Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 193184, 7 pages
doi:10.1155/2011/193184
Research Article
Integral Means and Arc length of Starlike
Log-harmonic Mappings
Z. Abdulhadi and Y. Abu Muhanna
Department of Mathematics, American University of Sharjah, P.O. Box 26666 Sharjah,
United Arab Emirates
Correspondence should be addressed to Z. Abdulhadi, zahadi@aus.edu
Received 13 January 2011; Accepted 17 February 2011
Academic Editor: Gaston Mandata N’Guerekata
Copyright q 2011 Z. Abdulhadi and Y. Abu Muhanna. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
We use star functions to determine the integral means for starlike log-harmonic mappings.
Moreover, we include the upper bound for the arc length of starlike log-harmonic mappings.
1. Introduction
Let HU be the linear space of all analytic functions defined on the unit disk U {z : |z| < 1}.
A Log-harmonic mapping is a solution to the nonlinear elliptic partial differential equation
fz
f
a
fz
,
f
1.1
where the second dilatation function a ∈ HU such that |az| < 1 for all z ∈ U. It has been
shown that if f is a nonvanishing Log-harmonic mapping, then f can be expressed as
fz hzgz,
1.2
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:
fz z|z|2β hzgz,
1.3
where Re β > −1/2, and h and g are analytic functions in U, g0 1, and h0 /
0 see 1.
Univalent Log-harmonic mappings have been studied extensively for details see 1–5.
2
Abstract and Applied Analysis
Let f z|z|2β hg be a univalent Log-harmonic mapping. We say that f is starlike Logharmonic mapping if
∂ arg f reiθ
zfz − zfz
Re
> 0,
∂θ
f
1.4
for all z ∈ U. Denote by ST∗Lh the set of all starlike Log-harmonic mappings, and by S∗ the set
of all starlike analytic mappings. It was shown in 4 that fz z|z|2β hzgz ∈ ST∗Lh if and
only if ϕz zhz/gz ∈ S∗ .
In Section 2, using star functions we determine the integral means for starlike Logharmonic mappings. Moreover, we include the upper bound for the arc length of starlike
Log-harmonic mappings.
2. Main Results
If f is univalent normalized starlike Log-harmonic mapping, then it was shown in 4 that
z
fz zhzgz Hzgz ϕz exp 2 Re
0
as ϕ s
ds ,
1 − as ϕs
2.1
where ϕz Hz/gz is starlike and a is analytic with a0 0 and |az| < 1 for z in the
unit disk,
Hz ϕz exp
z
0
z
gz exp
0
as ϕ s
ds ,
1 − as ϕs
as ϕ s
ds .
1 − as ϕs
2.2
2.3
Theorem 2.2 of this section is an application of the Baerstein star functions to starlike
Log-harmonic mapping. Star function was first introduced and properties were derived by
Baerstein 6, 7, Chapter 7. The first application was the remarkable result: if f ∈ S, then
p
p
it f re dt ≤ k reit dt,
2.4
where kz z/1 − z2 , 0 < r < 1, and p > 0.
If uz is a real L1 function in an annulus 0 < R1 < |z| < R2 , then the definition of the
star function of u, u∗ is
∗
u re
iθ
u reit dt,
sup
|E|2θ
E
for R1 < r < R2 .
2.5
Abstract and Applied Analysis
3
One important property is that when u is a symmetric even rearrangement, then
θ u reit dt.
u∗ reiθ −θ
2.6
Other properties 6, 7, Chapter 7 are that the star function is subadditive and the
star respects subordination. Respect means that the star of the subordinate function is less
than or equal to the star of the function. In addition, it was also shown that star function is
additive when functions are symmetric rearrangements. Here is a lemma, quoted in 6, 7,
Chapter 7, which we will be used later.
Lemma 2.1. For g, h real and L1 on −a, a, the following are equivalent:
a for every convex nondecreasing function Φ : R → R,
a
−a
Φ gx dx ≤
a
−a
Φhxdx;
2.7
hx − t dt;
2.8
b for every t ∈ R,
a
−a
gx − t dt ≤
a
−a
c for every x ∈ 0, a,
g ∗ x ≤ g ∗ x.
2.9
Here is the main result of the section.
Theorem 2.2. If fz zhzgz Hzgz is in ST∗Lh , then for each fixed r, 0 < r < 1, and as
a function of θ
a
log|Hz|
∗
∗
z exp Re 2z
≤ log ,
1−z
1 − z 2.10
b
∗
∗
2z
≤ log |1 − z| exp Re
,
log gz
1−z
2.11
∗
∗
4z
≤ log |z| exp Re
,
log fz
1−z
2.12
c
4
Abstract and Applied Analysis
the three results are sharp by the functions
4z
z1 − z
,
exp Re
1−z
1 − z
z
2z
zhz exp
,
1−z
1 − z
2z
.
gz 1 − z exp
1−z
fz 2.13
Proof. The proofs of the three parts are similar. We will emphasise the proof of part a.
By 2.2,
zΦ ρz
a ρz
dρ .
1 − a ρz Φ ρz
1
Hz Φz exp
0
2.14
Then
1
log|Hz| log|Φz| Re
0
zΦ ρz
a ρz
dρ ,
1 − a ρz Φ ρz
2.15
where z reiθ . Write aρz/1 − aρzρzΦ ρz/Φρz aρz/1 − aρz1 ωρz/ρ1−ωρz, where ω is analytic, |ω| < 1, and ω0 0, see 7. Then, as az/1−
azzΦ z/Φz is subordinate to z1 z/1 − z2 7,
az zΦ z ψz 1 ψz
2 ,
1 − az Φz
1 − ψz
2.16
for ψ analytic, |ψ| < 1 and ψ0 0.
