Radii of covering disks for locally univalent harmonic mappings

Abstract

For a locally univalent sense-preserving harmonic mapping f=h+g‾f=h+\overline{g}
defined on the unit disk \ID =\{z\in\mathbb C:\, |z|<1\}, let df(z)d_f(z) be the
radius of the largest (univalent) disk on the manifold f(\ID) centered at
f(z0)f(z_0) (∣z0∣<1|z_0|<1). One of the aims of the present investigation is to obtain
sharp upper and lower bounds for the quotient df(z0)/dh(z0)d_f(z_0)/d_h(z_0), especially,
for a family of locally univalent Q-quasiconformal harmonic mappings
f=h+g‾f=h+\overline{g} on \ID. In addition to several other consequences of our
investigation, the disk of convexity of functions belonging to certain linear
invariant families of locally univalent Q-quasiconformal harmonic mappings of
order α\alpha is also established.

Full text

arXiv:1501.04194v2 [math.CV] 21 Jan 2015
RADII OF COVERING DISKS FOR LOCALLY UNIVALENT
HARMONIC MAPPINGS
SERGEY YU. GRAF, SAMINATHAN PONNUSAMY, AND VICTOR V. STARKOV
Abstract. For a locally univalent sense-preserving harmonic mapping f=h+g
defined on the unit disk D={z∈C:|z|<1}, let df(z) be the radius of
the largest (univalent) disk on the manifold f(D) centered at f(z0) (|z0|<1).
One of the aims of the present investigation is to obtain sharp upper and lower
bounds for the quotient df(z0)/dh(z0), especially, for a family of locally univalent
Q-quasiconformal harmonic mappings f=h+gon D. In addition to several other
consequences of our investigation, the disk of convexity of functions belonging to
certain linear invariant families of locally univalent Q-quasiconformal harmonic
mappings of order αis also established.
1. Introduction and Main results
For a smooth univalent mapping hof the unit disk D={z∈C:|z|<1}
onto two-dimensional manifold M, we define dh(z) to be the radius of the largest
(univalent) disk centered at h(z) on the manifold M. If LU denotes the family
of functions hanalytic and locally univalent (h′(z)6= 0) in D, then the classical
Schwarz lemma for analytic functions gives the following well-known sharp upper
estimate of the radius dh(z):
dh(z)≤ |h′(z)|(1 − |z|2).
The sharp and nontrivial lower estimate of the value dh(z) was obtained by Pom-
merenke [9] in a detailed analysis of what is called linear invariant families of locally
univalent analytic functions in D. Throughout we denote by Aut (D), the set of all
conformal automorphisms (M¨obius self-mappings) ω(z) = eiθ z+z0
1+z0z, where |z0|<1
and θ∈R, of the unit disk D.
Definition 1. (cf. [9]) A non-empty collection Mof functions from LU is called
a linear invariant family (LIF) if for each h∈M, normalized such that h(z) =
z+P∞
k=2 ak(h)zk, the functions Hω(z) defined by
Hω(z) = h(ω(z)) −h(ω(0))
h′(ω(0))ω′(0) =z+···,
belong to Mfor each ω∈Aut (D).
The order of the family Mis defined to be α:= ord M= suph∈M|a2(h)|. The
universal LIF, denoted by Uα, is defined to be the collection of all linear invariant
2000 Mathematics Subject Classification. Primary: 30C62, 31A05; Secondary: 30C45,30C75.
Key words and phrases. Locally univalent harmonic mappings, linear and affine invariant fam-
ilies, convex and close-to-convex functions, and covering theorems.
1

2 S. Yu. Graf, S. Ponnusamy, and V. V. Starkov
families Mwith order less than or equal to α(see [9]). An interesting fact about
the order of a LIF family is that many properties of it depend only on the order of
the family. It is well-known [9] that Uα6=∅ ⇐⇒ α≥1, and that U1is precisely the
family Kof all normalized convex univalent (analytic) functions whereas S ⊂ U2.
Here Sdenotes the classical family of all normalized univalent (analytic) functions
in Dinvestigated by a number of researchers (see [3, 5, 10]).
Note that the universal LIF is the largest LIF such that for each h∈ Uαthe
following inequality holds:
|h′(z)| ≤ (1 + |z|)α−1
(1 − |z|)α+1 .
In [9], Pommerenke has proved that for each h∈ Uαthe following sharp lower
estimate of dh(z) holds:
dh(z)≥1
2α|h′(z)|(1 − |z|2).
In the present paper the question about the alteration of the estimate of the
functional df(z) is explored in the case when instead of analytic functions h(z) we
consider harmonic locally univalent mappings of the form
(1.1) f(z) = h(z) + g(z) = ∞
X
k=1 akzk+a−kzk,
i.e. when the co-analytic part is added to the function h. We say that f=h+gis
sense-preserving if the Jacobian of Jf(z) = |h′(z)|2−|g′(z)|2of fis positive. Lewy’s
theorem [7] (see also for example [4, Chapter 2, p. 20] and [11]) implies that every
harmonic function fon Dis locally one-to-one and sense-preserving on Dif and
only if Jf(z)>0 in D. The condition Jf(z)>0 is equivalent to h′(z)6= 0 and the
existence of an analytic function µfin Dsuch that
(1.2) |µf(z)|<1 for z∈D,
where µf(z) = g′(z)/h′(z) and µfis referred to as the (complex) dilatation of the
harmonic mapping f=h+g. When it is convenient, we simply use the notation µ
instead of µf.
There are different generalizations of the notion of the linear invariant family to
the case of harmonic mappings. For example, the question about a lower estimate of
the radius df(0) of the univalent disk centered at the origin was examined by Sheil-
Small [13] in the linear and affine invariant families of univalent harmonic functions
f. The concept of linear and affine invariance was also discussed by Schaubroeck
[12] for the case of locally univalent harmonic mappings.
Definition 2. The family LUHof locally univalent sense-preserving harmonic func-
tions fin the disk Dof the form (1.1) is called linear invariant (LIF) if for each
f=h+g∈ LUHthe following conditions are fulfilled: a1= 1 and
f(ω(z)) −f(ω(0))
h′(ω(0))ω′(0) ∈ LUH

