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Probl. Anal. Issues Anal. Vol. 8 (26), No 2, 2019, pp. 37–50 37
DOI: 10.15393/j3.art.2019.6190
UDC 517.54
S. Yu. Graf
HARMONIC MAPPINGS ONTO R-CONVEX DOMAINS
Abstract. The plane domain Dis called R-convex if Dcontains
each compact set bounded by two shortest sub-arcs of the radius R
with endpoints w1, w2∈D, |w1−w2|62R. In this paper, we prove
the conditions of R-convexity for images of disks under harmonic
sense preserving functions. The coefficient bounds for harmonic
mappings of the unit disk onto R-convex domains are obtained.
Key words: harmonic mappings, R-convex domains, coefficient
bounds
2010 Mathematical Subject Classification: 30C45, 30C50,
30C55, 30C99
1. Preliminaries. For a given pair of points w1, w2∈Csuch that
|w1−w2|62Rwith 0< R < ∞we define the R-convex hull ER(w1,w2)
of w1, w2as a compact set bounded by the two shortest arcs of the circles
of radius Rwith endpoints w1, w2. The set ER(w1, w2)is strictly convex
for each R > 0and tends to the segment [w1, w2]when R→ ∞.
Definition 1.The set A⊂Cis called R-convex if Acontains each set
ER(w1, w2)provided that w1, w2∈Aand |w1−w2|62R.
It is clear that the Jordan domain D⊂Cis R-convex if and only if its
closure Dis R-convex. Of course, R-convex domains are strictly convex.
R-convex sets ∈Rnwere introduced and studied in [13,14]. R-convex
sets and domains play an important role in convex analysis and so have
applications in many branches of mathematics, physics and economic sci-
ences.
A.W. Goodman [8] defined the convex functions of bounded type in-
dependently in the geometrical function theory as an univalent analytic
function hin the unit disk D={z∈C:|z|<1}, such that lim inf
|z|→1kh(z)>
1/R > 0.Here kh(z) = Re{zh00 (z)/h0(z) + 1}/|zh0(z)|is the curvature of
c
Petrozavodsk State University, 2019
38 S. Yu. Graf
the image Γr=h(γr)of the circle γr={z∈D:|z|=r}under the
mapping hat the point h(z). The curvature khis defined by a standard
way as kh=dθ/ds, where sis the natural parameter on Γrand θis the
argument of the tangent vector to Γr.
Let R∈(0,+∞)be given and CRdenote a family of convex analytic
functions hin Dof bounded type, such that h(0) = h0(0) −1=0. These
classes were studied by A.W. Goodman [8,10] and K.-J. Wirths [19]. In
particular, growth, covering and distortion theorems in CRwere proved,
as well as coefficient bounds.
Interrelation between R-convex domains and analytic univalent func-
tions onto such domains was revealed and investigated in the paper [18]
by V. Starkov and N. Shmelev. They proved that for locally univalent
analytic function hin Dthe domain D=h(D)is R-convex if and only if
Re zh00(z)
h0(z)+ 1>|zh0(z)|
Rfor all z∈D.(1)
More than that, (1) is equivalent to the statement that domains
Dr=h(Dr)are R-convex for all r∈(0,1], where Dr={z∈D:|z|< r}.
So, the heredity property is valid for R-convexity of h(Dr)in the case of
analytic functions. Functions hthat satisfy condition (1) are univalent.
The theorem of Peschl (cf., [12,19]) claims that if h∈CR, R > 0, then
kh(z)has not local minimums in D\ {0}and kh(z)>1/R in D\ {0}.
Therefore, his an analytic convex function of bounded type if and only if
(1) is true in Dfor some R > 0. This result, together with criterion (1) of
R-convexity due by V. Starkov and N. Shmelev, immediately leads to
Proposition 1.CRconsists of univalent analytic functions hin Dsuch
that h(0) = h0(0) −1 = 0 and h(D)is R-convex domain.
In this paper, we obtain the conditions of R-convexity of f(D)for sense-
preserving harmonic functions and prove some estimations and coefficient
bounds for the class of normalised univalent harmonic mappings of the
disk Donto R-convex domains.
