Table Of ContentUNIFORMLY LOCALLY UNIVALENT HARMONIC MAPPINGS
SAMINATHAN PONNUSAMY, JINJING QIAO †, AND XIANTAO WANG
6
1
0
Abstract. Theprimaryaimofthispaperistocharacterizetheuniformlylocally
2
univalent harmonic mappings in the unit disk. Then, we obtain sharp distor-
n tion, growth and covering theorems for one parameter family H(λ) of uniformly
a B
locally univalent harmonic mappings. Finally, we show that the subclass of k-
J
6 quasiconformal harmonic mappings in BH(λ) and the class BH(λ) are contained
in the Hardy space of a specific exponent depending on the λ, respectively, and
] we also discuss the growth of coefficients for harmonic mappings in H(λ).
V B
C
.
h
1. Introduction
t
a
m
The class of complex-valued harmonic mappings f defined on a simply connected
[ domain D of the complex plane C has attracted the attention of function theorists
1 because it generalizes the class of analytic functions with a lot of rich applications
v in many different fields. Every such f has the canonical decomposition f = h+g,
9
where bothhandg areanalytic inD andg(z ) = 0for someprescribed point z D
3 0 0
∈
1 (cf. [10, 13]). For a complex-valued and continuously differentiable mapping f, we
1
let
0
. λ = f f and Λ = f + f
1 f | z|−| z| f | z| | z|
0 so that the Jacobian J of f takes the form
f
6
1 J = λ Λ = f 2 f 2.
f f f z z
: | | −| |
v Moreover, a necessary and sufficient condition for harmonic mappings f = h + g
i
X to be locally univalent and sense preserving in D is that J = h 2 g 2 > 0, or
f ′ ′
| | −| |
r equivalently, its dilatation the dilatation ω (z) = g (z)/h(z) satisfies the inequality
a f ′ ′
ω (z) < 1 for z D. (see [17] and [10, 13, 22]). Let k [0,1) be a constant.
f
| | ∈ ∈
Then, we say that a sense preserving harmonic mapping f = h + g in D is a k-
quasiconformal mapping if ω (z) k holds in D.
f
| | ≤
Throughout this paper, we consider harmonic mappings defined on the unit disk
D = z C : z < 1 . Denote by the class of harmonic mappings f = h+g in
D suc{h t∈hat h(|0)| = g(}0) = h(0) 1H= 0 and consider the family
′
−
= f : f is sense-preserving and univalent in D .
H
S { ∈ H }
Often it is convenient to work with
0 = f : f (0) = 0 .
SH { ∈ SH z }
2000 Mathematics Subject Classification. Primary: 30C65, 30C45; Secondary: 30C20, 30C50,
30C80.
Key words and phrases. Harmonic mapping, Pre-Schwarzianderivatives,uniformly locally uni-
valence, growth estimate, coefficient estimate, harmonic Bloch space, Hardy space.
† Corresponding author.
1
2 S. Ponnusamy,J. Qiao and X.Wang
Although both the families and 0 are known to be normal, only 0 is compact
(see [10]). We also denote thSeHclass SofHanalytic functions f in D with fS(H0) = f (0)
′
−
1 = 0 by so that reduces to when the co-analytic part g of f = h+g
vanishes iAdentically Hin D. Then tAhe set := 0 of all normalized univa∈lenHt
analytic functions in D is the central obSject iAn ∩thSeHstudy of geometric function
theory so that 0 .
We denote dS(z⊂,wS)Ha⊂s tShHe hyperbolic distance of z,w D, that is,
h
∈
1 1+ z−w
d (z,w) = log 1 zw .
h 2 1 z−w
(cid:12) − (cid:12)!
−(cid:12)1−zw(cid:12)
We say that a harmonic mapping f = h+g in D(cid:12) is un(cid:12)iformly locally univalent if f
(cid:12) (cid:12)
is univalent in each hyperbolic disk
D (a,ρ) = z D : d (z,a) < ρ
h h
{ ∈ }
with center a D and hyperbolic radius ρ (independent of the center), 0 < ρ .