Linear invariant families of locally univalent harmonic mappings 3
for each ω∈Aut (D). A family ALHis called linear and affine invariant (LAIF) if
it is LIF and in addition each f∈ ALHsatisfies the condition that
f(z) + εf(z)
1 + εfz(0) ∈ ALHfor every ε∈D.
The number ord ALH= supf∈ALH|a2|is known as the order of the LAIF ALH.
The order of LIF LUHwithout the assumption of affine invariance property is
defined in the same way: ord LUH= supf∈LUH|a2|.
Throughout the discussion, we suppose that the orders of these families, namely,
ord ALHand ord LUHare finite. The universal linear and affine invariant family,
denoted by ALH(α), is the largest LAIF ALHof order α=ord ALH. Thus, the
subfamily AL0
Hof LAIF ALHconsists of all functions f∈ ALHsuch that fz(0) = 0.
If f∈ AL0
His univalent in D, then according to the result of Sheil-Small [13] one
has the following sharp lower estimate:
(1.3) df(0) ≥1
2α.
For α > 0 and Q≥1, denote by H(α, Q) the set of all locally univalent Q-
quasiconformal harmonic mappings f=h+gin Dof the form (1.1) with the
normalization a1+a−1= 1 such that
h(z)/h′(0) ∈ Uα,|g′(z)/h′(z)| ≤ k, k = (Q−1)/(Q+ 1) ∈[0,1).
The family H(α, Q) was introduced and investigated in details by Starkov [15, 16].
In particular, he established double-sided estimates of the value df(z) for functions
belonging to the family H(α, Q).
We shall restrict ourselves to the case of finite Q. In [15, 16], it was also shown
that the family H(α, Q) possess the property of linear invariance in the following
sense: for each f=h+g∈ H(α, Q) and for every ω(z) = eiθ z+a
1+az ∈Aut (D), the
transformation
(1.4) f(ω(z)) −f(ω(0))
∂θf(ω(0))|ω′(0)|∈ H(α, Q),
where ∂θf(z) = h′(z)eiθ +g′(z)eiθ denotes the directional derivative of the complex-
valued function fin the direction of the unit vector eiθ.
In [17], it was also proved that for each f∈ H(α, Q) and z∈D,
1− |z|2
2αQ max
θ|∂θf(z)| ≤ df(z)≤Q(1 − |z|2) min
θ|∂θf(z)|
which is equivalent to
1− |z|2
2αQ (|h′(z)|+|g′(z)|)≤df(z)≤Q(1 − |z|2) (|h′(z)| − |g′(z)|),(1.5)
and the lower estimate is sharp in contrast to the upper one.
One of the main aims of this article is to establish the sharp estimations of the ratio
df(z)/dh(z) for Q-quasiconformal harmonic mappings f=h+gand, in particular,
the sharp upper estimate in (1.5) is also obtained. We now state our first result.

4 S. Yu. Graf, S. Ponnusamy, and V. V. Starkov
Theorem 1. Let f=h+g∈ H(α, Q)for some Q∈[1,∞], and µ(z) = g′(z)/h′(z)
be the complex dilatation of the mapping f. Then for z∈D,
(1.6) 1 −k≤m|µ(z)|
k, Q≤df(z)
dh(z)≤M|µ(z)|
k, k≤1 + k,
where k= (Q−1)/(Q+ 1) ∈[0,1]. Here the functions M(., k)and m(., Q)are
defined as follows:
M(x, k) = 






1 + k
x1−1
x−xlog (1 + x)when x∈(0,1]
lim
x→0+M(x, k) = 1 + k
2when x= 0
,(1.7)
and
1
m(x, Q)=