Consider a harmonic and sense-preserving function fin D. It is well-
known (cf., [5]) that every such function fhas a form f=h+g, where
h, g are analytic in Dand
h(z) =
∞
X
k=0
akzk, g(z) =
∞
X
k=1
bkzk.(2)
Harmonic mappings onto R-convex domains 39
The dilatation ω(z) = g0(z)/h0(z)of sense-preserving harmonic function
fis analytic in Dand |ω(z)|<1for all z∈D. The tangent vector τ(t)
to curve Γr=f(γr), r ∈(0,1),at the point f(z), z =reit,has a form
τ(t) = izh0(z)−zg0(z)and arg τ(t) = π
2+ Im ln zh0(z)−zg0(z).
So, direct calculations show that the curvature kf(z) = (arg τ(t))0/|τ(t)|
of Γrcan be computed as
kf(z) = 1
|zh0(z)−zg0(z)|Re (z2h00(z) + z2g00(z)+2zg0(z)
zh0(z)−zg0(z)+ 1).(3)
It is clear that harmonic sense-preserving function fshould be univa-
lent in Drif kf(z)>0for all z∈γrand domains Dr=f(Dr)will be
convex in this case. This is a corollary of the argument principle [5]. It is
natural to ask what conditions for the function fguarantee R-convexity
of Drfor r∈(0,1].
2. Conditions of R-convexity. It is well-known (cf., [3,5]) that
harmonic functions f=h+gdo not possess the heredity property in
the case of convexity of f(D). If domain f(D)is convex for a harmonic
univalent function fthen f(Dr)can be not convex for all r∈(r0(f),1).
So we can’t expect the R-convexity of f(Dr)when f(D)is R-convex and
function fis harmonic. The next result describes the R-convexity of f(Dr)
in terms of curvature of its boundary.
Theorem 1.Let f=h+gbe a sense-preserving harmonic mapping of
the unit disk Dand r∈(0,1). The domain Dr=f(Dr)is R-convex if
and only if
Rez2h00(z) + z2g00 (z) + zh0(z) + zg0(z)
zh0(z)−zg0(z)>|zh0(z)−zg0(z)|
R
for all zsuch that |z|=r.
Proof. To prove this criterion it is sufficient to note that the arbitrary
infinitely-smooth Jordan domain is R-convex if and only if the curvature
of its boundary is not less than 1/R at every point. This fact was proved
in [18] by V. Starkov and N. Shmelev in the course of deriving of the main
results.
40 S. Yu. Graf
In our case, if function fis sense-preserving harmonic in D, then the
curve ∂f (Dr)is infinitely smooth for any r∈(0,1) and kfis defined on
|z|=r. The condition of the Theorem 1 allows us to state that, if domain
Dr=f(Dr)is R-convex, then kf(z)>1/R for all z, |z|=r. And vice
versa, if kf(z)>1/R for |z|=r, then kfis positive, so fis univalent
convex in Drand Dris Jordan and R-convex. Then formula (3) provides
us desired criterion of R-convexity of Dr.
The natural question is to describe R-convexity of an open domain
D=f(D)in terms of harmonic mappings onto this domain. The sufficient
condition of R-convexity is given by
Theorem 2.Let f=h+gbe a sense-preserving harmonic mapping of
the unit disk D. The domain D=f(D)is R-convex if
lim inf
|z|→1kf(z)>1
R,(4)
where kfis given by (3).
Proof. It was proved in [18] that if domain Dis ˜
R-convex for any ˜
R > R,
then Dis R-convex. Let a harmonic function fbe sense-preserving in
the unit disk Dand satisfy condition (4) for some R > 0, but domain
D=f(D)be not R-convex. Then there exists ε > 0such that Dis
not Rε-convex, where Rε=R+ε. Hence, there exists a pair of points
w1, w2∈Dsuch that |w1−w2|62Rεand convex hull ERε(w1, w2)6⊂ D.
