CC BY
62DOI: 10.15393/j3.art.2019.6190
UDC 517.54
S. Yu. Graf
HARMONIC MAPPINGS ONTO R-CONVEX DOMAINS
Abstract. The plane domain D is called R-convex if D contains each compact set bounded by two shortest sub-arcs of the radius R with endpoints w1, w2 E D, \w1 — w2\ ^ 2R. 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 w\,w2 E c such that \wi — w2\ ^ 2R with 0 < R < ro we define the R-convex hull ER(wi,w2) of w1, w2 as a compact set bounded by the two shortest arcs of the circles of radius R with endpoints w1 ,w2. The set ER(w1,w2) is strictly convex for each R > 0 and tends to the segment [w1, w2] when R ^ ro.
Definition 1. The set A C c is called R-convex if A contains each set ER(w1,w2) provided that w1,w2 E A and \w1 — w2\ ^ 2R.
It is clear that the Jordan domain D C c is R-convex if and only if its closure D is R-convex. Of course, R-convex domains are strictly convex.
R-convex sets E were 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 sciences.
A.W. Goodman [8] defined the convex functions of bounded type independently in the geometrical function theory as an univalent analytic
function h in the unit disk d = {z E c : \z\ < 1}, such that liminf kh(z) ^
kK1
1/R > 0. Here kfc(z) = Re{zh//(z)/h'(z) + 1}/\zh'(z)\ is the curvature of
© Petrozavodsk State University, 2019
the image rr = h(Yr) of the circle Yr = {z E d : |z| = r} under the mapping h at the point h(z). The curvature kh is defined by a standard way as kh = d9/ds, where s is the natural parameter on rr and 9 is the argument of the tangent vector to rr.
Let R E (0, + to) be given and CR denote a family of convex analytic functions h in d of bounded type, such that h(0) = h'(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 CR were proved, as well as coefficient bounds.
Interrelation between R-convex domains and analytic univalent functions 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 h in d the domain D = h(d) is R-convex if and only if
^ r h"(z) , |zh'(z)| „ „ Re I+ 1} ^ 1—tt21 for all z E d. (1)
1 h'(z) J R
More than that, (1) is equivalent to the statement that domains Dr = h(dr) are R-convex for all r E (0,1], where dr = {z E d : |z| < r}. So, the heredity property is valid for R-convexity of h(dr) in the case of analytic functions. Functions h that satisfy condition (1) are univalent.
The theorem of Peschl (cf., [12,19]) claims that if h E CR, R > 0, then kh(z) has not local minimums in d \ {0} and kh(z) > 1/R in d \ {0}. Therefore, h is an analytic convex function of bounded type if and only if (1) is true in d for some R > 0. This result, together with criterion (1) of R-convexity due by V. Starkov and N. Shmelev, immediately leads to
Proposition 1. CR consists of univalent analytic functions h in d such that h(0) = h'(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 d onto R-convex domains.
Consider a harmonic and sense-preserving function f in d. It is well-known (cf., [5]) that every such function f has a form f = h + g, where h, g are analytic in d and
h(z) = Y^ akzk, g(z) = Y bkzk. k=0 k=1
The dilatation w(z) = g'(z)/h'(z) of sense-preserving harmonic function f is analytic in d and |w(z)| < 1 for all z E d. The tangent vector t(t) to curve rr = f (Yr),r E (0,1), at the point f (z),z = rert, has a form
t(t) = i ^zh'(z) — zg'(zand argt(t) = ^ + Im ln ^zh'(z) — zg'(z)^ .
So, direct calculations show that the curvature kf (z) = (arg t(t))'/|t(t)| of rr can be computed as
kf (z) =_^Re z 2h'' (z) + ^W.g'M + !
|zh'(z) — zg'(z)| ^ zh'(z) — zg'(z)
It is clear that harmonic sense-preserving function f should be univalent in dr if kf (z) ^ 0 for all z E Yr and 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 f guarantee R-convexity of Dr for r E (0,1].