Z1
0
1 + ϕ−1(Q−1ϕ(t))x
1−kx +ϕ−1(Q−1ϕ(t))(x−k)dt when Q < ∞
0when Q=∞
,(1.8)
with
ϕ(t) = π
2
K′(t)
K(t)(t∈(0,1))
where Kdenotes the (Legendre) complete elliptic integral of the first kind given by
K(t) = Zπ/2
0
dx
p1−t2sin2x=Z1
0
dx
p(1 −x2)(1 −t2x2)
and K′(t) = K(√1−t2). The argument tis sometimes called the modulus of the
elliptic integral K(t).
Estimations in (1.6) are sharp for the family H(α, Q)for Q < ∞and for each
α≥1. When Q=∞, estimations in (1.6) are sharp in the sense that for each
z∈D,
inf
f∈H(α,∞)
df(z)
dh(z)=m(x, ∞) = 0 and sup
f∈H(α,∞)
df(z)
dh(z)=M(1,1) = 2.
Remark 1. In some neighbourhood of the origin, it is also possible to obtain a simple
lower estimate in the inequality (1.6) without the involvement of elliptic integrals.
For example, the well-known theorem of Mori [8] reveals that for Q-quasiconformal
automorphism Fof the disk Dsuch that F(0) = 0, one has
|F(z)| ≤ 16|z|1/Q.
Using this result in the estimation of the value of |ℓ0(t)|in Part 3 of the proof of
Theorem 1, one can easily obtain that
|ℓ0(t)|=|f−1(At)|=(16t1/Q when 0 ≤t < 1/16−Q
1 when 16−Q≤t < 1.

Linear invariant families of locally univalent harmonic mappings 5
The last relation provides an opportunity to estimate the ratio dh(z)/df(z) by means
of an integral of an elementary function, namely,
dh(z)
df(z)≤1
m(x, Q)
≤1
1−k 1−16−Q+ (1 −k)Z16−Q
0
1 + y x
1−kx +y(x−k)dt!
≤1
1−k,
where y= 16t1/Q ≤1 for t∈[0,16−Q].Here x=|µ(z)|/k, z ∈D.
Remark 2. For fixed ζ∈D, the least value of the upper estimation in (1.6) is
attained when x= 0; that is when µ(ζ) = 0. In this case the estimation in (1.6)
takes the form
df(ζ)
dh(ζ)≤1 + k
2.
Suppose that f=h+g∈ H(α, Q), α ∈[1,∞], and f1(z) = C·f(z) = h1(z)+g1(z),
where Cis a complex constant. Then the following relations hold:
df1(z) = |C|df(z) and dh1(z) = |C|dh(z), z ∈D.
Moreover, after appropriate normalization, every Q-quasiconformal harmonic map-
ping in Dhits the family H(α, Q) for some α. Therefore an equivalent formulation
of Theorem 1 may now be stated.
Theorem 2. Let f=h+gbe a locally univalent Q-quasiconformal harmonic map-
ping of the disk D,Q∈[1,∞], and µ(z) = g′(z)/h′(z). Then the inequalities (1.6)
continue to hold and the estimations in (1.6) are sharp.
Next, we consider f=h+g∈ H(α, Q) and introduce H(z) = h(z)/h′(0) from
Uα. Then we have (see [15, 16])
1
1 + k≤ |h′(0)| ≤ 1
1−k
and thus,
dH(z)
1 + k≤dh(z) = |h′(0)| · dH(z)≤dH(z)
1−k.
These inequalities and (1.6) give the following.
Corollary 1. Let f=h+g∈ H(α, Q)and h(z) = h′(0)H(z). Then we have
dH(z)
Q≤df(z)≤Q dH(z)for z∈D.
The sharpness of the last double-sided inequalities at the point z= 0 follows from
the proof of Theorem 1.
We now state the remaining results of the article.

6 S. Yu. Graf, S. Ponnusamy, and V. V. Starkov
Theorem 3. Let f=h+gbe a locally quasiconformal harmonic mapping belonging
to the family ALHwith ord (ALH) = α < ∞,µ(z) = g′(z)/h′(z)and |µ(z)|<1.
Then
(1.9) df(z)≥1− |µ(z)|
2α1− |z|
1 + |z|α
for z∈D.
The estimation df(0) is sharp for example in the universal LAIF ALH(α).
Recall that a locally univalent function fis said to be convex in the disk D(z0, r) :=
{z:|z−z0|< r}if fmaps D(z0, r) univalently onto a convex domain. The radius
of convexity of the family Fof functions defined on the disk Dis the largest number
r0such that every function f∈ F is convex in the disk D(0, r0).
Theorem 4. If f∈ H(α, Q), then for every z∈D, the function fis convex in the
disk D(z, R(z)), where
(1.10) R(z) = 1
2R0+R−1
0−qR0−R−1
02+ 4|z|2,
and
(1.11) R0=α+k−1−√k−2−1−rα+k−1−√k−2−12−1.
In particular, the radius of convexity of the family H(α, Q)is no less than R0.
The proofs of Theorems 1, 3 and 4 will be presented in Section 2.
2. Proofs of the Main results
2.1. Proof of Theorem 1. The proof of the theorem is divided into three parts.
Part 1: Let f=h+gsatisfy the assumptions of Theorem 1. In compliance with
the definition of the value df(0), there exists a boundary point Aof the manifold
f(D) such that A∈ {w:|w|=df(0)}. Consider the smooth curve ℓ0=f−1([0, A)),
namely, the preimage of the half-interval [0, A) with the starting point 0 in the disk
D. Then
df(0) = |A|=Zℓ0
df (z)
= min
γZγ
df (z)
,
where the minimum is taken over all smooth paths γ(t), t ∈[0,1), such that γ(0) =
0,|γ(t)|<1 and limt→1−|γ(t)|= 1.
Similarly we define the value
dh(0) = |B|=Zℓ
dh(z)
= min
γZγ
dh(z)
,
where the simple smooth curve ℓ=h−1([0, B)) is emerging from the origin, the
preimage of the half-interval [0, B) under the mapping h. Consider the following