From the other side, condition (4) means that for any ε > 0we can find
rε∈(0,1) such that kf(z)>1/Rεon the whole circle
γr={z∈D:|z|=r}for any r∈(rε,1). So, the curvature of the
image f(γr)of the circle γris not less than 1/Rε. Theorem 1implies
that Dr=f(Dr)are Rε-convex for all r∈(rε,1). Hence, the harmonic
function fis sense-preserving and convex in all such Drand, therefore, f
is univalent in D. It is clear that D=∪r∈(rε,1)Dr. So, both points w1, w2
belong to Drfor all sufficiently large r∈(rε,1). The Rε-convexity of do-
mains Drfor such rimplies that ERε(w1, w2)⊂Dr⊂Din contradiction
with assumption ERε(w1, w2)6⊂ D. Therefore, domain Dis R-convex.
Remark. The converse statement to Theorem 2is not true. Even for the
harmonic sense-preserving automorphisms of the unit disk Dthe values
lim inf|z|→1kf(z)can be negative.
Harmonic mappings onto R-convex domains 41
To illustrate this remark we consider function
θ(t) = (2t, for t∈(0,π),
0,for t∈[−π,0].
This function induces the continuous mapping w(eit) = eiθ(t)of the
unit circle onto itself such that w(eit)runs once monotonically (but not
strictly monotonically) unit circle while truns from −πto π. It is known
from Rad´o-Knezer-Choquet theorem [5] that the Poisson integral
fθ(z) = 1
2π
π
Z
−π
1− |z|2
|eit −z|2eiθ(t)dt
defines univalent harmonic mapping of Donto itself with boundary func-
tion eiθ(t). The boundary behaviour of fθis such that closed lower half of
the unit circle corresponds to the single point 1, while open upper half of
the unit circle is mapped onto whole circle without point 1. The image of
polar grid in Dunder mapping fθis presented on the left part of the Fig. 1.
Note that the images fθ(γr)of circles γrare not convex for sufficiently large
r < 1. The geometrical picture of fθ(γr)in the neighbourhood of the point
1is presented on the right part of Fig 1.
Figure 1: Image of polar grid in Dunder mapping fθ(left). The local non-
convex structure of the images of polar circles γrin the neighbourhood of
the point 1(right).
Using Wolfram Mathematica, it is possible to calculate the curvature
kfθ(z)of fθ(γr)at points z=reit. Let z=rtend from the origin to 1.
42 S. Yu. Graf
The image of this radius under fθis marked in the left-hand part of Fig. 1
by the bold line. Fig. 2. illustrates the values kfθ(r)when rtends to 1. It
can be seen from this dependence that curvatures kfθ(z)become negative
if zgoes to 1along some trajectories. So, lim inf |z|→1kfθ(z)<0. But,
from the other side, it is clear that the unit disk is an R-convex domain
with R= 1.
Figure 2: The curvature kfθ(r)for r→1−.
This counter-example leads us to the important fact that in contrast
to the analytic case, the set of harmonic univalent functions of the disk
Donto R-convex domains is wider than the family of harmonic sense-
preserving functions satisfying condition (4).
The next question is to find a maximal radius r0(R)such that sense-
preserving harmonic function maps every disk Dronto R-convex domain
for all r6r0(R). It is known [5] that near the origin every f=h+g
with h, g of form (2) and |b1|<|a1|maps infinitesimal disks |z|< ε onto
interiors of convex curves close to the infinitesimal ellipses {a0+a1z+b1z:
|z|=ε}. So, f(Dε)should be R-convex for sufficiently small ε > 0.
The linear hull L(f)of a sense-preserving harmonic function f=h+g
in Dwith a0=a1−1=0is defined as the linear-invariant family of all
harmonic sense-preserving functions
fΦ(z) = f◦Φ(z)−f◦Φ(0)
Φ0(0) ·h0◦Φ(0) ,
where Φ(z) = eiα(z−ζ)/(1 −ζz)runs over the family of the conformal
automorphisms of D.
The affine hull A(f)of a harmonic function fis defined as the family
Harmonic mappings onto R-convex domains 43
of all harmonic sense-preserving functions
fε(z) = f(z) + εf (z)
1 + εb1
,
where εruns over the disk D.