2. Conditions of R-convexity. It is well-known (cf., [3,5]) that harmonic functions f = h + g do not possess the heredity property in the case of convexity of f (d). If domain f (d) is convex for a harmonic univalent function f then f (dr) can be not convex for all r E (r0(f),1). So we can't expect the R-convexity of f (dr) when f (d) is R-convex and function f is harmonic. The next result describes the R-convexity of f (dr) in terms of curvature of its boundary.
Theorem 1. Let f = h + g be a sense-preserving harmonic mapping of the unit disk d and r E (0,1). The domain Dr = f (dr) is R-convex if and only if
n z2h''(z) + z2g''(z) + zh'(z) + zg'(z) |zh'(z) — zg'(z)|
Re-- ^ -5-
zh'(z) — zg'(z) R
for all z such 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.
In our case, if function f is sense-preserving harmonic in d, then the curve df (dr) is infinitely smooth for any r E (0,1) and kf is 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 kf is positive, so f is univalent convex in dr and Dr is 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 + g be a sense-preserving harmonic mapping of the unit disk d. The domain D = f (d) is R-convex if
limmf kf (z) ^ 1, (4)
|z|k1 r
where kf is given by (3).
Proof. It was proved in [18] that if domain D is R-convex for any R > R, then D is R-convex. Let a harmonic function f be sense-preserving in the unit disk d and satisfy condition (4) for some R > 0, but domain D = f (d) be not R-convex. Then there exists e > 0 such that D is not R£-convex, where R£ = R + e. Hence, there exists a pair of points w1 ,w2 E D such that \w1 — w2\ ^ 2R£ and convex hull er (w1,w2) C D.
From the other side, condition (4) means that for any e > 0 we can find r£ E (0,1) such that kf (z) ^ 1/R£ on the whole circle Yr = {z E d : \z\ = r} for any r E (r£, 1). So, the curvature of the image f (Yr) of the circle Yr is not less than 1/R£. Theorem 1 implies that Dr = f (dr) are R£-convex for all r E (r£, 1). Hence, the harmonic function f is sense-preserving and convex in all such dr and, therefore, f is univalent in d. It is clear that D = Ur€(r£,1)Dr. So, both points w1,w2 belong to Dr for all sufficiently large r E (r£, 1). The R£-convexity of domains Dr for such r implies that ERs (w1,w2) C Dr C D in contradiction with assumption er (w1,w2) C D. Therefore, domain D is R-convex. □
Remark. The converse statement to Theorem 2 is not true. Even for the harmonic sense-preserving automorphisms of the unit disk d the values liminf|z|K1 kf (z) can be negative.
To illustrate this remark we consider function
m=i2t, for t e (0,n),
( ) \0, for t e [—n,0].
This function induces the continuous mapping w(elt) = ei0(t) of the unit circle onto itself such that w(elt) runs once monotonically (but not strictly monotonically) unit circle while t runs from —n to n. It is known from Rado-Knezer-Choquet theorem [5] that the Poisson integral
n
/' (z) = 2lJ —rj ^ dt
—n
defines univalent harmonic mapping of d onto itself with boundary function ei0(t). The boundary behaviour of /<? is 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 d under mapping /0 is presented on the left part of the Fig. 1. Note that the images /0 (Yr) of circles 7r are not convex for sufficiently large r < 1. The geometrical picture of /0 (Yr) in the neighbourhood of the point 1 is presented on the right part of Fig 1.
Figure 1: Image of polar grid in d under mapping /0 (left). The local non-convex structure of the images of polar circles Yr in the neighbourhood of the point 1 (right).
Using Wolfram Mathematica, it is possible to calculate the curvature kfe (z) of /0 (Yr) at points z = relt. Let z = r tend from the origin to 1.