Linear invariant families of locally univalent harmonic mappings 7
parametrization of the curve ℓ:ℓ(t) = h−1(Bt), t ∈[0,1). Then h′(ℓ(t))ℓ′(t) = B
and
df(0) = Z1
0
df (ℓ0(t))≤Z1
0
df (ℓ(t))
=Z1
0nh′(ℓ(t))ℓ′(t) + g′(ℓ(t))ℓ′(t)odt
=|B|Z1
0(1 + g′(ℓ(t))ℓ′(t)
h′(ℓ(t))ℓ′(t))dt
≤dh(0) 1 + Z1
0|µ(ℓ(t))|dt.(2.1)
At first we consider the case k= supz∈D|µ(z)|<1. Since |µ(z)| ≤ kfor z∈D,
we have
µ(0)/k =a−1/(k a1) =: u∈D.
If |u|= 1 for k < 1, then we have the inequality
df(0) ≤dh(0)(1 + k) = dh(0)M(1, k)
which proves the upper estimate in the inequality (1.6) for z= 0.
Let us now assume that |u|<1 for some k < 1. Then, from a generalized version
of the classical Schwarz lemma (see for example [5, Chapter VIII, §1]), it follows
that
(2.2) |µ(z)|
k≤|z|+|u|
1 + |u||z|.
Consequently, by (2.1), one has
(2.3) df(0) ≤dh(0) 1 + kZ1
0
|ℓ(t)|+|u|
1 + |u||ℓ(t)|dt.
Also, the function h−1(Bζ) maps biholomorphically Donto some sub domain
of the disk D. Applying the classical Schwarz lemma, we obtain the inequality
|h−1(Bζ )| ≤ |ζ|and hence, |ℓ(t)| ≤ tholds. Using the last estimate and the inequal-
ity (2.3), one can obtain, after evaluating the integral, the inequality
df(0) ≤dh(0) 1 + kZ1
0
t+|u|
1 + |u|tdt=dh(0)M(|u|, k),
where M(x, k) is defined by (1.7). The function M(x, k) is strictly increasing on
(0,1] with respect to the variable xand for each fixed k∈[0,1]. This follows from
the observation that (see (1.7))
∂M (x, k)
∂x =−k
x2+2k
x3log(1 + x)−k1−x
x2,
which is positive, since log(1 + x)> x −x2/2. Hence
(2.4) df(0) ≤dh(0)M(|u|, k)≤dh(0)M(1, k) = (1 + k)dh(0).

8 S. Yu. Graf, S. Ponnusamy, and V. V. Starkov
Let us now set k= 1. According to Lewy’s theorem [7] for locally univalent
harmonic mapping f, we obtain that |µ(z)| 6= 1 for all z. Next we obtain the
inequality (2.4) in the case k= 1 by repeating the argument of the case k < 1.
Let us now begin to prove that the upper estimate in (1.6) is true for all ζ∈D.
As mentioned above, the family H(α, Q) is linear invariant in the sense of [15, 16]
(see (1.4) above). Hence, for each fixed ζ=reiθ ∈D(r∈[0,1), θ ∈R), the function
Fdefined by
F(z) = feiθ z+r
1+rz −f(reiθ )
∂θf(reiθ)(1 −r2)=H(z) + G(z)
belongs to the family H(α, Q), where Hand Gare analytic in Dsuch that H(0) =
G(0) = 0.Therefore, in view of (2.4) for k∈[0,1], we have
dF(0) = df(ζ)
|∂θf(ζ)|(1 − |ζ|2)≤dH(0)M(x, k),
where x=|G′(0)/H′(0)|/k =|µ(ζ)|/k ∈[0,1] if k∈[0,1), and x=|G′(0)/H ′(0)|=
|µ(ζ)| ∈ [0,1) when k= 1. Note that
H(z) = heiθ z+r
1+rz −h(reiθ )
∂θf(reiθ )(1 −r2).
Consequently,
dH(0) = dh(ζ)
|∂θf(ζ)|(1 − |ζ|2)
so that
df(ζ)≤dh(ζ)M(x, k)≤(1 + k)dh(ζ)
and we complete the proof of the upper estimate in (1.6).
Part 2: We now deal with the sharpness of the upper estimate in (1.6). Consider the
case k∈[0,1). For every α∈Nand every ζ∈D, we shall indicate functions from
the families H(α, Q) such that df(ζ)/dh(ζ) = M(x) = 1 + k, where x=|µ(ζ)|/k.
Since the families H(α, Q) are enlarging with increasing values of α, the sharpness
of the upper estimate in (1.6) will be shown for every ζ∈Dand each α∈[1,∞].
Consider the sequence of {kn}∞
n=1 functions from Undefined by
kn(z) = i
2n1−iz
1 + iz n
−1.
Then we have dkn(0) = 1/2n(see [9]) and observe that knmaps the unit disk D
univalently onto the Riemann surface kn(D) whose boundary
∂kn(D) = i
2n[(iλ)n−1] : λ∈R=i
2n[s e±iπn/2−1] : s≥0
consists of two rays. Then the univalent image of the disk Dunder the mapping
fn(z) = hn(z) + gn(z) = 1
1−k[kn(z)−kkn(z)] ∈ H(n, Q), k ∈[0,1),