∈ ≤ ∞
The subscript h in d and D should not be confused with the analytic part h of
h h
the harmonic mapping f.
If f is analytic in the above definition, then it reduces to the uniformly locally
univalent (analytic) functions. We know that a holomorphic universal covering map
of a plane domain D is uniformly locally univalent if and only if the boundary of D
is uniformly perfect (cf. [20, 23]). Also, in [24], Yamashita showed that an analytic
function f in D is uniformly locally univalent in D if and only if the pre-Schwarzian
derivative T = f /f of f is hyperbolically bounded, i.e., the norm
f ′′ ′
T = sup(1 z 2) T (z)
f f
k k z D −| | | |
∈
is finite and this means that logf belongs to the Bloch space B (cf. [3, 12]).
′
In Section 2 (see Theorem 2.1), we characterize the uniformly locally univalent
harmonic mappings f = h+g in terms of the pre-Schwarzian derivative of h+eiθg
for each θ [0,2π]. This result and the corresponding results in [15] helps to obtain
∈
sharp distortion, growth and covering theorems (see Section 3) for the class (λ),
H
B
where λ is a positive real number, and
(λ) = f = h+g : T 2λ
H f
B { ∈ H k k ≤ }
with
h (z)+eiθg (z)
(1.1) T := sup (1 z 2) ′′ ′′ .
k fk z D, θ [0,2π] −| | (cid:12) h′(z)+eiθg′(z) (cid:12)
∈ ∈
(cid:12) (cid:12)
Henceforth, T is defined by (1.1) in the ca(cid:12)se of harmonic m(cid:12)appings f = h + g
inD. k fk (cid:12) (cid:12)
It is known that for λ > 1, the class (λ) and the subclass (λ) are contained
B B ∩S
in the Hardy space Hp with 0 < p < 1/(λ2 1) and 0 < p < 1/(λ 1), respectively
− −
(cf. [14, 16]).
In Section 4, we consider relationships between the space (λ) and the harmonic
H
B
Hardy space. We also prove that a k-quasiconformal harmonic mapping f (λ)
H
∈ B
(λ > 1) is contained in the harmonic Hardy space hp with 0 < p < 1/(λ 1), and
−
Uniformly locally univalent harmonic mappings 3
also obtain that (λ) hp with 0 < p < 1/(λ2 1). Finally, in the last section,
H
B ⊂ −
as applications of distortion estimate obtained in Section 3, we discuss the growth
of coefficients for harmonic mappings in (λ) (λ > 1).
H
B
In [15], the authors discussed the set (λ) := (λ) and obtained distortion
H
B A∩B
estimates for analytic functions in (λ) in terms of λ, and characterization for
B
functions in (λ) (cf. [15, Proposition 1.1]). As a consequence of Theorem 2.1 in
B
Section2and[15, Proposition1.1], wecaneasilyobtainthefollowingcorollarywhich
characterizes harmonic mappings in (λ). We omit its proof and this particular
H
B
case is indeed a generalization of earlier known result (see [15, Proposition 1.1]) to
the case of harmonic mappings.
Corollary 1.1. A locally univalent harmonic mapping f = h + g belongs to
(λ) if and only if, for each pair of points z , z in D and θ [0,2∈π]H,
H 1 2
B ∈
u (z ) u (z ) 2λd (z ,z ),
θ 1 θ 2 h 1 2
| − | ≤
where u (z) = log h(z)+eiθg (z) .
θ ′ ′
(cid:0) (cid:1)
2. Characterizations of Uniformly locally univalent harmonic
mappings
We now state our first result which is indeed a generalization of [24, Theorem 1]
to the case of harmonic mappings.
Theorem 2.1. A harmonic mapping f = h+g is uniformly locally univalent in D
if and only if T < .
f
k k ∞
For the proof of the sufficiency of Theorem 2.1, we need the following classical
result due to Noshiro [19].
Lemma A. Let f(z) = z + ∞k=2akzk be analytic for |z| < R and |f′(z)| < M
for z < R. Then the disk z < R/M is mapped on a starlike domain with respect
to t|he| origin by f and also| b|Py all its polynomial sections f (z) = z + n a zk
n k=2 k
(n = 2,3,...).
