Coefficient Estimates and Bloch’s Constant in Some Classes of Harmonic Mappings
Bull. Malays. Math. Sci. Soc.
Coefficient Estimates and Bloch's Constant in Some Classes of Harmonic Mappings
S. Kanas 0 1
D. KlimekSme¸t 0 1
Mathematics Subject Classification 0 1
B S. Kanas 0 1
0 Department of Mathematics, Maria CurieSklodowska University in Lublin , Pl. Marii CurieSkłodowskiej 5, 20031 Lublin , Poland
1 Faculty of Mathematics and Natural Sciences, University of Rzeszow , ul. S. Pigonia 1, 35959 Rzeszów , Poland
Following Clunie and SheilSmall, the class of normalized univalent harmonic mappings in the unit disk is denoted by SH. The aim of the paper is to study the properties of a subclass of SH, such that the analytic part is a convex function. We establish estimates of some functionals and bounds of the Bloch's constant for coanalytic part.
Univalent harmonic mappings; Convex functions; Bloch's constant; Normal family

30C75 · 30C45 · 30D45
1 Introduction
A complexvalued harmonic function f that is harmonic in a simply connected domain
⊂ C has the canonical representation
Communicated by V. Ravichandran.
f = h + g,
where h and g are analytic in with g(z0) = 0 for some prescribed point z0 ∈
. According to a theorem of Lewy [
17
], f is locally univalent, if and only if its
Jacobian J f (z) =  fz (z)2 −  fz¯ (z)2 = h (z)2 − g (z)2 does not vanish, and is
sensepreserving if the Jacobian is positive. Then h (z) = 0 and the analytic function
ω = g / h , called the second complex dilatation of f , has the property ω < 1 in .
Throughout this paper, we will assume that f is locally univalent and sensepreserving,
and we call f a harmonic mapping. Also, we assume = D ⊂ C, and z0 = 0, where
D is the open unit disk on the complex plane. The class of all sensepreserving univalent
harmonic mappings of D with h(0) = g(0) = h (0) − 1 = 0 is denoted by SH, and
0 (cf. [
8
]). Fundamental informations about harmonic
imtsaspupbincglasssinf othregp(l0a)ne=ca0nbbyeSfHound in [
11
]. Note that each f satisfying (1.1) in D is
uniquely determined by coefficients of the following power series expansions
then
ω(z) = c0 + c1z + c2z2 + · · · ,
cn ≤ 1 − c02,
∞
n=0
∞
n=1
h(z) =
an zn, g(z) =
bn zn (z ∈ D),
with an ∈ C, n = 0, 1, 2, ..., and bn ∈ C, n = 1, 2, 3, .... When f ∈ SH, then
a0 = 0, a1 = 1.
α of SH, consisting of all
In [
14
], the authors studied the properties of a subclass SH
univalent antianalytic perturbations of the identity in the unit disk with b1 = α, and
in [
15
], the authors studied the class Sα of all f ∈ SH, such that b1 = α ∈ (0, 1) and
h ∈ CV, where CV denotes the wellknown family of normalized, univalent functions
which are convex.
The classical Schwarz–Pick estimate for an analytic function ω which is bounded
by one on the unit disk of the complex plane is the inequality
ω (z) ≤
1 − ω(z)2
1 − z2
(z < 1).
Ruscheweyh [
21
] has obtained the bestpossible estimates of higher order derivatives
of bounded analytic functions on the disk. Similar estimates were derived by other
methods and for different classes of analytic functions in one and several variables by
Anderson and Rovnyak [
1
]
(1 − z2)n−1 ω(n)(z)
n!
≤
1 − ω(z)2
1 − z2
(n = 1, 2, ...).
The case z = 0 in (1.4) asserts that if
(1.1)
(1.2)
(1.3)
(1.4)
(1.5)
(1.6)
for every n ≥ 1. This result is classical and due to Wiener; see [
2,16
].
2 Bounds of the Fekete–Szegö and Other Functionals
Theorem 2.1 Let f ∈ Sα, f = h + g¯ with the power series (1.2). Then
bn ≤ α +
(1 − α2)(n − 1)
2
(n = 2, 3, ...).
Proof Making use of a relation g = ωh and the power series expansions (1.2) and
(1.5), we obtain
n−1
p=0
nbn =
( p + 1)a p+1cn− p−1 (n = 2, 3, ...).
Since h ∈ CV, ak  ≤ 1 (k = 1, 2, ...). Applying this for (2.2), we have
For the case n = 2, the inequality is sharp, with the equality realized by the function
f (z) = 1 −z z + 1 −z z − 11 −+ αα log 11+−αzz
.
We note that for α close to 1, the above bounds are better than that obtained in [
15
].
1 n−1
p=0
bn ≤ n
( p + 1)cn− p−1.
bn ≤ α + n1 np−=20( p + 1)(1 − α2)
= α +
(1 − α2)(n − 1)
2
.
b2 ≤ α +
, b3 ≤ 1 + α − α2.