Linear invariant families of locally univalent harmonic mappings 9
(a1=1
1−k, a−1=−k
1−k) represents the manifold with the boundary
∂fn(D) = i
2n(1 −k)[s(e±iπn/2+k e∓iπn/2)−1−k] : s≥0,
which consists of two rays parallel to the coordinate axes and arising from the point
−i
2nQ. Note that the function fnmaps the half-interval [0,−i) bijectively onto
[0,−i
2nQ) and thus, we conclude that dfn(0) = Q
2n; that is,
dfn(0) = dkn(0)Q=dhn(0)(1 + k),
where hn(z) = kn(z)/(1 −k). The sharpness of the upper estimate in (1.6) is proved
for ζ= 0 and k < 1.
Next we let 0 6=ζ∈D,k < 1 and consider a conformal automorphism ω(z) =
(z+ζ)/(1 + ζz) of the unit disk D. Then the inverse mapping is given by ω−1(z) =
(z−ζ)/(1 −ζz). From the condition (1.4) of the linear invariance property of the
family H(α, Q), it follows that the function fdefined by
f(z) = fn(ω−1(z)) −fn(−ζ)
∂0fn(−ζ)(1 − |ζ|2)=h(z) + g(z)
belongs to H(α, Q), where hand ghave the same meaning as above. Taking into ac-
count of the normalization condition for functions in the family H(α, Q), we deduce
that f(ω(z)) −f(ζ)
∂0f(ζ)(1 − |ζ|2)=fn(z) = hn(z) + gn(z).
Therefore,
dfn(0) = df(ζ)
|∂f0(ζ)|(1 − |ζ|2)=dhn(0)(1 + k).
On the other hand, a direct computation gives
hn(z) = h(ω(z)) −h(ζ)
∂0f(ζ)(1 − |ζ|2)and dhn(0) = dh(ζ)
|∂f0(ζ)|(1 − |ζ|2)
showing that
dfn(0)|∂f0(ζ)|(1 − |ζ|2) = df(ζ) = dh(ζ)(1 + k),
which completes the proof of the upper estimation in Theorem 1 for k∈[0,1).
If k= 1 then for j∈N, we consider the sequence {fn,j }of functions
fn,j(z) = hn,j (z) + gn,j (z) = j kn(z)−(j−1) kn(z).
We see that fn,j ∈ H(n, 2j−1) ⊂ H(n, ∞) for each j∈N. Therefore,
dfn,j (0) = dhn,j (0)M(1,1−1/j).
Hence
sup
j∈N
dfn,j (0)
dhn,j (0) =M(1,1) = 2.
The sharpness of the upper estimation in (1.6) for k= 1, ζ 6= 0, can be proved
analogously. So, we omit the details.

