)+f (z) ≺
z
z
=⇒ f (z) ≺
.
1 − hz, ui
1 − hz, ui
2nd Order Differential Subordinations
of Holomorphic Mappings
Yu-Can Zhu and Ming-Sheng Liu
vol. 8, iss. 4, art. 104, 2007
Title Page
Contents
JJ
II
J
I
Page 22 of 30
Go Back
Let ρ(z) be differentiable in Ω − {0}. For M > 0, we define Ψ(M ) to be the
class of maps ψ : Cn × Cn × Cn × Ω → Cn that satisfy the following conditions:
Full Screen
Close
1. ρ(ψ(0, 0, 0, 0)) < M and
D
2. ρ(ψ(α, β, γ, z)) ≥ M for all ρ(α) = M, 2 β,
D
E
(α)
≥ (t2 − t)M, and t ≥ 1.
2 Re γ, ∂ρ
∂z
E
∂ρ
(α)
∂z
= tM,
Theorem 2.13. Let ψ ∈ Ψ(M ). If w ∈ H(Ω, Cn ) with w(0) = 0 satisfies
(2.16)
ρ(ψ(w(z), Dw(z)(z), D2 w(z)(z, z), z)) < M
for all z ∈ Ω − {0}, then ρ(w(z)) < M for z ∈ Ω.
Proof. Suppose that the conclusion of Theorem 2.13 is false. Then there exists a
point z0 ∈ Ω − {0} such that ρ(w(z0 )) = M and ρ(w(z)) ≤ M for ρ(z) ≤ ρ(z0 ). It
implies w0 = w(z0 ) 6= 0.
Let
z0
∂ρ
ξ ∈ C.
ϕ(ξ) = 2 w
ξ , (w0 ) ,
∂z
ρ(z0 )
2nd Order Differential Subordinations
of Holomorphic Mappings
Yu-Can Zhu and Ming-Sheng Liu
vol. 8, iss. 4, art. 104, 2007
Then ϕ(ξ) is an analytic function in |ξ| < 1. By Lemma 2.5, we have
z
∂ρ
z
0
0
ξ
≤ ρ(w(z0 ))
ξ , (w0 ) ≤ ρ w
|ϕ(ξ)| ≤ 2 w
∂z
ρ(z0 )
ρ(z0 )
JJ
II
for all |ξ| ≤ ρ(z0 ), and
J
I
∂ρ
ϕ(ρ(z0 )) = 2 w(z0 ), (w0 ) = ρ(w(z0 )) = max |ϕ(ξ)|.
|ξ|≤ρ(z0 )
∂z
By a simple calculation, we have
∂ρ
ρ(z0 )ϕ (ρ(z0 )) = 2 Dw(z0 )(z0 ), (w0 ) ,
∂z
∂ρ
2 00
2
ρ(z0 ) ϕ (ρ(z0 )) = 2 D w(z0 )(z0 , z0 ), (w0 ) .
∂z
0
Title Page
Contents
Page 23 of 30
Go Back
Full Screen
Close
Using Lemma A in [10] (also see [15, p. 19]), there exists a real number t ≥ 1 such
that
∂ρ
2 Dw(z0 )(z0 ), (w0 ) = tM,
∂z
∂ρ
2
2 Re D w(z0 )(z0 , z0 ), (w0 ) ≥ (t2 − t)M.
∂z
By the definition of ψ, we have
2nd Order Differential Subordinations
of Holomorphic Mappings
Yu-Can Zhu and Ming-Sheng Liu
2
ρ(ψ(w(z0 ), Dw(z0 )(z0 ), D w(z0 )(z0 , z0 ), z0 )) ≥ M,
vol. 8, iss. 4, art. 104, 2007
which contradicts (2.16). Hence ρ(w(z)) < M for z ∈ Ω, and the proof is complete.
Title Page
Theorem 2.14. Let ρ(z) be differentiable in Ω−{0}. Suppose that A(z), B(z), C(z) ∈
H(Ω, C) with A(z) 6= 0 for all z ∈ Ω satisfy
B(z)
1
C(z) ρ(ϕ(z))
(2.17)
Re
≥ max −1,
− Re
+
,
A(z)
|A(z)|
A(z)
|A(z)|
or
C(z)
1
ρ(ϕ(z))
(2.18)
Re
≥
+
+1
A(z)
|A(z)|
|A(z)|
s
C(z)
1
ρ(ϕ(z))
B(z)
and 1 − 2 Re
−
−
≤ Re
≤ −1
A(z) |A(z)|
|A(z)|
A(z)
for all z ∈ Ω. If w(z) ∈ H(Ω, Cn ) with w(0) = 0 satisfies
ρ(A(z)D2 w(z)(z, z) + B(z)Dw(z)(z) + C(z)w(z) + ϕ(z)) < 1
for all z ∈ Ω, then ρ(w(z)) < 1 for z ∈ Ω.