The fact g = ωh , for the case z = 0, implies that c0 = b1, so that by (1.6), we obtain
cn− p−1 ≤ 1 − b12 = 1 − α2. Therefore,
Specially, we get
(2.1)
(2.2)
(2.3)
In conclusion, we obtain
Corollary 2.2 Let f ∈ Sα, f = h + g¯ with the power series (1.2). Then
bn ≤ min α +
(1 − α2)(n − 1), α + (n − α2)(n − 1)
2 n
(n = 2,3,...).
Theorem 2.3 Let f ∈ Sα, f = h + g¯ with the power series (1.2). Then for μ ∈ R
1 − α2 3
b3 − μb22 ≤ 3 1 + 4μ(1 − α2) + 2 − 3μb1 + α max 31,1 − μb1 ,
and
Then
Proof From the relation (2.2), we have
bn+1 − bn ≤ 2α + 1 − α2 2n2− 1.
2b2 = c1 + 2a2c0, 3b3 = c2 + 2a2c1 + 3a3c0.
(2.4)
(2.5)
(2.6)
1 2 1
b3 − μb22 = 3c2 + 3a2c1 + a3c0 − μ 2c1 + a2c0
2
Apply now the estimate that holds for the coefficients of convex functions: an ≤
1 (n = 2,3,...),a3 − νa22 ≤ max{1/3,1 − ν} (ν ∈ R), and the relation (1.6). We
obtain then
b3 − μb22 ≤ 3 c2 + 43 μc12 + a2c1 3 − μc0 + αa3 − μb1a22
1 2
1 − α2 3
≤ 3 1 + 4μ(1 − α2) + (1 − α2) 23 − μb1
+α max{1/3,1 − μb1}.
Next, by (2.2), we have
bn+1 − bn =
1
n + 1
n+1
p=1
1 n
pa pcn+1− p − n
1
≤ an+1c0 + anc0 + n + 1
1 − α2 n
≤ 2α + n + 1
p=1
= 2α + (1 − α2) 2n 2− 1 .
p +
The proof is now complete; however, the results are not sharp, for example, the function
that realizes the accuracy of b2 in the previous theorem gives b2−b1 = (1−α2)/2 ≤
2α + (1 − α2)/2, for any α ∈ (0, 1). The righthand side is obtained from (2.6) for
the case when n = 1.
Theorem 2.4 For f ∈ Sα, f = h + g¯ and z = r < 1, it holds
r 1 − α log 1 + r
1 + r − 1 + α 1 − αr
2r r (1 − α2)
1 + r − r − α(1 − αr ) ≤
zg (z)
Re 1 + >
g (z)
≤ g(z) ≤ 1 −r r + 11 +− αα log 11+−αrr , (2.7)
zg (z) 2r r (1 − α2) , (2.8)
g (z) ≤ 1 − r + r − α(1 − αr )
r (α2 − 1) 1 − r
α − r (1 − αr ) + 1 + r
.
(2.9)
Proof Applying the relation g = ωh , we estimate g (z) as follows [
15
]:
α − r  α + r
(1 − αr )(1 + r )2 ≤ g (z) ≤ (1 + αr )(1 − r )2
(z = r < 1).
Then integrating along a radial line ζ = t eiθ , the righthand side of (2.7) is obtained
immediately [
15
].
In order to prove the lefthand side of (2.7), we note first that g is univalent. Let
= g({z : z = r }) and let ξ1 ∈ be the nearest point to the origin. By a rotation
we may assume that ξ1 > 0. Let γ be the line segment 0 ≤ ξ ≤ ξ1 and suppose that
z1 = g−1(ξ1) and L = g−1(γ ). With ζ as the variable of integration on L, we have
that dξ = g (ζ )dζ > 0 on L. Hence
ξ1 =
dξ =
g (ζ )dζ =
g (ζ )dζ  ≥
g (t eiθ )dt
z1
0
z1
0
0
r
α − r 
(1 − αr )(1 + r )2 dr =
Moreover, ω satisfies [12, p. 320]
from which it follows
From the relation g = ωh , we obtain
Since h is convex, so is univalent, then it holds [13, p. 118]
zg (z)
g (z) =
zωω((zz)) + zhh ((zz)) .
2r
1 + r ≤
zh (z)
h (z)
We note that ω(0) = c0 = b1 = α, so that by (2.13) we have
1r −−ααr ≤ ω(z) ≤ 1r ++ααr .
Taking into account (2.10), (2.11), and (2.14) and the Schwarz–Pick inequality
(1.3), we obtain for z = r < 1,
zg (z)
g (z)
zω (z) zh (z)
≤ ω(z) + h (z)
r (1 − ω(z)2) 2r
≤ ω(z)(1 − r 2) + 1 − r
r (1 − r 2)(1 − α2) 2r
≤ (1 − r 2)r − α(1 − αr ) + 1 − r
r (1 − α2) 2r .
