Volume 2010, Article ID 896087, 6 pagesdoi:10.1155/2010/896087 Research Article Application of the Subordination Principle to the Harmonic Mappings Convex in One Direction with Shear Con
Trang 1Volume 2010, Article ID 896087, 6 pages
doi:10.1155/2010/896087
Research Article
Application of the Subordination Principle to
the Harmonic Mappings Convex in One Direction with Shear Construction Method
Department of Mathematics and Computer Science, ˙Istanbul K ¨ult ¨ur University, ˙Istanbul 31456, Turkey
Correspondence should be addressed to H Esra ¨Ozkan,e.ozkan@iku.edu.tr
Received 3 June 2010; Accepted 26 July 2010
Academic Editor: N Govil
Copyrightq 2010 Yas¸ar Polato˘glu et al 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
Any harmonic function in the open unit discD {z | |z| < 1} can be written as a sum of an analytic and antianalytic functions f hz gz, where hz and gz are analytic functions inD and are
called the analytic part and the coanalytic part of f, respectively Many important questions in the
study of the classes of functions are related to bounds on the modulus of functionsgrowth or the modulus of the derivativedistortion In this paper, we consider both of these questions
1 Introduction
Let U be a simply connected domain in the complex plane A harmonic function f has the representation f hz gz, where hz and gz are analytic in U and are called the analytic and coanalytic parts of f, respectively Let hz a0 a1z a2z2 · · · , and gz
b0b1z b2z2· · · be analytic functions in the open unit disc D If J f z |hz|2−|gz|2> 0,
then f hzgz is called the sense-preserving harmonic univalent function in D The class
of all sense-preserving harmonic univalent functions is denoted bySH , with a0 b0 0,
a1 1, and |b1| < 1, and the class of all sense-preserving harmonic univalent functions is
denoted by S0
H with a0 b0 b1 0, a1 1 For convenience, we will examine
sense-preserving functions, that is, functions for which J f z > 0 If f has J f z < 0, then f is sense preserving The analytic dilatation of the harmonic functions is given by wz gz/hz.
We also note that if f is locally univalent and sense preserving then |wz| < 1.
In this paper we examine the class of functions that are convex in one direction The shear construction is essential to the present work as it allows one to study harmonic functions through their related analytic functions as shown in1 by Hengartner and Schober
Trang 2The shear construction produces a univalent harmonic function that mapsD to the region that
is convex in the direction of the real axis This construction relies on the following theorem of Clunie and Sheil-Small
Theorem 1.1 see 2 A harmonic function f hz gz locally univalent in D is a univalent
mapping of D onto a domain convex in the direction of the real axis if and only if hz − gz is a
conformal univalent mapping of D onto a domain convex in the direction of the real axis.
Theorem 1.1leads to the construction of univalent harmonic function with analytic
dilatation wz Hengartner and Schober 1 studied the analytic functions ψz that are
convex in the direction of the imaginary axis They used a normalization which requires, in
essence, that right and left extremes of ψD be the image of 1 and −1 This normalization
is that there exist pointsz
n converging to z 1 and z
n converging to z −1 such that
lim
n→ ∞
Re ψ
zn
Sup
|z|<1
Re ψz,
lim
n→ ∞
Re ψ
zn
Inf
|z|<1
Re ψz.
1.1
If CIA is the class of domains,D, that are convex in the direction of the imaginary axis and
that admit a mapping ψz so that ψD D and satisfies the normalization 1.1, then we have the following result
Theorem 1.2 see 1 Suppose that ψz is analytic and nonconstant for |z| < 1, then one has
Re1 − z2ψz > 0 if and only if
i ψz is univalent on D,
ii ψz ∈⊂ IA,
iii ψz is normalized by 1.1.
Using this characterization of functions, Hengartner and Schober proved the following theorem.
Theorem 1.3 see 1 If ψz is analytic for |z| < 1 and satisfies Re1 − z2ψz ≥ 0, then
1 − rψ0
1 r1 r2 ≤ ψ0
To be able to obtain this result for functions that are in the direction of the real axis,
let us consider the following situation Suppose that ϕz is a function that is analytic and convex in the direction of the real axis Furthermore, suppose that ϕz is normalized by the
following
Let there exist pointsz
n converging to z e iα andz
n converging to z e i απ, such that
lim
n→ ∞
Im ϕ
zn
Sup
|z|<1
Im ϕz,
lim
n→ ∞
Re ϕ
zn
Inf
|z|<1
Im ϕz.