P
Proof of Theorem 2.1. Let f = h+g be harmonic in D and assume that T <
f
. Define F(ξ) = (f T)(ξ) for ξ D, where k k
∞ ◦ ∈
Rξ +a
w = T(ξ) =
1+aRξ
with R = tanhρ, the constants a D and ρ (0 < ρ ). Then F = H + G is
harmonic in D. Elementary compu∈tations yield ≤ ∞
H (ξ)+eiθG (ξ) h (w)+eiθg (w) T (ξ)
′′ ′′ ′′ ′′ ′′
= T (ξ)+ ,
′
H (ξ)+eiθG(ξ) h(w)+eiθg (w) T (ξ)
′ ′ ′ ′ ′
where
R(1 a 2) 2a(1 a 2)R2
T (ξ) = −| | and T (ξ) = −| |
′ ′′
(1+aRξ)2 − (1+aRξ)3
4 S. Ponnusamy,J. Qiao and X.Wang
so that
T (ξ) 2aR
′′
= .
T (ξ) −1+aRξ
′
Since
(1 a 2)(1 ξ 2R2)
1 w 2 = −| | −| | ,
−| | 1+aRξ 2
| |
we easily have T (ξ) (1 ξ 2) 1 w 2 and therefore, it follows that
′
| | −| | ≤ −| |
H (ξ)+eiθG (ξ) h (w)+eiθg (w)
(1 ξ 2) ′′ ′′ (1 w 2) ′′ ′′
−| | H (ξ)+eiθG(ξ) ≤ −| | h(w)+eiθg (w)
(cid:12) ′ ′ (cid:12) (cid:12) ′ ′ (cid:12)
(cid:12) (cid:12) (cid:12) 2aR (cid:12)
(cid:12) (cid:12) +(1 ξ (cid:12)2) . (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
−| | 1+aRξ
(cid:12) (cid:12)
(cid:12) (cid:12)
This inequality implies that (cid:12) (cid:12)
(cid:12) (cid:12)
H (ξ)+eiθG (ξ)
(2.1) sup(1 ξ 2) ′′ ′′ k < ,
ξ D −| | (cid:12) H′(ξ)+eiθG′(ξ) (cid:12) ≤ 0 ∞
∈
(cid:12) (cid:12)
where (cid:12) (cid:12)
(cid:12) h (w)+eiθg ((cid:12)w) 2R
k = sup(1 w 2) ′′ ′′ + .
0
w∈D −| | (cid:12) h′(w)+eiθg′(w) (cid:12) 1−R
Let ϕ be an analytic branch of log(cid:12) H (ξ)+eiθG(ξ)(cid:12) in D. Then
(cid:12) ′ ′ (cid:12)
(cid:12) (cid:12)
H(cid:0)′′(ξ)+eiθG′′(ξ) (cid:1)
ϕ(ξ) = .
′
H (ξ)+eiθG(ξ)
′ ′
This choice is clearly possible, because H (ξ)+eiθG(ξ) = 0 for ξ D, by (2.1). It
′ ′
6 ∈
then follows from (2.1) that
H (ξ)+eiθG(ξ) k 1+ ξ
′ ′ 0
(2.2) log ϕ(ξ) ϕ(0) log | | .
H (0)+eiθG(0) ≤ | − | ≤ 2 1 ξ
(cid:12) (cid:12) ′ ′ (cid:12)(cid:12) (cid:18) −| |(cid:19)
(cid:12) (cid:12)
Now, we int(cid:12)rodu(cid:12)(cid:12)ce Hθ(ξ) by (cid:12)(cid:12)(cid:12)
(cid:12) (cid:12)
H(ξ)+eiθG(ξ)
H (ξ) = .
θ H (0)+eiθG(0)
′ ′
We see that H is analytic in D and is normalized so that H (0) 1 = 0. We infer
θ θ′ −
from (2.2) that
k 1
0
log H (ξ) log3 for ξ < ,
θ′ ≤ 2 | | 2
whence (cid:12) (cid:12)
(cid:12) (cid:12) 1
H (ξ) < 3k0/2 for ξ < .