The image of this radius under f is marked in the left-hand part of Fig. 1 by the bold line. Fig. 2. illustrates the values kf (r) when r tends to 1. It can be seen from this dependence that curvatures kfe (z) become negative if z goes to 1 along some trajectories. So, liminf^^ kfg(z) < 0. But, from the other side, it is clear that the unit disk is an R-convex domain with 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 d onto 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 Dr onto R-convex domain for all r ^ r0 (R). It is known [5] that near the origin every f = h + g with h,g of form (2) and |bi| < |ai| maps infinitesimal disks |z| < e onto interiors of convex curves close to the infinitesimal ellipses {a0 + a1z + b1z : |z| = e}. So, f (de) should be R-convex for sufficiently small e > 0.
The linear hull L(f) of a sense-preserving harmonic function f = h+g in d with a0 = a1 — 1 = 0 is defined as the linear-invariant family of all harmonic sense-preserving functions
f o $(z) — f o $(0) /#(z) $'(0) ■ h' o $(0) ,
where $(z) = eia(z — ()/(1 — (z) runs over the family of the conformal automorphisms of d.
The affine hull A(f) of a harmonic function f is defined as the family
of all harmonic sense-preserving functions
f (z)+ gf(z) Mz) = 1+ gbi '
where g runs over the disk d.
Let AL(f) := A(L(f)) denote the affine and linear hull of f. The subfamily AL0(f) C AL(f) consists of functions f such that fz(0) = 0. Define ao = ao(f) = sup |/Zz(0)|/2_and ^o = &(f) = sup |/z(0)|/2, where the suprema are taken over all f E AL0(f). It is known [3] that a0 is finite for all univalent f and ^ 1/2. The sharp upper bound of a0 for an arbitrary univalent harmonic function f is still unknown (though, conjectured) [3]. However, the sharp upper bounds of a0 have 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 + g of the unit disk d have form (2) and a0 = ai — 1 = 0, a0 < Then the domain Dr = f (dr) is R-convex for every r ^ r0(R), where r0(R) E (0,a0 + — \J(a0 + ^0)2 — 1) is the smallest positive root of the equation
1 — |bi| (1 — r\a0+3/2 r2 — 2r(a0 + AO + 1 1 ,,,
— (5)
(1 + |bi|)2 V1 + r) r R'
Proof. Let f satisfy the conditions of Theorem 3. The upper and lower bounds of curvature kf (z) of images f (Yr) of the circle Yr,r E (0,1), in the linear- and affine-invariant families of harmonic sense-preserving in d functions f were published in [11]. In particular, it was proved that f is convex in the disk |z| < a0 + — \J(a0 + ^0)2 — 1 and
1 — |bi| (1 — r y°+3/2 r2 — 2r(a0 + ft) + 1 kf(z) ^ 11
(1 + |bi|)2 v 1 + rt
for all z such that |z| ^ r ^ a0 + — \J(a0 + ^0)2 — 1 and |bi| = |fz(0)|. The validity of Theorem 3 and condition (5) follows from this result immediately. □
The values a0 and are known to be 3/2 and 1/2 if a harmonic function f is convex, i.e., if f(d) is a convex domain. Then Theorem 3 leads to
Corollary 1. Let / = h + g be a sense-preserving harmonic mapping of the unit disk d onto a convex domain, such that a0 = a1 — 1 = b\ = 0. Then the domains Dr = /(dr) are R-convex for every r ^ r* (R) ^ 2 — v^, where r* (R) is the smallest positive root of the equation
(1 — r \3 r2 — 4r + 1 _ 1 \1 + r) r = R.
Note that the radius r*(R) is not the best possible, because every convex harmonic function / such that ao — ai — 1 = b1 = 0 is convex in any disk dr for r ^ ^2 — 1 (cf., [5]) and 2 — < V2 — 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 / = h+g, where h, g have form (2), and / be sense-preserving. It is clear that the value a0 = h(0) does not influence the radius of R-convexity of /(d). Also, if / is harmonic and the domain /(d) is R-convex for some R e (0, + to), then /(d)/a1 is R-convex for R = R/|a1|. Therefore let us assume that a0 = a1 — 1 = 0.