10 S. Yu. Graf, S. Ponnusamy, and V. V. Starkov
Part 3: Finally, we deal with the correctness of the lower estimation of df(z). If
k= 1, then the lower estimation in (1.6) is trivial because m(x, ∞) = 0. So, we may
assume that k∈[0,1). As in Part 1, we define the boundary points Aand Bof the
manifolds f(D) and h(D), respectively, and smooth curves ℓ0=f−1([0, A)) and ℓin
the same manner as in Part 1. Consider the parametrization of the curve ℓ0:
ℓ0(t) = f−1(At), t ∈[0,1).
Then df(ℓ0(t)) = Adt and thus,
dh(0) = Z1
0
dh(ℓ(t))
≤Z1
0
h′(ℓ0(t))ℓ′
0(t)dt
=Z1
0nh′(ℓ0(t))ℓ′
0(t) + g′(ℓ0(t))ℓ′
0(t)o
× 1−g′(ℓ0(t))ℓ′
0(t)
h′(ℓ0(t))ℓ′
0(t) + g′(ℓ0(t))ℓ′
0(t)!dt
=Z1
0
h′(ℓ0(t))ℓ′
0(t)
h′(ℓ0(t))ℓ′
0(t) + g′(ℓ0(t))ℓ′
0(t)df (ℓ0(t))
≤ |A|Z1
0
dt
1− |µ(ℓ0(t))|.(2.5)
In view of the inequality (2.2), we find that
(2.6) |µ(ℓ0(t))| ≤ k|ℓ0(t)|+x
1 + x|ℓ0(t)|,
where x=|µ(0)|/k.
It is possible to obtain an estimation of the value |ℓ0(t)|=|f−1(At)|, t ∈[0,1),
with the help of the analog of the Schwarz lemma for Q-quasiconformal automor-
phisms of the disk. Let Fbe a Q-quasiconformal automorphism of D, and F(0) = 0.
It is known (see for example [1, Chapter 10, equality (10.1)]) that the sharp estima-
tion
|F(z)| ≤ ϕ−1Q−1ϕ(|z|)
holds, where ϕand Qare as in the statement. The function f−1(Aw) defined on
the unit disk {w:|w|<1}satisfies the conditions f−1(0) = 0 and |f−1(Aw)|<1.
Let Φ be the univalent conformal mapping of the domain f−1(AD) onto the unit
disk Dand Φ(0) = 0.Then the composition Φ ◦f−1(Az) is a Q-quasiconformal
automorphism of Dand Φ−1satisfies the conditions of the classical Schwarz lemma
for analytic functions. Hence, we have
|ℓ0(t)|=|Φ−1(Φ ◦f−1(At))| ≤ |Φ◦f−1(At)| ≤ ϕ−1(Q−1ϕ(t)).
As a result of it and taking into account of the last estimation, inequalities (2.5)
and (2.6), and the fact that the function (1 + y x)/(1 −kx +y(x−k)) is strictly

Linear invariant families of locally univalent harmonic mappings 11
increasing with respect to yon (0,1), we conclude that
dh(0) ≤df(0) Z1
0
1 + y x
1−kx +y(x−k)dt ≤df(0)
1−k,
where y=ϕ−1(Q−1ϕ(t)) ≤1 for t∈(0,1). Therefore the lower estimate in (1.6) is
sharp at the origin.
The proof of the lower estimation in (1.6) for 0 6=ζ∈Dfollows easily if we
proceed with the same manner as in Part 1 and use the linear invariance property
of the family H(α, Q).
For the sharpness of the left side of the inequality in (1.6) for k∈[0,1), we
consider the functions (see[15, 16])
(2.7) hα(z) = 1
2iα 1 + iz
1−iz α
−1∈ Uα
and
f(z) = h(z) + g(z) := hα(z)
1 + k+khα(z)
1 + k.
Then it is a simple exercise to see that
df(0) = 1
2αQ and dh(0) = 1
2α(1 + k).
If k→1−then from the last equality we obtain
lim
k→1−
df(0) = 0 and lim
k→1−
dh(0) = 1
4α,
so that
inf
f∈H(α,∞)
df(0)
dh(0) = 0.
Thus the last equality is sharp not only at the origin but also at points z∈D, in
view of the degeneration of functions
f(z) = hα(z) + khα(z)
1 + k
when k→1−. The proof of the theorem is complete. 
2.2. Proof of Theorem 3. Let us first prove the inequality (1.9) for z= 0. As
in the proof of Theorem 1, consider on the circle {w:|w|=df(0)}the boundary
point Aof the manifold f(D) and define a curve ℓ0=f−1([0, A)) with the starting
point 0 in D. Then
(2.8) df(0) = |A|=Zℓ0
df (ζ)
=Zℓ0|df(ζ)| ≥ Zℓ0
(|h′(ζ)| − |g′(ζ)|)|dζ|.
In view of the affine invariance property of the family ALH, the function Fdefined
by
F(ζ) = H(ζ) + G(ζ) = f(ζ)−εf(ζ)
1−εfz(0)
belongs ALHfor every εwith |ε|<1 .