θ′ | | 2
Therefore, byLemma AofN(cid:12)oshiro(cid:12), H (ξ) H (0)isunivalent inthedisk ξ < 3−k0/2
(cid:12) (cid:12) θ − θ | | 2
for each θ. The radius of convexity for univalent functions is known to be 2 √3
−
(cf. [13, Theorem 2.13]). Thus, H (ξ) H (0) and H(ξ) + eiθG(ξ) are convex in
θ θ
ξ < (2 √3)3−k0/2 = ρ . This implies−that F is harmonic (convex) univalent in
| | − 2 0
ξ < (2 √3)3−k0/2 (cf. [10]).
| | − 2
Uniformly locally univalent harmonic mappings 5
Consequently, f is univalent in the hyperbolic disk D (a,ρ ) with tanhρ = (2
h 0 0
√3)3−k0/2 tanhρ if (2 √3)3−k0/2 1, and ρ = ρ if (2 √3)3−k0/2 > 1. Hence,−f
2 − 2 ≤ 0 − 2
is uniformly locally univalent.
To prove the necessity, we assume that f is uniformly locally univalent in D, that
is, f is univalent in each hyperbolic disk D (a,ρ), where a D and 0 < ρ .
h
∈ ≤ ∞
Again, as above, we consider w = T(ξ) and let
F(ξ) = (f T)(ξ) = f(w) for ξ D.
◦ ∈
Then F = H + G is univalent in D. By Lewy’s Theorem (cf. [13]), the Jacobian
J is different from 0 for all z D, and then, without loss of generality, we assume
F
∈
that F is sense-preserving. Let
H(ξ) H(0) G(ξ) G(0)
F (ξ) = − + − = H (ξ)+G (ξ).
0 0 0
H (0) H (0)
′ ′
Obviously, F . For ξ D, set
0 H
∈ S ∈
F z+ξ F (ξ)
0 0
1+ξz −
F (z) = = H (z)+G (z),
1 (1(cid:16) ξ(cid:17)2)H (ξ) 1 1
−| | 0′
which again belongs to . The analytic function H (z) has the form
H 1
S
H (z) = z +A (ξ)z2 +A (ξ)z3 +
1 2 3
···
and a direct computation shows that
1 H (ξ) 1 H (ξ)
A (ξ) = (1 ξ 2) 0′′ 2ξ = (1 ξ 2) ′′ 2ξ .
2
2 −| | H (ξ) − 2 −| | H (ξ) −
(cid:26) 0′ (cid:27) (cid:26) ′ (cid:27)
Let α = sup{|a2| : f(z) = ∞k=1akzk + ∞k=1bkzk ∈ SH}. For f ∈ SH, we have
f(z1)−bb11f2(z) ∈ SH0 . It is knowPn that for f∗P(z) = ∞k=1a∗kzk + ∞k=1b∗kzk ∈ SH0 , the
coe−ffi|c|ient a < 49 and b < 1(cf. [13]). Using this estimate, by computations, it
| ∗2| | ∗2| 2 P P
is possible to get a < 98. It has been recently shown by Abu-Muhanna et al. [2]
2
| |
that a 16.5 which indeed the best known upper bound for a . Since F ,
| ∗2| ≤ | ∗2| 1 ∈ SH
we must have A (ξ) α and therefore,
2
| | ≤
H (ξ)
(1 ξ 2) ′′ < 2(α+1), ξ D.
−| | H (ξ) ∈
(cid:12) ′ (cid:12)
(cid:12) (cid:12)
For each c D, the compositio(cid:12)n of sen(cid:12)se preserving affine mapping φ(w) = w+cw
∈ (cid:12) (cid:12)
with F, namely, the function F +cF, is univalent and sense-preserving in D. Then
by what we have just proved, we obtain
H (ξ)+cG (ξ)
(1 ξ 2) ′′ ′′ < 2(α+1), ξ D,
−| | H (ξ)+cG(ξ) ∈
(cid:12) ′ ′ (cid:12)
(cid:12) (cid:12)
which in particular implies (cid:12)that, for each θ (cid:12) [0,2π],
(cid:12) ∈(cid:12)
H (ξ)+eiθG (ξ)
(1 ξ 2) ′′ ′′ < 2(α+1), ξ D.