Definition 2. Let R e (0, + to) be fixed and CH,R denote the set of all harmonic sense-preserving functions in the disk d such that /(d) is R-convex and a0 = a1 — 1 = 0. In addition, CHR denotes the subset of CH,R that consists of all / with 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 1 claims that in the analytic case the family CR of normalized analytic convex function of bounded type consists of analytic mappings onto R-convex domains. So, Cr C CHR C Ch,R.
It is known [8] that CR is empty for R < 1 and C1 consists of one element / = z only. 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,1 is / = z.
Proof. As we have mentioned above, every R-convex domain D is an image of d under some analytic convex function of bounded type. It was proved [9] that D is contained in some disk of radius R in this case.
Therefore, if f G CH,R then f (d) is strictly convex subdomain of some disk of radius R. Then (cf., [5]) f has a continuous boundary function f (e^) = p(t)ei0(i) on dd with continuous real p(t),0(t),t G [0,2n); so, the generalisation of the Rado-Knezer-Choquet theorem claims that f is the Poisson integral:
2n
f (z) = C +2n/ lii-T^P^e*^ 0
where p(t) ^ R for all t G [0,2n). It is easy to see that
2n 2n
|ai| = Ifz (0)|
1 i p(t)ei(0(i)-i)di
^ 2~ y P(t)dt.
0
Therefore, |ai| < 1 if p(t) ^ R < 1 and CH,R is empty in this case. If R =1, then the equality |a1| = 1 implies p(t) = 1 on [0,2n). Therefore, F(z) = f (z) — c maps the disk d exactly onto itself, if a1 = 1. It is known [4] that the only harmonic automorphism of d with a1 = 1 is F = z. Thus f = z + c = z because 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 G CH,R where h, g have form (2). Then
1
~ C r2k-1(i r2) R2
2Ek2M2/V+T^< ^. (6)
k=1 0
Particularly, if f G CH R then
1 ^ k
^ + V ~r_k— k|2 ^ R2. (7)
2 ^ k + 11 k 1 v 1 k=2
Proof. As we have indicated above [9], every R-convex domain D is contained in some disk of radius R. Therefore, area of D = f (d) is not greater than nR2 for any harmonic function f G CH,R. Jacobian of a harmonic function f is equal to |h'|2 — |g'|2. Also, the dilatation u = g'/h' is analytic in d and meets the condition of the Schwarz lemma (cf., [7]). Therefore,
w(z) — w(0)
1 — w(0)w(z)
ii ii / \i Iz| + ^ |z| and |w(z)| ^
1 + |w(0)z |
as a consequence. For f G CH,R, the coefficient &i = w(0) and the following estimations of area of D are true:
nR2 ^ JJ dudv ^y(|h/(z)|2 - |g'(z)|2)dxdy
D D
= [[ |h/ (z)|2(1 -|^(z)|2)dxdy >[[ |h/(z)|^1 -
z| +
ny
2n 1
,2 (1 — I bl |2 )(1 - r2 )
|h/(re )|2
0 0
(1 + |bi|r)2 l
1 + |biz| J
r drdt
dxdy =
(1 — |bi|2)
2N />2n
r(1 — r2) (1 + |bi|r)^o
|h/(re )|2dtdr.
For h(z) = z + akzk the series k=2
|h/(reit)|2
^ kark-1 ei(k-1)t
: ak r e
k=1
2 œ
ark+l-2ei(k-l)t
k l ak aj r" 1 " "e"
k,l=1
converges uniformly by t for a fixed r as a product of two uniformly converging series rfc-1ei(fc-1)i and /01r1-1e-i(1-1)i. The system of exponential functions |eifci} is orthogonal on [0, 2n]. Then integration gives
2n
2n
|h/(reit)|2dt
£ k
k=1
akrk-1ei(k-1)t
dt = 2^k2|ak |2r2k-2
k=1
Continuing the lower estimation of the area, we obtain:
œ
nR2 ^ 2n(1 — |b1|2) J] k2|akf J ^
k=1 n
r2k-1(1 — r2) + |b1|r)2
dr
and inequality (6) is proved. If a function f E CHR, then bi = 0 and the second inequality (7) in Theorem 4 follows from (6). □
The area theorem allows us to obtain coefficient estimations for harmonic mappings onto R-convex domains.
