Journal of Inequalities in Pure and
Applied Mathematics
STARLIKE LOG-HARMONIC MAPPINGS OF ORDER α
Z. ABDULHADI AND Y. ABU MUHANNA
Department of Mathematics
American University of Sharjah
Sharjah, Box 26666, UAE
volume 7, issue 4, article 123,
2006.
Received 20 September, 2005;
accepted 13 December, 2005.
Communicated by: Q.I. Rahman
EMail: zahadi@aus.ac.ae
EMail: ymuhanna@aus.ac.ae
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
281-05
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Abstract
In this paper, we consider univalent log-harmonic mappings of the form f =
zhg defined on the unit disk U which are starlike of order α . Representation
theorems and distortion theorem are obtained.
2000 Mathematics Subject Classification: Primary 30C35, 30C45; Secondary
35Q30.
Key words: log-harmonic, Univalent, Starlike of order α.
Starlike log-harmonic Mappings
of Order α
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Representation Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Distortion Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
Z. AbdulHadi and Y. Abu Muhanna
3
5
9
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J. Ineq. Pure and Appl. Math. 7(4) Art. 123, 2006
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1.
Introduction
Let H(U ) be the linear space of all analytic functions defined on the unit disk
U = {z : |z| < 1}. A log-harmonic mapping is a solution of the nonlinear
elliptic partial differential equation
(1.1)
fz
fz
=a ,
f
f
where the second dilation function a ∈ H(U ) is such that |a(z)| < 1 for all
z ∈ U . It has been shown that if f is a non-vanishing log-harmonic mapping,
then f can be expressed as
f (z) = h(z)g(z),
f (z) = z|z|2β h(z)g(z),
where Re β > −1/2, and h and g are analytic functions in U , g(0) = 1 and
h(0) 6= 0 (see [3]). Univalent log-harmonic mappings have been studied extensively (for details see [1] – [5]).
Let f = z|z|2β hg be a univalent log-harmonic mapping. We say that f is a
starlike log-harmonic mapping of order α if
∂ arg f (reiθ )
zfz − zfz
= Re
> α,
∂θ
f
Z. AbdulHadi and Y. Abu Muhanna
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where h and g are analytic functions in U . On the other hand, if f vanishes at
z = 0 but is not identically zero, then f admits the following representation
(1.2)
Starlike log-harmonic Mappings
of Order α
0≤α<1
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J. Ineq. Pure and Appl. Math. 7(4) Art. 123, 2006
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for all z ∈ U . Denote by STLh (α) the set of all starlike log-harmonic mappings of order α. If α = 0, we get the class of starlike log-harmonic mappings.
Also, let ST (α) = {f ∈ STLh (α) and f ∈ H(U )}. If f ∈ STLh (0) then
F (ζ) = log(f (eζ )) is univalent and harmonic on the half plane {ζ : Re{ζ} <
0}. It is known that F is closely related with the theory of nonparametric minimal surfaces over domains of the form −∞ < u < u0 (v), u0 (v + 2π) = u0 (v),
(see [7]).
In Section 2 we include two representation theorems which establish the
linkage between the classes STLh (α) and ST (α). In Section 3 we obtain a
sharp distortion theorem for the class STLh (α).
Starlike log-harmonic Mappings
of Order α
Z. AbdulHadi and Y. Abu Muhanna
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2.
Representation Theorems
In this section, we obtain two representation theorems for functions in STLh (α).