Let AL(f) := A(L(f)) denote the affine and linear hull of f. The sub-
family AL0(f)⊂AL(f)consists of functions ˜
fsuch that ˜
fz(0) = 0.Define
α0=α0(f) = sup |˜
fzz (0)|/2and β0=β0(f) = sup |˜
fzz (0)|/2, where the
suprema are taken over all ˜
f∈AL0(f). It is known [3] that α0is finite
for all univalent fand β061/2. The sharp upper bound of α0for an
arbitrary univalent harmonic function fis still unknown (though, conjec-
tured) [3]. However, the sharp upper bounds of α0have been obtained for
harmonic functions with some special geometric properties (cf., [3,5]). For
more results on linear- and affine-invariant families of harmonic functions,
see [6,11,16,17].
Theorem 3.Let a sense-preserving harmonic mapping f=h+gof the
unit disk Dhave form (2)and a0=a1−1 = 0, α0<∞. Then the domain
Dr=f(Dr)is R-convex for every r6r0(R), where
r0(R)∈(0, α0+β0−p(α0+β0)2−1) is the smallest positive root of
the equation
1− |b1|
(1 + |b1|)21−r
1 + rα0+3/2r2−2r(α0+β0)+1
r=1
R.(5)
Proof. Let fsatisfy the conditions of Theorem 3. The upper and lower
bounds of curvature kf(z)of images f(γr)of the circle γr, r ∈(0,1),in
the linear- and affine-invariant families of harmonic sense-preserving in D
functions fwere published in [11]. In particular, it was proved that fis
convex in the disk |z|< α0+β0−p(α0+β0)2−1and
kf(z)>1− |b1|
(1 + |b1|)21−r
1 + rα0+3/2r2−2r(α0+β0)+1
r
for all zsuch that |z|6r6α0+β0−p(α0+β0)2−1and |b1|=|fz(0)|.
The validity of Theorem 3and condition (5) follows from this result im-
mediately.
The values α0and β0are known to be 3/2and 1/2if a harmonic
function fis convex, i. e., if f(D)is a convex domain. Then Theorem 3
leads to
44 S. Yu. Graf
Corollary 1.Let f=h+gbe a sense-preserving harmonic mapping
of the unit disk Donto a convex domain, such that a0=a1−1 = b1= 0.
Then the domains Dr=f(Dr)are R-convex for every r6r∗(R)62−√3,
where r∗(R)is the smallest positive root of the equation
1−r
1 + r3r2−4r+ 1
r=1
R.
Note that the radius r∗(R)is not the best possible, because every
convex harmonic function fsuch that a0=a1−1 = b1= 0 is convex in
any disk Drfor r6√2−1(cf., [5]) and 2−√3<√2−1.
3. Coefficient bounds. In this section we introduce the families
of normalized univalent harmonic functions onto R-convex domains and
investigate their properties.
Let f=h+g, where h, g have form (2), and fbe sense-preserving. It is
clear that the value a0=h(0) does not influence the radius of R-convexity
of f(D). Also, if fis harmonic and the domain f(D)is R-convex for some
R∈(0,+∞), then f(D)/a1is ˜
R-convex for ˜
R=R/|a1|. Therefore let us
assume that a0=a1−1 = 0.
Definition 2.Let R∈(0,+∞)be fixed and CH,R denote the set of
all harmonic sense-preserving functions in the disk Dsuch that f(D)is
R-convex and a0=a1−1 = 0.In addition, C0
H,R denotes the subset of
CH,R that consists of all fwith b1= 0.
As has been mentioned above, the family of harmonic functions onto
R-convex domains is wider than the family of harmonic convex functions
of bounded type (satisfying inequality (4)). Proposition 1claims that in
the analytic case the family CRof normalized analytic convex function of
bounded type consists of analytic mappings onto R-convex domains. So,
CR⊂C0
H,R ⊂CH,R.
It is known [8] that CRis empty for R < 1and C1consists of one
element f≡zonly. In the harmonic case, the same property also holds.
Proposition 2.The family CH,R is empty for any R < 1. The only
member of the family CH,1is f≡z.
Proof. As we have mentioned above, every R-convex domain Dis an
image of Dunder some analytic convex function of bounded type. It
was proved [9] that Dis contained in some disk of radius Rin this case.