−| | H (ξ)+eiθG(ξ) ∈
(cid:12) ′ ′ (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
6 S. Ponnusamy,J. Qiao and X.Wang
Thus, for f = h+g, we have
h (z)+eiθg (z)
A(θ) := sup(1 z 2) ′′ ′′ < .
z D −| | (cid:12) h′(z)+eiθg′(z) (cid:12) ∞
∈
(cid:12) (cid:12)
Since A(θ) is a continuous function of θ (cid:12)in [0,2π], it follow(cid:12)s from A(θ) < that
(cid:12) (cid:12) ∞
h (z)+eiθg (z)
sup (1 z 2) ′′ ′′ < .
z D, θ [0,2π] −| | (cid:12) h′(z)+eiθg′(z) (cid:12) ∞
∈ ∈
(cid:12) (cid:12) (cid:3)
The proof of the theorem is complete. (cid:12) (cid:12)
(cid:12) (cid:12)
3. Growth estimate for the class (λ)
H
B
For a nonnegative real number λ, we consider
z 1+t λ
H (z) = dt.
λ
1 t
Z0 (cid:16) − (cid:17)
It is easy to verify that T = 2λ, and thus H (λ). If λ 0, then it is
known that H is univalekntHinλkD if and only if 0 λλ ∈ B1H(see [15, Lem≥ma 2.1]). We
λ
≤ ≤
will see later that H is extremal in the class (λ). It follows from Theorem 2.1
λ H
B
that if f = h+g (λ), then h+eiθg (λ). This fact and [15, Theroem 2.3] give
∈ BH 1+eiθb1 ∈ B
the following result.
Theorem 3.1 (Distortiontheorem). Let λ be a nonnegative real number and f(z) =
h(z)+g(z) = ∞n=1anzn + ∞n=1bnzn ∈ BH(λ). Then for z ∈ D, we have
λ
P P 1 z
λ (z) = h(z) g (z) 1 b −| | = 1 b H ( z ),
f | ′ |−| ′ | ≥ | −| 1|| 1+ z | −| 1|| λ′ −| |
(cid:18) | |(cid:19)
(cid:12) (cid:12) (cid:12) (cid:12)
λ
(cid:12) (cid:12) (cid:12) (cid:12) 1+ z
Λ (z) = h(z) + g (z) (1+ b ) | | = (1+ b )H ( z )
f | ′ | | ′ | ≤ | 1| 1 z | 1| λ′ | |
(cid:18) −| |(cid:19)
and f(z(cid:12)) ((cid:12)1+ b )H ( z ). Furthermore, if f 0 (λ), then
| (cid:12) | ≤ (cid:12) | 1| λ | | ∈ SH ∩BH
H ( z ) f(z) H ( z ).
λ λ
− −| | ≤ | | ≤ | |
Equality occurs in each case when f(z) = µH (µz) for a unimodular constant µ.
λ
Corollary 3.1. For λ > 1, each f(z) = h(z) + g(z) = ∞n=1anzn + ∞n=1bnzn ∈
(λ) satisfies the growth condition
H
B P P
f(z) = O (1 z )1 λ
−
−| |
as z 1. On the other hand, for λ < 1, each mapping f (λ) is bounded with
(cid:0) (cid:1) H
the|b|ou→nd (1+ b )H (1). Moreover, if λ > 0 and f 0∈ B (λ) in D, then the
image f(D) con|ta1i|ns tλhe disk w : w < H ( 1) . ∈ SH ∩BH
λ
{ | | − − }
By [4, 5], for λ 1/2, (λ) and so, by Theorem 2.1, for λ 1/2, f (λ)
H
must be univalent≤in D. WBe al⊂soSnote that, for 0 λ 1, we hav≤e ∈ B
≤ ≤
H ( 1) H ( 1) = 2log2 1 = 0.38629...,
λ 1
− − ≥ − − −
and therefore the result is an improvement of the covering theorem for harmonic
mappings in 0 .