In the first one we establish the connection between the classes STLh (α) and
ST (α). The second one is an integral representation theorem.
Theorem 2.1. Let f (z) = zh(z)g(z) be a log-harmonic mapping on U , 0 ∈
/
zh(z)
hg(U ). Then f ∈ STLh (α) if and only if ϕ(z) = g(z) ∈ ST (α).
Proof. Let f (z) = zh(z)g(z) ∈ STLh (α), then it follows that
∂ arg f (reiθ )
zfz − zfz
= Re
f
∂θ
zh0
−
= Re 1 +
h
zh0
= Re 1 +
−
h
Z. AbdulHadi and Y. Abu Muhanna
zg 0
g
zg 0
> α.
g
Setting
ϕ(z) =
we obtain
Re
Starlike log-harmonic Mappings
of Order α
zh(z)
,
g(z)
zfz − zfz
zϕ0
= Re
> α.
f
ϕ
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Since f is univalent, we know that 0 ∈
/ fz (U ). Furthermore,
J. Ineq. Pure and Appl. Math. 7(4) Art. 123, 2006
ϕ ◦ f −1 (w) = q1 (w) = w|g ◦ f −1 (w)|−2 ,
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is locally univalent on f (U ).
0 z)
Indeed, we have zϕ
= (1 − a(z)) zff z 6= 0 for all z ∈ U . From Lemma 2.3
ϕ(z)
in [4] we conclude that ϕ is univalent on U . Hence ϕ ∈ ST (α).
Conversely, let ϕ ∈ ST (α) and a ∈ H(U ) such that |a(z)| < 1 for all
z ∈ U be given. We consider
Z z
a(s)ϕ0 (s)
(2.1)
g(z) = exp
ds ,
0 ϕ(s)(1 − a(s))
0
z)
where zϕ
= (1 − α)p(z) + α, and p ∈ H(U ) such that p(0) = 1 and
ϕ(z)
Re(p) > 0.
Also, let
ϕ(z)g(z)
h(z) =
z
and
(2.2)
f (z) = zh(z)g(z) = ϕ(z)|g(z)|2 .
Then h and g are non-vanishing analytic functions defined on U , normalized by
h(0) = g(0) = 1 and f is a solution of (1.1) with respect to a.
Simple calculations give that
iθ
0
∂ arg f (re )
zfz − zfz
zϕ z)
= Re
= Re
> α.
∂θ
f
ϕ(z)
Starlike log-harmonic Mappings
of Order α
Z. AbdulHadi and Y. Abu Muhanna
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Using the same argument we conclude that
J. Ineq. Pure and Appl. Math. 7(4) Art. 123, 2006
f ◦ ϕ−1 (w) = q2 (w) = w|g ◦ ϕ−1 (w)|2
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is locally univalent on ϕ(U ) and that f is univalent from Lemma 2.3 in [4]. It
follows that f ∈ STLh (α), which completes the proof of Theorem 2.1.
The next result is an integral representation for f ∈ STLh (α) for the case
a(0) = 0. For ϕ ∈ ST (α), we have
zϕ0 z)
= (1 − α)p(z) + α,
ϕ(z)
where p ∈ H(U ) is such that p(0) = 1 and Re(p) > 0. Hence, there is a
probability measure µ defined on the Borel σ−algebra of ∂U such that
Z
1 + ζz
zϕ0 z)
(2.3)
= (1 − α)
dµ(ζ) + α,
ϕ(z)
∂U 1 − ζz
Z. AbdulHadi and Y. Abu Muhanna
Title Page
and therefore,
(2.4)
Starlike log-harmonic Mappings
of Order α
Z
ϕ(z) = z exp −2(1 − α)
log(1 − ζz)dµ(ζ) .
∂U
On the other hand, let a ∈ H(U ) be such that |a(z)| < 1 for all z ∈ U and
a(0) = 0. Then there is a probability measure ν defined on the Borel σ−algebra
of ∂U such that
Z
a(z)
ξz
(2.5)
=
dν(ξ).
1 − a(z)
∂U 1 − ξz
Substituting (2.3), (2.4), and (2.5), into (2.1) and (2.2) we get
Z
f (z) = z exp −2(1 − α)
log(1 − ζz)dµ(ζ) + L(z),
∂U
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where
Z
Z
L(z) =
∂U x∂U
0
z
ξ
1 + ζs
(1 − α)
+ α ds dµ(ζ)dν(ξ).
1 − ξs
1 − ζs
Integrating and simplifying implies the following theorem:
Theorem 2.2. f = zhg ∈ STLh (α) with a(0) = 0 if and only if there are two
probability measures µ and ν such that
Z
f (z) = z exp
K(z, ζ, ξ)dµ(ζ)dν(ξ) ,
∂U x∂U
where
Z. AbdulHadi and Y. Abu Muhanna
K(z, ζ, ξ) = (1 − α) log
Starlike log-harmonic Mappings
of Order α
1 + ζz
1 − ζz
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+ T (z, ζ, ξ);
(2.6) T (z, ζ, ξ)