Harmonic mappings onto R-convex domains 45
Therefore, if f∈CH,R then f(D)is strictly convex subdomain of some
disk of radius R. Then (cf., [5]) fhas a continuous boundary function
f(eit) = ρ(t)eiθ(t)on ∂Dwith continuous real ρ(t), θ(t), t ∈[0,2π); so, the
generalisation of the Rad´o-Knezer-Choquet theorem claims that fis the
Poisson integral:
f(z) = c+1
2π
2π
Z
0
1− |z|2
|eit −z|2ρ(t)eiθ(t)dt,
where ρ(t)6Rfor all t∈[0,2π).It is easy to see that
|a1|=|fz(0)|=
1
2π
2π
Z
0
ρ(t)ei(θ(t)−t)dt
61
2π
2π
Z
0
ρ(t)dt.
Therefore, |a1|<1if ρ(t)6R < 1and CH,R is empty in this case. If
R= 1, then the equality |a1|= 1 implies ρ(t)≡1on [0,2π).Therefore,
F(z) = f(z)−cmaps the disk Dexactly onto itself, if a1= 1. It is
known [4] that the only harmonic automorphism of Dwith a1= 1 is
F≡z. Thus f≡z+c≡zbecause of f(0) = 0.
The next theorem is an analogue of an area principle for harmonic
mappings onto R-convex domains.
Theorem 4.Let f=h+g∈CH,R where h, g have form (2). Then
2
∞
X
k=1
k2|ak|2
1
Z
0
r2k−1(1 −r2)
(1 + |b1|r)2dr 6R2
1− |b1|2.(6)
Particularly, if f∈C0
H,R then
1
2+
∞
X
k=2
k
k+ 1|ak|26R2.(7)
Proof. As we have indicated above [9], every R-convex domain Dis con-
tained in some disk of radius R. Therefore, area of D=f(D)is not greater
than πR2for any harmonic function f∈CH,R. Jacobian of a harmonic
function fis equal to |h0|2−|g0|2. Also, the dilatation ω=g0/h0is analytic
in Dand meets the condition of the Schwarz lemma (cf., [7]). Therefore,
ω(z)−ω(0)
1−ω(0)ω(z)
6|z|and |ω(z)|6|z|+|ω(0)|
1 + |ω(0)z|
46 S. Yu. Graf
as a consequence. For f∈CH,R, the coefficient b1=ω(0) and the following
estimations of area of Dare true:
πR2>ZZ
D
du dv =ZZ
D
(|h0(z)|2− |g0(z)|2)dx dy =
=ZZ
D
|h0(z)|2(1 − |ω(z)|2)dx dy >ZZ
D
|h0(z)|21−|z|+|b1|
1 + |b1z|2dxdy =
=
2π
Z
0
1
Z
0
|h0(reit)|2(1 − |b1|2)(1 −r2)
(1 + |b1|r)2r drdt =
= (1 − |b1|2)
1
Z
0
r(1 −r2)
(1 + |b1|r)2Z2π
0|h0(reit)|2dt dr.
For h(z) = z+
∞
P
k=2
akzkthe series
|h0(reit)|2=
∞
X
k=1
k akrk−1ei(k−1)t
2
=
∞
X
k,l=1
k l akalrk+l−2ei(k−l)t
converges uniformly by tfor a fixed ras a product of two uniformly con-
verging series P∞
k=1 kakrk−1ei(k−1)tand P∞
l=1 l alrl−1e−i(l−1)t. The system
of exponential functions {eikt}is orthogonal on [0,2π]. Then integration
gives
2π
Z
0
|h0(reit)|2dt =
2π
Z
0
∞
X
k=1
k akrk−1ei(k−1)t
2
dt = 2π
∞
X
k=1
k2|ak|2r2k−2.
Continuing the lower estimation of the area, we obtain:
πR2>2π(1 − |b1|2)
∞
X
k=1
k2|ak|2
1
Z
0
r2k−1(1 −r2)
(1 + |b1|r)2dr
and inequality (6) is proved. If a function f∈C0
H,R, then b1= 0 and the
second inequality (7) in Theorem 4follows from (6).
The area theorem allows us to obtain coefficient estimations for har-
monic mappings onto R-convex domains.
Harmonic mappings onto R-convex domains 47
Corollary 1.If f∈C0
H,R, then
|ak|6k+ 1
kR2−1
21/2
for k>2.
Indeed, the coefficient bounds for akfollow immediately from (7), be-
cause 1/2 + k/(k+ 1)|ak|26R2.