2
1
Corollary 1. If f G CH, R, then
k| ^ (R2 - 1/2 for k > 2.
Indeed, the coefficient bounds for ak follow immediately from (7), because 1/2 + k/(k + l)|afc|2 ^ R2.
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 SH of harmonic univalent mapping from d into c such that a0 — a i — 1 = b1 — 0. However, for some special subclasses of SH the sharp coefficient estimations are known. One of such subclasses was defined as the family SH (S) of all f — h + g G SH such that F — h + e%eg G S for some constant 9 G r. Here S denotes the famous class of univalent analytic functions F in d such that F(0) — F'(0) — 1 — 0. Several years ago S. Ponnusamy and A. Sairam Kaliraj [15] obtained the sharp coefficient estimations in SH (S) and conjectured that SH(S) — SH. 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 C% R.
Definition 3. Let CH R(CR) denote the subclass of C°H R consisting of functions f — h + g such that F — h + e%eg G CR for some constant 9 G r.
K.-J. Wirths [19] obtained the sharp upper bounds for coefficients A2,A3 for functions F(z) — z + A2z2 + A3z3 + ... in the families CR of analytic convex functions of bounded type:
|A2| ^/l—R, | A31 ^ 1 — R. (8)
Here we prove similar estimations in CH R(CR). Theorem 5. Let f — h + g G C0H, r(Cr). Then
, , 1 I 1 . . 4 1 2 I 1 , ,
— R, |«3|< 3 — R + 3V1 — S- (9)
Both estimations are sharp when R ^ to.
Proof. Analytic and co-analytic parts of harmonic function f e CH R satisfy the equality g' = u ■ h', where u is the dilatation of f and |u| < 1, u(0) = 0. Let u(z) = E0= ckZk in d. Then
OO / OO \ / oo
Y kbkzk-1 = Y CkzM ■ Y kakzk-1 k=2 ^ k=1 ' ^ k=1
where a1 = 1 and, hence, b2 = c1/2, b3 = (c2 + 2c1a2)/3. Analytic function u meets the conditions of the Schwarz lemma (cf., [7]). Therefore, |c1| ^ 1 and |c2| ^ 1 — |c1|2 (see [1], for instance).
If f = h + g G c°,r(cR)> then, by ^fimti^ there exists some 9 G r such that F(z) = h(z) + e%eg(z) e CR. The second coefficient A2 of an analytic convex function F of bounded type has the form A2 = a2 + e%db2. Then, using (8), we have
M ^ |A2| + |b21 ^ |A2| + M ^ ± + ^1 — R.
To estimate the third coefficient, we note that A3 = a3 + e%eb3, where b3 = (C2 + 2c1a2)/3, H ^ |A2| + |cx|/2 and |C21 ^ 1 — |c112. Therefore, applying the second inequality (8), we conclude that
M ^ |A31 + 1 |C2 + 2^ ^ |A31 + 1 (1 — |C1|2 + |cx|(2|a2| + |d|)) = 1 4 12/ 1
= W + ^ + W) < 4 — R + 3V1 — 1
If R —y to, then (9) becomes |a2| ^ 3/2 and |a3| ^ 2. This estimations coincide with the known sharp coefficient bounds for the convex harmonic mappings f such that ao — a1 — 1 = b1 = 0. So, the sharpness of (9) for R — to is proved. □
Note that inequalities (9) take the form |a2| ^ 1/2, |a3| ^ 1/3 when 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.
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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: [email protected]