ζ+ξ
1−ξz

−2(1
−
α)
Im
arg

ζ−ξ
1−ζz




−2α log |1 − ξz|;
if |ζ| = |ξ| = 1, ζ =
6 ξ
=


4ζz

(1
−
α)
Re

1−ζz


−2α log |1 − ζz|;
if |ζ| = |ξ| = 1, ζ = ξ
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





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





Remark 1. Theorem 2.2 can be used in order to solve extremal problems for
the class STLh (α) with a(0) = 0. For example see Theorem 3.1.
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3.
Distortion Theorem
The following is a distortion theorem for the class STLh (α) with a(0) = 0.
Theorem 3.1. Let f (z) = zh(z)g(z) ∈ STLh (α) with a(0) = 0. Then for
z ∈ U we have
|z|
−4|z|
(3.1)
exp (1 − α)
(1 + |z|)2α
1 + |z|
|z|
4|z|
≤ |f (z)| ≤
exp (1 − α)
.
(1 − |z|)2α
1 − |z|
Starlike log-harmonic Mappings
of Order α
Z. AbdulHadi and Y. Abu Muhanna
The equalities occur if and only if f (z) = ζf0 (ζz), |ζ| = 1, where
1
4z
1−z
exp (1 − α) Re
.
(3.2)
f0 (z) = z
1 − z (1 − z)2α
1−z
Proof. Let f (z) = zh(z)g(z) ∈ STLh (α) with a(0) = 0. It follows from (2.1)
and (2.2) that f admits the representation
Z z
a(s)ϕ0 (s)
(3.3)
f (z) = ϕ(z) exp 2 Re
ds ,
0 ϕ(s)(1 − a(s))
where ϕ ∈ ST (α) and a ∈ H(U ) such that |a(z)| < 1 for all z ∈ U.
For |z| = r, the well known facts
0 zϕ (z) 1+r
ϕ(z) ≤ (1 − α) 1 − r + α,
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a(z)
1
z(1 − a(z)) ≤ 1 − r ,
and
|ϕ(z)| ≤
r
(1 − r)2(1−α)
,
imply that
Z r
1+t
1
(1 − α)
+ α dt
|f (z)| ≤
exp 2
1−t
(1 − r)2(1−α)
0 1−t
r
4r
=
exp (1 − α)
.
1−r
(1 − r)2α
r
Starlike log-harmonic Mappings
of Order α
Z. AbdulHadi and Y. Abu Muhanna
Equality occurs if and only if, a(z) = ζz and ϕ(z) = (1−ζz)z 2−2α , |ζ| = 1, which
leads to f (z) = ζf0 (ζz).
For the left-hand side, we have
Z
f (z) = z exp
K(z, ζ, ξ)dµ(ζ)dν(ξ) ,
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∂U x∂U
where
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1 + ζz
K(z, ζ, ξ) = (1 − α) log
+ T (z, ζ, ξ);
1 − ζz