Contents
JJ
II
J
I
Page 24 of 30
Go Back
Full Screen
Close
Proof. Let
ψ(α, β, γ, z) = A(z)γ + B(z)β + C(z)α + ϕ(z),
D
E
D
E
∂ρ
where ρ(α) = 1, 2 β, ∂ρ
(α)
=
t,
2
Re
γ,
(α)
≥ (t2 − t) and t ≥ 1. From
∂z
∂z
(2.9) and (2.10), we have
−iθ ∂ρ
ρ(ψ(α, β, γ, z)) ≥ 2 ψ(α, β, γ, z)e , (α) ∂z
∂ρ
∂ρ
−iθ
= |A(z)|2 γ, (α) + B(z)e 2 β, (α)
∂z
∂z
∂ρ
∂ρ
−iθ
−iθ
+ C(z)e 2 α, (α) + 2e
ϕ(z), (α) ∂z
∂z
B(z)
C(z) ρ(ϕ(z))
2
≥ |A(z)| t + t Re
− 1 + Re
−
,
A(z)
A(z)
|A(z)|
where θ = arg A(z). Let
B(z)
C(z) ρ(ϕ(z))
2
L(t) = t + t Re
− 1 + Re
−
A(z)
A(z)
|A(z)|
for t ≥ 1. Then we have
Yu-Can Zhu and Ming-Sheng Liu
vol. 8, iss. 4, art. 104, 2007
Title Page
Contents
JJ
II
J
I
Page 25 of 30
Go Back
Full Screen
B(z)
L0 (t) = 2t + Re
− 1.
A(z)
If Re B(z)
≥ −1 for z ∈ Ω, then L0 (t) ≥ Re B(z)
+ 1 ≥ 0. Hence we obtain
A(z)
A(z)
min L(t) = L(1) = Re
t≥1
2nd Order Differential Subordinations
of Holomorphic Mappings
B(z)
C(z) ρ(ϕ(z))
1
+ Re
−
≥
.
A(z)
A(z)
|A(z)|
|A(z)|
Close
It follows that ρ(ψ(α, β, γ, z)) ≥ 1.
If Re B(z)
≤ −1 for z ∈ Ω, then
A(z)
1
B(z)
min L(t) = L
1 − Re
t≥1
2
A(z)
2
1
B(z)
C(z) ρ(ϕ(z))
=−
Re
− 1 + Re
−
4
A(z)
A(z)
|A(z)|
1
≥
.
|A(z)|
It also follows that ρ(ψ(α, β, γ, z)) ≥ 1.
Hence we have ψ ∈ Ψ(1). From Theorem 2.13, we obtain ρ(w(z)) < 1 for
z ∈ Ω.
2nd Order Differential Subordinations
of Holomorphic Mappings
Yu-Can Zhu and Ming-Sheng Liu
vol. 8, iss. 4, art. 104, 2007
Title Page
Contents
Remark 4. Setting n = 1, ϕ(z) ≡ 0, A(z) ≡ A and C(z) = 1 − B(z) in Theorem
2.14, we get Theorem 4 in [13].
JJ
II
Corollary 2.15. Suppose that B(z) ∈ H(B, C) and A ≥ 0 satisfy Re B(z) ≥ 0 for
all z ∈ B. If w(z) ∈ H(B, Cm ) with w(0) = 0 satisfy
J
I
Page 26 of 30
kAD2 w(z)(z, z) + B(z)Dw(z)(z) + w(z)k < 1
Go Back
Full Screen
for all z ∈ B, then kw(z)k < 1 for z ∈ B.
Example 2.3. Let k and n1 be positive integers and let α = (α1 , α2 , . . . , αm ) ∈ Cm .
k
We define αk = (α1k , α2k , . . . , αm
). Suppose A1 , A2 , . . . , An1 ∈ H(B, C) (n1 ≥ 2)
satisfy
n1
X
Re A1 (z) ≥
|Ak (z)|
k=2
Close
for z ∈ B, where B is the unit ball in Cn . If w(z) = (w1 (z), w2 (z), . . . , wm (z)) ∈
H(B, Cm ) with w(0) = 0 satisfies
2
n1
m X
n
n
2
X
X
X
∂
w
(z)
∂w
(z)
υ
υ
zj zl +
zj +
Ak (z)[wυ (z)]k < 1
∂z
∂z
∂z
j
l
j
υ=1 j,l=1
j=1
k=1
for all z ∈ B, then
In fact, if we let
Pm
υ=1
2nd Order Differential Subordinations
of Holomorphic Mappings
|wυ (z)|2 < 1 for all z ∈ B.
Yu-Can Zhu and Ming-Sheng Liu
ψ(α, β, γ, z) = γ + β +
n1
X
vol. 8, iss. 4, art. 104, 2007
Ak (z)αk
k=1
Title Page
for kαk = 1, hβ, αi = t, Rehγ, αi ≥ t2 − t and t ≥ 1, then we have
n1
X
k
kψ(α, β, γ, z)k ≥ hγ, αi + hβ, αi + A1 (z) +
Ak (z)hα , αi
Contents
k=2
≥ Re{hγ, αi + hβ, αi + A1 (z)} −
n1
X
k=2
≥ t2 + Re A1 (z) −
n1
X
|Ak (z)|
m
X
!
|αj |k+1
j=1
|Ak (z)| ≥ 1.
k=2
qP
Hence ψ ∈ Ψ(1) for ρ(z) =
Pm
2
υ=1 |wυ (z)| < 1 for all z ∈ B.
n
j=1
|zj |2 . According to Theorem 2.13, we have
JJ
II
J
I
Page 27 of 30
Go Back
Full Screen
Close
References
[1] J.B. CONWAY, Functions of One Complex Variable (Second Edition),
Springer-Verlag, New York, 1978.