The coefficient problem is one of the most attractive and complicated
in the theory of univalent harmonic mappings (cf., [5]). The sharp bound
is still not proved even in the case of |a2|in the family S0
Hof harmonic
univalent mapping from Dinto Csuch that a0=a1−1 = b1= 0. However,
for some special subclasses of S0
Hthe sharp coefficient estimations are
known. One of such subclasses was defined as the family S0
H(S)of all
f=h+g∈S0
Hsuch that F=h+eiθg∈Sfor some constant θ∈R.
Here Sdenotes the famous class of univalent analytic functions Fin D
such that F(0) = F0(0) −1=0. Several years ago S. Ponnusamy and
A. Sairam Kaliraj [15] obtained the sharp coefficient estimations in S0
H(S)
and conjectured that S0
H(S) = S0
H. However, recently this conjecture was
proved to be wrong [2]. Here we follow the same manner to obtain the
coefficient bounds in the analogous subclass of C0
H,R.
Definition 3.Let C0
H,R(CR)denote the subclass of C0
H,R consisting of
functions f=h+gsuch that F=h+eiθg∈CRfor some constant θ∈R.
K.-J. Wirths [19] obtained the sharp upper bounds for coefficients
A2, A3for functions F(z) = z+A2z2+A3z3+... in the families CR
of analytic convex functions of bounded type:
|A2|6r1−1
R,|A3|61−1
R.(8)
Here we prove similar estimations in C0
H,R(CR).
Theorem 5.Let f=h+g∈C0
H,R(CR). Then
|a2|61
2+r1−1
R,|a3|64
3−1
R+2
3r1−1
R.(9)
Both estimations are sharp when R→ ∞.
48 S. Yu. Graf
Proof. Analytic and co-analytic parts of harmonic function f∈C0
H,R sat-
isfy the equality g0=ω·h0, where ωis the dilatation of fand
|ω|<1, ω(0) = 0. Let ω(z) = P∞
k=1 ckzkin D. Then
∞
X
k=2
k bkzk−1=∞
X
k=1
ckzk·∞
X
k=1
k akzk−1
where a1= 1 and, hence, b2=c1/2, b3= (c2+ 2c1a2)/3.Analytic function
ωmeets the conditions of the Schwarz lemma (cf., [7]). Therefore, |c1|61
and |c2|61− |c1|2(see [1], for instance).
If f=h+g∈C0
H,R(CR), then, by definition, there exists some θ∈R
such that F(z) = h(z) + eiθg(z)∈CR.The second coefficient A2of an
analytic convex function Fof bounded type has the form A2=a2+eiθb2.
Then, using (8), we have
|a2|6|A2|+|b2|6|A2|+|c1|
261
2+r1−1
R.
To estimate the third coefficient, we note that A3=a3+eiθb3, where
b3= (c2+ 2c1a2)/3,|a2|6|A2|+|c1|/2and |c2|61− |c1|2. Therefore,
applying the second inequality (8), we conclude that
|a3|6|A3|+1
3|c2+ 2c1a2|6|A3|+1
31− |c1|2+|c1|(2|A2|+|c1|)=
=|A3|+1
3(1 + 2|c1||A2|)64
3−1
R+2
3r1−1
R.
If R→ ∞, then (9) becomes |a2|63/2and |a3|62. This estimations
coincide with the known sharp coefficient bounds for the convex harmonic
mappings fsuch that a0=a1−1 = b1= 0. So, the sharpness of (9) for
R→ ∞ is proved.
Note that inequalities (9) take the form |a2|61/2,|a3|61/3when
R→1.There are the best possible estimations [4] in the family of non-
normalised harmonic automorphisms of the unit disk D.
Acknowledgement. The Author expresses gratitude to prof. V. Star-
kov for fruitful discussion and comments.
This work was supported by the Russian Science Foundation, project
17-11-01229.
Harmonic mappings onto R-convex domains 49
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Received April 22, 2019.
In revised form, June 6, 2019.
Accepted June 6, 2019.
Published online June 9, 2019.
Tver State University,
33, Zheliabova str., Tver, Russia.
Petrozavodsk State University,
33, Lenina pr., Petrozavodsk, Russia.
E-mail: Sergey.Graf@tversu.ru