SH
Uniformly locally univalent harmonic mappings 7
In Corollary 3.1, the case λ = 1 is critical. By Theorem 3.1, we have that, for
f (1),
H
∈ B
f(z) (1+ b )H ( z ) = (1+ b ) 2log(1 z ) z ,
1 1 1
| | ≤ | | | | | | − −| | −| |
which shows thatfunctions in (1)need notbebounded. The next theorem, which
H (cid:0) (cid:1)
B
follows from Theorem 2.1 and [15, Proposition 2.5], gives a boundedness criterion
for mappings in (1).
H
B
Theorem 3.2. If a harmonic mapping f = h+g in D satisfies the condition
h (z)+eiθg (z) 1
lim (1 z 2) ′′ ′′ 2 log < 2
|z|→1−(cid:26) −| | (cid:12) h′(z)+eiθg′(z) (cid:12)− (cid:27) 1−|z|2 −
(cid:12) (cid:12)
for each θ [0,2π], then f is bounded. Here the constant 2 on the right hand side
(cid:12) (cid:12)
∈ (cid:12) (cid:12) −
is sharp.
We conclude this section with the H¨older continuity of mappings in (λ).
H
B
Theorem 3.3. Let 0 λ < 1. Then each mapping f (λ) is H¨older continuous
H
of exponent 1 λ in D≤. ∈ B
−
The proof follows from Theorem 2.1 and [15, Theorem 2.6] and so, we omit its
detail.
4. The space (λ) and the Hardy space
H
B
We begin this section with the following concepts.
Definition 4.1. For 0 < p < , the Hardy space Hp is the set of all functions f
analytic in D for which ∞
1 2π 1/p
M (r,f) = f(reiθ) pdθ
p
2π | |
(cid:26) Z0 (cid:27)
is bounded on 0 < r < 1.
The space hp consists of all harmonic mappings f in D for which M (r,f) (0 <
p
r < 1) are bounded (cf. [13]).
For a harmonic mapping f = h + g in D, the Bloch seminorm is given by (cf.
Colonna [11])
f B = sup(1 z 2) h′(z) + g′(z) ,
k k H z D −| | | | | |
∈
and f is called a Bloch mapping when f B(cid:0) < . In the(cid:1)recent years, the class
k k H ∞
of harmonic Bloch mappings has been studied extensively together with its higher
dimensional analog (see for example, [7, 8, 9, 11] and the references therein).
By Theorem 3.1, we have, for f (λ),
H
∈ B
z 1+t λ
| |
f(z) (1+ b ) dt,
1
| | ≤ | | 1 t
Z0 (cid:18) − (cid:19)
which shows that
f is bounded when λ < 1,
•
f(z) = O( log(1 z )) ( z 1) when λ = 1, and
• − −| | | | →
8 S. Ponnusamy,J. Qiao and X.Wang
f(z) = O((1 z )1 λ) ( z 1) when λ > 1.
−
• −| | | | →
Let BMOA (resp. BMOH) denote the class of analytic functions (resp. harmonic
mappings) that have bounded mean oscillation on the unit disk D (cf. [1]). In [14],
Kim proved the following result for analytic functions.
Theorem B.
(1) If λ < 1, (λ) H ,
∞
B ∩S ⊂
(2) If λ = 1, (λ) BMOA,
B ∩S ⊂
(3) If λ > 1, (λ) Hp for every 0 < p < 1/(λ 1).
B ∩S ⊂ −
In order to state a generalization of this result for harmonic mappings, we intro-
duce
= f = h+g : f is k-quasiconformal
SHk { ∈ SH }
for 0 k < 1. We now state the analog of Theorem 2.
≤
Theorem 4.1. (1) If λ < 1, then (λ) h .
H H ∞
B ∩S ⊂
(2) If λ = 1, then (λ) BMOH.
H H
B ∩S ⊂
(3) If λ > 1, then (λ) hp for every 0 < p < 1/(λ 1).
BH ∩SHk ⊂ −
For the proof of Theorem 4.1, we need some preparation.