[2] I. GRAHAM AND G. KOHR, Geometric Function Theory in One and Higher
Dimensions, Dekker, New York, 2003.
[3] S. GONG, Convex and Starlike Mappings in Several Complex Variables, Science Press/Kluwer Academic Publishers, 1998.
2nd Order Differential Subordinations
of Holomorphic Mappings
Yu-Can Zhu and Ming-Sheng Liu
vol. 8, iss. 4, art. 104, 2007
[4] S. GONG, A note on partial differential inequalities, Chin. Ann. of Math., 5A(6)
(1984), 771–780 (Chinese).
Title Page
[5] S. GONG AND T. LIU, Distortion theorems for biholomorphic convex mappings on bounded convex circular domains, Chin. Ann. of Math., 20B(3)
(1999), 297–304.
[6] S. GONG AND S.S. MILLER, Partial differential subordinations and inequalities define on complete circular domains, Comm. Partial Diff. Equation, 11(11)
(1986), 1243–1255.
[7] S. GONG AND T. LIU, Criterion for the family of ε starlike mappings, J. Math.
Anal. Appl., 274 (2002), 696–704.
[8] H. HAMADA, G. KOHR AND M. KOHR, First order partial differential subordinations on bounded balanced pseudoconvex domains in Cn , Mathematica
(Cluj), 41(64) (1999), 161–175.
[9] T. LIU AND G. REN, The growth theorem of convex mapping on bounded
convex circular domains, Science in China (Series A), 41(2) (1998), 123–130.
Contents
JJ
II
J
I
Page 28 of 30
Go Back
Full Screen
Close
[10] S.S. MILLER AND P.T. MOCANU, Second order differential inequalities in
the complex plane, J. Math. Anal. Appl., 65 (1978), 289–305.
[11] S.S. MILLER AND P.T. MOCANU, Differential subordinations and univalent
functions, Michigan Math. J., 28 (1981), 157–171.
[12] S.S. MILLER AND P.T. MOCANU, On some classes of first-order differential
subordinations, Michigan Math. J., 35 (1985), 185–195.
[13] S.S. MILLER AND P.T. MOCANU, Differential subordinations and inequalities in the complex plane, J. Diff. Equation, 67(2) (1987), 199–211.
[14] S.S. MILLER AND P.T. MOCANU, Averaging operator and generalized Robinson differential inequality, J. Math. Anal. Appl., 173 (1993), 459–467.
[15] S.S. MILLER AND P.T. MOCANU, Differential Subordinations Theory and
Applications, New York, Marcel Dekker Inc. 2000.
[16] P.T. MOCANU, Second order averaging operators for analytic functions, Rev.
Roumaine Math. Appl., 33(10) (1988), 875–881.
[17] K. ROPER AND T. SUFFRIDGE, Convex mappings on the unit ball of Cn , J.
d’Analyse Math., 65 (1995), 333–347.
[18] T.J. SUFFRIDGE, Starlike and convex maps in Banach spaces, Pacific J. Math.,
46(2) (1973), 475–489.
[19] T.J. SUFFRIDGE, Starlikeness, convexity and other geometric properties of
holomorphic maps in higher dimensions, Lecture Notes in Math., 599 (1976),
146–159.
[20] A.E. TAYLOR AND D.C. LAY, Introduction to Functional Analysis, New York:
John Wiley & Sons Inc., 1980, 111–115.
2nd Order Differential Subordinations
of Holomorphic Mappings
Yu-Can Zhu and Ming-Sheng Liu
vol. 8, iss. 4, art. 104, 2007
Title Page
Contents
JJ
II
J
I
Page 29 of 30
Go Back
Full Screen
Close
[21] YUCAN ZHU AND HEZENG LIN, First-order differential subordinations,
Acta Sci. Nature Unvi. Sunyatseni, 30(4) (1991), 114–118 (Chinese).
[22] YUCAN ZHU, Differential subordinations of holomorphic mappings in Banach spaces, J. of Fuzhou University, 30(4) (2002), 430–434 (Chinese).
[23] YUCAN ZHU, Criteria of biholomorphic convex mappings on bounded convex
balanced domains, Acta Math. Sinica, 46(6) (2003), 1153–1162 (Chinese).
2nd Order Differential Subordinations
of Holomorphic Mappings
Yu-Can Zhu and Ming-Sheng Liu
vol. 8, iss. 4, art. 104, 2007
Title Page
Contents
JJ
II
J
I
Page 30 of 30
Go Back
Full Screen
Close