12 S. Yu. Graf, S. Ponnusamy, and V. V. Starkov
For a fixed ζ, we introduce θ(z) = arg h′(z)−arg g′(z) when g′(z)6= 0, and
θ(z) = arg h′(z) otherwise. Consider then ε=seiθ(z)for s∈[0,1). Therefore,
taking into account of the relation fz(0) = µ(0), we obtain that
H′(ζ) = h′(ζ)−sg′(ζ)eiθ(z)
1−εµ(0)
and thus,
(2.9) |H′(ζ)| ≤ |h′(ζ)| − s|g′(ζ)|
1−s|µ(0)|.
For the other side of the inequality for functions in the family ALH, the inequality
(2.10) |H′(ζ)| ≥ (1 − |ζ|)α−1
(1 + |ζ|)α+1
holds, where α=ord (ALH) is defined as in the sense of Definition 2. The inequality
(2.10) was obtained in [13] for LAIF of univalent harmonic mappings, but the proof
is still valid without a change for any LAIF ALHof finite order α. Using inequalities
(2.9) and (2.10), we obtain the inequality
|h′(ζ)| − s|g′(ζ)| ≥ (1 −s|µ(0)|)(1 − |ζ|)α−1
(1 + |ζ|)α+1
for every s∈(0,1). Allowing in the last inequality s→1−and substituting the
resulting estimate into (2.8), we easily obtain that
df(0) ≥(1 − |µ(0)|)Zℓ0
(1 − |ζ|)α−1
(1 + |ζ|)α+1 |dζ|
≥(1 − |µ(0)|)Z1
0
(1 −t)α−1
(1 + t)α+1 dt =1− |µ(0)|
2α.
If 0 <|z|<1,then as in the proof of Theorem 1, we may use the linear invariance
property of the family ALHin accordance with the function F1∈ ALH, where
F1(ζ) =
fζ+z
1+zζ −f(z)
h′(z)(1 − |z|2).
In this way, applying the estimation of df(0) to the function F1, we see that
dF1(0) ≥1− |µf(0)|
2α.
Also, we have
dF1(0) = df(z)
|h′(z)|(1 − |z|2).
It remains to note that |µF1(0)|=|µf(z)|and apply the inequality (2.10) to the
function h′(z).
In order to prove the sharpness of the estimate of df(0), we first note that the
functions p(z) = hα(z) + khα(z), where each hαhas the form (2.7), belong to

Linear invariant families of locally univalent harmonic mappings 13
ALH(α) for every k=|µ(0)| ∈ [0,1). Indeed, for each α, the function pis lo-
cally univalent and meet the normalization condition of the family ALH(α), and
|pzz (0)/2|=|h′′
α(0)/2|=α. Affiliation of the functions
q(z) = p(ω(z)) −p(ω(0))
h′
α(ω(0))ω′(0) =hα(ω(z)) −hα(ω(0))
h′
α(ω(0))ω′(0) +khα(ω(z)) −hα(ω(0))
h′
α(ω(0))ω′(0) ,
and
w(z) = q(z) + εq(z)
1 + εqz(0)
=hα(ω(z)) −hα(ω(0))
h′
α(ω(0))ω′(0)
+hα(ω(z)) −hα(ω(0))
h′
α(ω(0))ω′(0) k+εh′
α(ω(0))ω′(0)/(h′
α(ω(0))ω′(0))
1 + εk h′
α(ω(0))ω′(0)/(h′
α(ω(0))ω′(0))!
to the family ALH(α) for every conformal automorphism ωof the disk Dand every
ε∈D, follow from the membership of the function hαto the universal LIF Uα.
The analogous reasoning is true after the change of order of the linear and affine
transforms of the function p.
Therefore, p=hα+khα∈ ALH(α) for each k∈[0,1) and at the same time
dp(0) = 1− |µ(0)|
2α,
which proves the sharpness of the established estimate in the universal LAIF ALH(α).
The proof of the theorem is complete. 
Remark 3. (a) Recall that a domain D⊂Cis called close-to-convex if its comple-
ment C\Dcan be written as an union of disjoint rays or lines. The family CHof all
univalent sense-preserving harmonic mappings fof the form (1.1) such that a1= 1
and f(D) is close-to-convex, is LAIF (cf. [13]). Also, the inequality in Theorem 3
is sharp in the LAIF CH. The order of the family CHis proved to be 3 ([2]). The
harmonic analog of the analytic Koebe function k(z) = z/(1 −z)2(see for example
[4, Chapter 5, p. 82]) is given by
F(z) = z−1
2z2+1
6z3
(1 −z)3+1
2z2+1
6z3
(1 −z)3,
where F∈ CHand F(D) = C\(−∞,−1/6] which is indeed a domain starlike with
respect to the origin. From the affine invariance property of the family CH, we
deduce that for every b∈[0,1), the affine mapping
f(z) = F(z)−bF(z)
belongs to CHsuch that µ(0) = fz(0)/fz(0) = −b. The function fis a composition
of the univalent harmonic mapping Fof the disk Donto C\(−∞,−1/6] and affine
transformation ψ(w) = w−bw. The plane Cwith a slit (−∞,−1/6] under the
transformation ψis the plane with a slit along the ray emanating from the point
ψ(−1/6) = −(1 −b)/6 through the point ψ(−1) = b−1<(b−1)/6, since b∈[0,1).