Lemma 4.1. If f = h+g ∈ BH(1), then kfkBH ≤ 4(1+|b1|).
Proof. For f = h+ g (1), by Theorem 2.1, we have h+eiθg (1) for each
H
∈ B ∈ B
θ [0,2π]. It follows from [14, Theorem 2.1] that h+eiθg B 4(1+ b1 ), which
im∈plies that kfkBH ≤ 4(1+|b1|). k k ≤ | | (cid:3)
In the next lemma, we shall consider the problem of how the integral means of
k-quasiconformal harmonic univalent mappings f behaves. Here the integral means
of f is defined by
1 2π
(4.1) I (r) = I (r,f) = f(reiθ) pdθ.
p p
2π | |
Z0
The following lemma is regarded as a generalization of [21, Proposition 8.1] to the
case of harmonic mappings.
Lemma 4.2. Let f and p > 0. Then
∈ SHk
2(1+k2)( p 2 +1) r
I (r) | − | M(ρ)pρ 1dρ (0 r < 1),
p −
≤ 1 k2 ≤
− Z0
where
M(r) := M(r,f) = max f(reiθ) .
0 θ 2π| |
≤ ≤
Proof. Let f = h+g and write z = reiθ, where 0 r < 1. Writing
∈ SHk ≤
p/2
f(z) p = h(z)+g(z) h(z)+g(z) ,
| |
h i
(cid:0) (cid:1)(cid:0) (cid:1)
Uniformly locally univalent harmonic mappings 9
elementary computations give
∂
r f(z) p = p f(z) p 2Re zh(z)+zg (z) f(z)
− ′ ′
∂r | | | |
and (cid:0) (cid:1) (cid:8)(cid:0) (cid:1) (cid:9)
∂
f(z) p = p f(z) p 2Re izh(z)+izg (z) f(z) .
− ′ ′
∂θ | | | |
Further computat(cid:0)ions yie(cid:1)ld (cid:8)(cid:0) (cid:1) (cid:9)
∂ 2 2
r f(z) p = p(p 2) f(z) p 4 Re zh(z)+zg (z) f(z)
− ′ ′
∂r | | − | |
(cid:16) (cid:17) (cid:16) (cid:17)
(cid:0) (cid:1) +p f(z) p 2Re z2h (cid:8)(z(cid:0))+z2g (z)+zh(cid:1)(z)+(cid:9) zg (z) f(z)
− ′′ ′′ ′ ′
| |
+p f(z) p−2 zh(cid:8)′((cid:0)z)+zg′(z) 2 (cid:1) (cid:9)
| |
and (cid:12) (cid:12)
(cid:12) (cid:12)
∂ 2 2
f(z) p = p(p 2) f(z) p 4 Re izh(z)+izg (z) f(z)
− ′ ′
∂θ | | − | | }
(cid:16) (cid:17) (cid:0) (cid:1) +p f(z) p 2Re (cid:16) z2(cid:8)h(cid:0) (z) z2g (z) (cid:1)zh(z) (cid:17)zg (z) f(z)
− ′′ ′′ ′ ′
| | − − − −
+p f(z) p−2 izh(cid:8)′(cid:0)(z)+izg′(z) 2. (cid:1) (cid:9)
| |
Adding the last two expressions s(cid:12)hows that (cid:12)
(cid:12) (cid:12)
∂ 2 ∂ 2
r f(z) p + f(z) p
∂r | | ∂θ | |
(cid:16) (cid:17) (cid:16) (cid:17)
(cid:0) (cid:1) (cid:0) (cid:1)
2
= p(p 2) f(z) p 4 Re zh(z)+zg (z) f(z)
− ′ ′
− | |
(cid:20)
(cid:16) (cid:17)
(cid:8)(cid:0) 2 (cid:1) (cid:9)
+ Re izh(z)+izg (z) f(z)
′ ′
(cid:21)
(cid:16) (cid:17)
+p f(z(cid:8))(cid:0)p 2 zh(z)+zg ((cid:1)z) 2 +(cid:9) izh(z)+izg (z) 2
− ′ ′ ′ ′
|
2p((cid:12)1+k2)( ph(cid:12) 2 +1)r2 f(z)(cid:12) p 2(cid:12)h(z) 2. (cid:12) i
− ′
≤ (cid:12) | (cid:12)− | | (cid:12)| |(cid:12) | (cid:12)
It follows that
1 2π ∂ 2 ∂ 2
r f(reiθ) p + f(reiθ) p dθ
2π ∂r | | ∂θ | |
Z0 (cid:20)(cid:16) (cid:17) (cid:16) (cid:17) (cid:21)
(cid:0) (cid:1) (cid:0) (cid:1)
1 2π ∂ 2
= r f(reiθ) p dθ
2π ∂r | |
Z0
(cid:16) (cid:17)
d (cid:0) (cid:1)
= r rI (r)
dr p′
(cid:0) (cid:1) 1 2π
p(1+k2)( p 2 +1)r2 f(reiθ) p 2 h(reiθ) 2dθ,
− ′
≤ | − | π | | | |
Z0
where we have used the fact that the integral corresponding to the second term
above vanishes because of the periodicity of the function involved in the integrand.