Then 2.15 becomes
1
log|Hz| log|Φz| Re
0
1 ψ ρz 1 ψ ρz
2 dρ .
ρ
1 − ψ ρz
2.17
As the star function is subadditive,
∗ ∗
log|Hz| ≤ log|Φz| 1 0
∗
1 ψρz1 ψ ρz
Re
dρ.
ρ 1 − ψ ρz2
2.18
Consequently, by 2.4 and the fact that star functions respect subordination,
log |Hz|
∗
∗
z ∗ 1 1
ρz 1 ρz
Re log
dρ.
2
1 − z2 0 ρ
1 − ρz
≤
2.19
Abstract and Applied Analysis
5
Hence, as star functions are additive when functions are symmetric re-arrangements,
log |Hz|
∗
∗
z ∗ 1 1
ρz 1 ρz
log
dρ
Re 2
1 − z2 0 ρ
1 − ρz
≤
∗
z 1 1
ρz 1 ρz
Re log
2 dρ
1 − z2 0 ρ
1 − ρz
1 ∗
z ρz 1 ρz
1
Re log exp
2 dρ
1 − z2 0 ρ
1 − ρz
2.20
∗
z exp Re 2z
log .
1−z
1 − z Therefore,
log |Hz|
∗
∗
z exp Re 2z
≤ log ,
1−z
1 − z 2.21
which is part a.
By 2.4, 2.15 and in similar fashion to the upper part,
log |gz|
∗
∗
ρz 1 ρz
Re ≤
dρ
2
0
1 − ρz
∗
2z
log |1 − z| exp Re
.
1−z
1
1
ρ
2.22
2.21 and 2.22 give part c.
Now by using Lemma 2.1, we have the following corollary.
Corollary 2.3. If fz zhzgz is starlike Log-harmonic, then, for z eiθ ,
p
|z| exp Re 4z dθ, ∀p > 0,
1−z 4z
dθ ≤ C < ∞,
log fz dt ≤ log |z|dθ Re
1−z
fzp dt ≤
2.23
the later implies that f ∈ N hence has radial limits.
Proof. If we choose Φx exppx which is nondecreasing convex, then part a of
Lemma 2.1 and part c of Theorem 2.2 give the first integral mean. The choice Φx log x
gives the second integral mean.
In the next theorem, we establish an upper for the arc length of starlike Log-harmonic
mappings.
6
Abstract and Applied Analysis
Theorem 2.4. If fz zhzgz ∈ ST∗Lh and for r, 0 < r < 1, |freiθ | ≤ Mr, then Lr ≤
4πMr/1 − r 2 .
Proof. Let Cr denote the closed curve which is the image of the circle |z| r < 1 under the
mapping w fz. Then
df Lr Cr
2π
zfz − zfz dθ
0
2π
zfz − zfz dθ
f f
0
2π zfz − zfz dθ.
≤ Mr
f
0
2.24
Now using 2.1, we have
zfz − zfz
zϕ
Re
i Im
f
ϕ
1a
1−a
zϕ
ϕ
.
2.25
Therefore,
Lr ≤ Mr
2π
0
2π zϕ zϕ
1a
dθ
dθ Mr
Re
Im 1 − a
ϕ
ϕ 0
2.26
MrI1 MrI2 .
Since Rezϕ /ϕ is harmonic, and by the mean value theorem for harmonic functions,
I1 2π. Moreover, 1 a/1 − azϕ /ϕ is subordinate to 1 z/1 − z2 ; therefore, we
have
2π ∞
1 z 2
1 r2
2n
2π
.
I2 ≤
1 − z dθ 2π 1 2 r
1 − r2
0
n1
2.27
Substituting the bounds for I1 and I2 in 2.26, we get
Lr ≤ 2πMr 2πMr
1 r2
1 − r2
≤
4πMr
.
1 − r2
2.28
References
1 Z. Abdulhadi and D. Bshouty, “Univalent functions in HH,” Transactions of the American Mathematical
Society, vol. 305, no. 2, pp. 841–849, 1988.
2 Z. Abdulhadi, “Close-to-starlike logharmonic mappings,” International Journal of Mathematics and
Mathematical Sciences, vol. 19, no. 3, pp. 563–574, 1996.
Abstract and Applied Analysis
7
3 Z. Abdulhadi, “Typically real logharmonic mappings,” International Journal of Mathematics and
Mathematical Sciences, vol. 31, no. 1, pp. 1–9, 2002.
4 Z. Abdulhadi and W. Hengartner, “Spirallike logharmonic mappings,” Complex Variables. Theory and
Application. An International Journal, vol. 9, no. 2-3, pp. 121–130, 1987.
5 Z. Abdulhadi and W. Hengartner, “One pointed univalent logharmonic mappings,” Journal of
Mathematical Analysis and Applications, vol. 203, no. 2, pp. 333–351, 1996.
6 A. Baernstein, II, “Integral means, univalent functions and circular symmetrization,” Acta Mathematica,
vol. 133, pp. 139–169, 1974.
7 P. L. Duren, Univalent Functions, vol. 259 of Grundlehren der Mathematischen Wissenschaften, Springer,
New York, NY, USA, 1983.
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