= r − α(1 − αr ) + 1 − r
Similarly, we have
Moreover
and h is convex, therefore
By the above, (1.3) and (2.14), we have
zg (z)
Re 1 + g (z)
zω (z) zh (z)
= Re ω(z) + Re 1 + h (z)
> Re zωω((zz)) + 11 −+ rr .
zg (z)
Re 1 + g (z)
> α −r(αr2(1−−1)αr ) + 11 −+ rr ,
as asserted.
3 Estimates of the Bloch’s Constant
A harmonic function f is called the Bloch function if where
B f =
sup
z,w∈D,z=w
 f (z) − f (w) < ∞,
(z, w)
(z, w) = 21 log
z−w ⎞
⎛ 1 + 1−z¯w
⎝ 1 − 1−z¯w
z−w ⎠
= artanh
z − w
1 − z¯w
denotes the hyperbolic distance in D, and B f is called the Bloch’s constant of f . The
harmonic Bloch’s constant was studied by Colonna [
9
]. Colonna established that the
(2.15)
(2.16)
(2.17)
(3.1)
Bloch’s constant B f of a harmonic mapping f = h + g¯ can be expressed in terms of
moduli of the derivatives of h and g
B f = sup(1 − z2) h (z) + g (z)
z∈D
= sup(1 − z2)h (z)(1 + ω(z)),
z∈D
which agrees with the wellknown notion of the Bloch’s constant for analytic functions.
Moreover, the function f is Bloch if and only if both h and g are, and
max(Bh , Bg) ≤ B f ≤ Bh + Bg.
Colonna also obtained the bestpossible estimate of the Bloch’s constant for the family
of harmonic mappings of D into itself. Recently, the Bloch’s constant was studied by
many authors, see, for example [
3,4,19
]. Very interesting results in this direction
were obtained in [
5–7,18,20,22
]. Our aim is to determine the bounds for the Bloch’s
constant in the classes Sα and Sα.
Theorem 3.1 Let f = h + g¯ with h(z) = z/(1 − Bz), −1 < B < 1, and let
B = A, 0 ≤ A < 1. Then the Bloch’s constant B f is bounded by
,
where r0 is given by
r0 =
α(1 + 3 A) − 3 − A + √(1 + α)(1 + A)(9 − 7 A + α(−7 + 9 A))
4α + 2 A(α − 1)
.
Proof Applying the distortion theorem 1
h (z) ≤ (1 − Ar )2
(z = r ),
and (3.2), we find
Setting
(3.2)
(3.3)
(3.4)
B f = sup(1 − z2)h (z)(1 + ω(z)) ≤ (1 + α) sup
z∈D 0≤r<1 (1 − Ar )2(1 + αr )
.
(1 + r )(1 − r 2)
(1 + r )(1 − r 2)
q(r ) = (1 − Ar )2(1 + αr ) ,
we observe that q (r ) = 0, if and only if
(1 + r )[(2α + α A − A)r 2 + (3 + A − α − 3α A)r + α − 1 − 2 A] = 0.
Remark By the fact that the Bloch’s constant is finite, we already have that the family
of harmonic mappings with h(z) ≡ z, and b1 = α, is a normal family. A function f
is normal, if the constant σ f is finite, where
,
r0 =
α − 3 +
√
.
σ f = sup
z∈D
(1 − z2) f (z)
1 +  f (z)
,
see [
10
]. Indeed, since the quantity  f (z) in Sα is bounded [
15
]
therefore, by (3.5) and (3.7), we obtain
⎧ (1 + α) (1+r)(1−r2)
σ f ≤ ⎨⎪ 1+αr
⎪ (1 + α) (1+r)(1−r2)
⎩ 1+αr
The last equation has solution in the interval (0, 1) at the point r0 given by (3.4), and
the function q attains its maximum at r0.
Setting B = 0 in the above theorem, we obtain the estimate of B f in the class S¯α,
below.
Corollary 3.2 For f ∈ S¯α, f = h + g¯, the Bloch’s constant B f is bounded by
where r = r0 is given by (3.6), and we see that in both cases σ f is finite.
Remark The univalent Bloch functions can be described in terms of geometry of
their images; they are precisely those functions whose images do not contain disks of
arbitrarily large radius [
10
]. Therefore, we suppose that the functions from the class
Sα may not be the Bloch functions. Indeed, reasoning similarly as in the Theorem 3.1
we note that in the class Sα we have h(z) = z/(1 − z), then h (z) ≤ 1/(1 − r )2.
Thus
B f = (1 + α) sup
0≤r<1 (1 − r )(1 + αr )
,
(1 + r )2
for α = 0,
for α = 0,
for α = 0,
(3.5)
(3.6)
(3.7)
(3.8)
and the function p(r ) = (1 + r )2/[(1 − r )(1 + αr )] increases in the whole interval
(0, 1), with infinity as the supremum.
Acknowledgments This work was partially supported by the Centre for Innovation and Transfer of
Natural Sciences and Engineering Knowledge.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0
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