10 S. Ponnusamy,J. Qiao and X.Wang
As J (z) = h(z) 2 g (z) 2 and
f ′ ′
| | −| |
h(z) 2 1
h(z) 2 = | ′ | J (z) J (z),
| ′ | h(z) 2 g (z) 2 f ≤ 1 k2 f
′ ′
| | −| | −
we may integrate the last expression on both sides and obtain the inequality
rI (r) p(1+k2)( p 2 +1) f(z) p 2 h(z) 2dσ(z)
p′ ≤ | − | | | − | ′ |
ZZz r
| |≤
p(1+k2)( p 2 +1)
| − | f(z) p 2J (z)dσ(z),
≤ 1 k2 | | − f
− ZZ|z|≤r
where dσ(z) = (1/π)dxdy denotes the normalized area element. Now, we substitute
w = f(z). Since f is univalent in D and M(r) = max f(reiθ) , the last
0 θ 2π
≤ ≤ | |
inequality gives
p(1+k2)( p 2 +1)
rI (r) | − | w p 2dσ(w)
p′ ≤ 1 k2 | | −
− ZZ|w|≤M(r)
2p(1+k2)( p 2 +1) M(r)
= | − | tp 1dt
−
1 k2
− Z0
2(1+k2)( p 2 +1)
= | − | M(r)p,
1 k2
−
which upon integration on both sides shows that
2(1+k2)( p 2 +1) r
I (r) | − | M(ρ)pρ 1dρ.
p ≤ 1 k2 −
− Z0
(cid:3)
The desired conclusion follows.
4.1. The proof of Theorem 4.1. Let f (λ) for some λ < 1. Then, by
H
∈ B
Corollary 3.1, f is bounded.
Next we assume that f = h+ g (1) . Then, by Lemma 4.1, it follows
H H
∈ B ∩S
that f is Bloch and thus, h is Bloch, since, for f = h+ g , h is Bloch if and
H
∈ S
only if h is BMOA if and only if f is BMOH (cf. [1]). Consequently, f BMOH.
∈
Finally, we assume that f (λ) for some λ > 1. Then, by Theorem
∈ BH ∩ SHk
3.1, we deduce that f(z) = O((1 z )1 λ) and thus, M(r) = O((1 z )1 λ).
− −
− | | − | |
Furthermore, using Lemma 4.2, we find that
2(1+k2)( p 2 +1) r
I (r) | − | M(ρ)pρ 1dρ
p ≤ 1 k2 −
− Z0
2C(1+k2)( p 2 +1) r 1
| − | ρ 1dρ
−
≤ 1 k2 (1 z )p(λ 1)
− Z0 −| | −
for some positive constant C. Hence, f hp if 0 < p < 1/(λ 1). (cid:3)
∈ −
Obviously, the assertion (1) in Theorem B remains valid if we replace (λ) by
B
(λ). By [16, Theorem 1], we see that the assertion (3) does not hold for (λ).
H H
B B
On the other hand, we will show that (λ) is contained in some Hardy space.
H
B