14 S. Yu. Graf, S. Ponnusamy, and V. V. Starkov
Therefore, f(D) = C\(−∞,−(1 −b)/6] and thus, df(0) = (1 −b)/6 and the lower
estimate of df(0) is sharp in the LAIF CH.
(b) In the first part of the present paper, we are concerned with the question
about the covering of the manifold f(D) by the disks. Now we turn our attention
on the problem related with the covering of f(D) by convex domains.
Sheil-Small [13] proved that the radius of convexity of the univalent subfamily of
the linear and affine invariant family ALHof harmonic mappings is equal to
(2.11) r0=α−√α2−1,
where α=ord (ALH). Later this result was generalized to the families of locally
univalent harmonic mappings [6]. Now we will show the radius of convexity will be
altered under the assumption of Q-quasiconformality of functions f.
Lemma 1. Let LUH(α, Q)denote the LIF of locally univalent Q-quasiconformal
harmonic mappings of the order α < ∞, where Q≤ ∞. Then the affine hull
ALH=(F(z) = f(z) + εf(z)
1 + εa−1
:f∈ LUH(α, Q), ε ∈D)
of the family LUH(α, Q)is linear and affine invariant of the order no greater than
α+1−√1−k2
k, where k= (Q−1)/(Q+ 1).
Proof. In [14], it was shown that the affine hull of the linear invariant in the sense
of Definition 2 of the family of the locally univalent harmonic mappings is the LAIF
ALH. Thus, it remains to determine the estimate of the order of the family ALH.
We begin with F=H+G∈ ALH. Then there exists an f=h+g∈ LUH(α, Q)
of the form (1.1) with the normalization f(0) = 0, h′(0) = 1,and ε∈Dsuch that
F(z) = f(z) + εf(z)
1 + εg′(0) =H(z) + G(z).
It is easy to compute that
A2=H′′(0)
2=a2+εa−2
1 + εg′(0) ,
where a2=h′′(0)/2 and a−2=g′′(0)/2. Taking into account of the relation g′(z) =
µ(z)h′(z), where µis the complex dilatation of fwith |µ(z)|< k, we see that
g′(0) = µ(0) and g′′(0) = h′′(0)µ(0) + h′(0)µ′(0),
so that
a−2=a2µ(0) + µ′(0)/2.
If we apply the Schwarz-Pick lemma (see for example [5, Chapter VIII, §1]) to the
function µ(z)/k, then the inequality (1.3) in this case leads to
|µ′(0)|
k≤1−|µ(0)|2
k2.

Linear invariant families of locally univalent harmonic mappings 15
Using the expression for a2, we deduce that
|A2|=
a2(1 + εµ(0)) + εµ′(0)/2
1 + εµ(0) ≤ |a2|+k
2
1− |µ(0)/k|2
1− |µ(0)|=|a2|+k2− |µ(0)|2
2k(1 − |µ(0)|)
(since |ε|<1). Calculating the maximum of the function u(t) = (k2−t2)/(1 −t)
over the interval [0, k], we obtain the estimate
|A2| ≤ |a2|+1−√1−k2
k≤α+1−√1−k2
k< α + 1.
The proof of the lemma is complete. 
Using Lemma 1 and the equality (2.11), one obtains the estimate of the radius of
convexity of functions in the family H(α, Q).
2.3. Proof of Theorem 4. Let f0=h0+g0∈ H(α, Q). It is easy to see that
function f0is convex in the same disks as the normalized function
f(z) = f0(z)/h′
0(0) = h(z) + g(z)
that belongs to some LIF LUH(α, Q). So it is enough to prove the statement of the
theorem for such functions f. We first show that function fis convex in the disk
centered at the origin with radius R0defined by (1.11).
Clearly, the function fbelongs to the affine hull ALHof the family LUH(α, Q).
In view of Lemma 1, the family ALHhas the order α1≤α+1−√1−k2
k.Taking into
consideration of the equality (2.11), we conclude that the function fis convex in
the disk of radius R0=α1−√α12−1 centered at the origin.
We now let 0 6=z0∈D. Consider a conformal automorphism Φ of the unit disk
Dgiven by
Φ(ζ) = eiarg z0ζ+|z0|
1 + |z0|ζ.
We see that Φ maps the disk D(0, R0) onto the disk D(z0, R(z0)),where R(z0) is
defined in (1.10). In view of the linear invariance property of the family LUH(α, Q),
the function Fdefined by
F(ζ) = f(Φ(ζ)) −f(z0)
h′(z0))Φ′(0)
belongs to LUH(α, Q) and as remarked above, the function Fmaps the disk D(0, R0)
onto a convex domain. Therefore, the function
f(z) = F(Φ−1(z)) ·h′(z0))Φ′(0) + f(z0)
is convex and univalent in the disk D(z0, R(z0)). The proof of the theorem is com-
plete. 
Acknowledgements. The research was supported by the project RUS/RFBR/P-
163 under Department of Science & Technology (India) and the Russian Foundation
for Basic Research (project 14-01-92692). The second author is currently on leave
from Indian Institute of Technology Madras, India. The third author is also sup-
ported by Russian Foundation for Basic Research (project 14-01-00510).

16 S. Yu. Graf, S. Ponnusamy, and V. V. Starkov
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S. Yu. Graf, Tver State University, ul. Zhelyabova 33, Tver, 170000 Russia.
E-mail address:sergey.graf@tversu.ru
S. Ponnusamy, Indian Statistical Institute (ISI), Chennai Centre, SETS (Society
for Electronic Transactions and Security), MGR Knowledge City, CIT Campus,
Taramani, Chennai 600 113, India.
E-mail address:samy@isichennai.res.in, samy@iitm.ac.in
V. V. Starkov, Department of Mathematics, University of Petrozavodsk, ul.
Lenina 33, 185910 Petrozavodsk, Russia
E-mail address:vstarv@list.ru