ζ+ξ
1−ξz

−2(1
−
α)
Im
arg

ζ−ξ
1−ζz


−2α log |1 − ξz|;
if |ζ| = |ξ| = 1, ζ =
6 ξ
T (z, ζ, ξ) =



 (1 − α) Re 4ζz − 2α log |1 − ζz|; if |ζ| = |ξ| = 1, ζ = ξ
1−ζz
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.
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For |z| = r we have
Z
f (z) log = Re
K(z, ζ, ξ)dµ(ζ)dν(ξ)
z ∂U x∂U
Z
≥ min min Re
K(z, ζ, ξ)dµ(ζ)dν(ξ)
µ,ν
|z|=r
∂U x∂U
Z
ζ +ξ
1
≥ log
+ min min
−2(1 − α) Im
µ,ν
|z|=r
|1 + r|2α
ζ −ξ
∂U x∂U
1 − ξz
× arg
dµ(ζ)dν(ξ)
1 − ζz
1
1 + e2il
= log
+ min minπ min −2(1 − α) Im
0<|l|≤ 2 |z|=r
1 − e2il
|1 + r|2α
1 − e2il z
−4r
× arg
; (1 − α)
,
1−z
1+r
where e2il = ζξ.
Let

2il 1−e z
 min −2(1 − α) Im 1+e2il
arg
, if 0 < |l| <
1−z
1−e2il
|z|=r
Φr (l) =
 (1 − α) −4r
if l = 0
1+r
Starlike log-harmonic Mappings
of Order α
Z. AbdulHadi and Y. Abu Muhanna
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π
2



Then Φr (l) is a continuous and even function on |l| < π2 . Hence
f (z) 1
1
≥ log
+
min
+ inf Φr (l).
log Φ
(l)
=
log
r
z
|1 + r|2α 0<|l|≤ π2
|1 + r|2α 0<l< π2
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Since
max arg
|z|=r
1 − e2il z
1−z
= 2 arctan
r sin(l)
1 + r cos(l)
,
we get
f (z) 1
r sin(l)
log ≥ log
+ inf −4(1 − α) cot(l) arctan
,
z |1 + r|2α 0<l< π2
1 + r cos(l)
and using the fact that | arctan(x)| ≤ |x|, we have
f (z) 1
r
cos(l)
≥ log
log + inf −4(1 − α)
1 + r cos(l)
z |1 + r|2α 0<l< π2
1
r
.
+ inf −4(1 − α)
≥ log
1+r
|1 + r|2α 0<l< π2
Starlike log-harmonic Mappings
of Order α
Z. AbdulHadi and Y. Abu Muhanna
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The case of equality is attained by the functions f (z) = ζf0 (ζz), |ζ| = 1.
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The next application is a consequence of Theorem 2.1.
Theorem 3.2. Let f (z) = zh(z)g(z) ∈ STLh (α). Then
f
(z)
arg
≤ 2(1 − α) arcsin(|z|).
z Proof. Let f (z) =
Theorem 2.1. The
from [6, p. 142].
zh(z)g(z) ∈ STLh (α). Then ϕ(z) = zh(z)
g(z)
f (z)
result follows immediately from arg z =
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∈ ST (α) by
arg
ϕ(z)
z
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References
[1] Z. ABDULHADI, Close-to-starlike logharmonic mappings, Internat. J.
Math. & Math. Sci., 19(3) (1996), 563–574.
[2] Z. ABDULHADI, Typically real logharmonic mappings, Internat. J. Math.
& Math. Sci., 31(1) (2002), 1–9.
[3] Z. ABDULHADI AND D. BSHOUTY, Univalent functions in HH, Tran.
Amer. Math. Soc., 305(2) (1988), 841–849.
[4] Z. ABDULHADI AND W. HENGARTNER, Spirallike logharmonic mappings, Complex Variables Theory Appl., 9(2-3) (1987), 121–130.
[5] Z. ABDULHADI AND W. HENGARTNER, One pointed univalent logharmonic mappings, J. Math. Anal. Appl., 203(2) (1996), 333–351.
[6] A.W. GOODMAN, Univalent functions, Vol. I, Mariner Publishing Company, Inc., Washington, New Jersey, 1983.
[7] J.C.C. NITSCHE, Lectures on Minimal Surfaces, Vol. I, NewYork, 1989.
Starlike log-harmonic Mappings
of Order α
Z. AbdulHadi and Y. Abu Muhanna
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