1.3
Trang 3Consequently, if ψz satisfies 1.1, then ϕz iψze −iα z satisfies 1.3 Knowing this, we
can apply ϕz and see that the result still holds, with ψz being replaced by ϕz In this
situation, Re−ieiα − e −iα z2ϕz > 0 We can now prove the derivative bounds for the
harmonic function convex in the direction of the real axis
Finally, letΩ be the family of functions φz which are analytic in D and satisfying the condition φ0 0, |φz| < 1 for every z ∈ D Denote by P the class of analytic functions pz given by pz 1p1z p z z2· · · which satisfy Re pz > 0 for all z ∈ D Let s1z zc2z2· · ·
and s2z z d2z2 · · · be analytic functions in D If s1z s2φz is satisfied for some
φ z ∈ Ω and every z ∈ D, then we say that s1z is subordinate to s2z, and we write
s1z ≺ s2z.
2 Main Results
Lemma 2.1 Let f hz gz be an element of S H , and let w z gz/hz be the analytic
dilatation of f, then
|b1| − r
1− |b1|r ≤ |wz| ≤ |b1| r
1 − |b1|1 − r
1 |b1|r ≤ 1 − |wz| ≤ 1 − |b1|1 r
1 |b1|1 − r
1− |b1|r ≤ 1 |wz| ≤ 1 |b1|1 r
1 − r2
1− |b1|2
1− |b1|2r2 ≤ 1 − |wz|2≤
1− r2
1− |b1|2
1 − |b1|r2 . 2.4
Proof Since f hz gz ∈ S H, then
w z
b1z b2z2 · · ·
z a2z2 · · · b1 2b2z · · ·
1 a2z · · · ⇒ w0 b1. 2.5 Now, we define the function
φ z w z − w0
1− w0wz
w z − b1
This function satisfies the conditions of the Schwarz lemma Then, we have
w z φ z b1
Using the principle of subordination and2.7, we see that the analytic dilatation wz
is subordinate tozb1/1b1z On the other hand, the transformation zb1/1b1z
Trang 4maps|z| r onto the circle with the centre Cr α11−r2/1−|b1|2r2, α21−r2/1−|b1|2r2
and the radius ρr 1 − |b1|2r/1 − |b1|2r2, where b1 α1 iα2 Thus, again using the subordination principle, we write
w z −
b1
1− r2
1− |b1|2r2
≤
1− |b1|2
r
Following some simple calculations from2.8, we get 2.1, 2.2, 2.3, and 2.4
Theorem 2.2 Let f hzgz be an element of S H , and let f be convex in the direction of the real axis, and let ϕ z hz−gz, wz gz/hz Furthermore, let ϕz satisfy the normalization
1.1, then for |z| < 1, one has
|1 − b1|1 |b1|r1 − r
1 |b1|1 r21 r2 ≤f z ≤ |1 − b1|1 |b1|r
1 − |b1|1 − r3,
|wz||1 − b1|1 − r1 |b1|r
1 |b1|1 r21 r2 ≤f z ≤ |1 − b1|1 |b1|rr
1 − |b1|1 − r3 .
2.9
Proof Since ϕ z hz − gz ⇒ ϕz hz − gz, gz hzwz, then we have
f z hz ϕz
1− wz ,
f z gz w zϕz
1− wz .
2.10
Since analytic dilatation wz satisfies the condition |wz| < 1 for every z ∈ D, then
we have
ϕz
1 |wz|≤f z ≤ ϕz
1− |wz| ,
|wz|ϕz
1 |wz| ≤f
z ≤ |wz|ϕz
1− |wz| .
2.11
Using2.2, 2.3, and 1.2 in 2.11, we get
1 − r1 |b1|rϕ0
1 |b1|1 r21 r2 ≤f z ≤ 1 |b1|rϕ0
1 − |b1|1 − r3,
|wz|1 − r1 |b1|rϕ0
1 |b1|1 r21 r2 ≤f z ≤ 1 |b1|rϕ0r
1 − |b1|1 − r3 .
2.12
On the other hand, ϕz hz−gz ⇒ ϕz hz−gz ⇒ ϕ0 1−b1therefore,
2.12 can be written in the form 2.9
Trang 5Corollary 2.3 If one lets b1 0, then ϕ0 1 therefore, one obtains
1− r
1 r21 r2 ≤f z ≤ 1
1 − r3,
|wz|1 − r
1 r21 r2 ≤f z ≤ r
1 − r3.
2.13
These distortions were found by Schaubroeck [ 3 ].
Theorem 2.4 Let f hzgz be convex in the direction of the real axis, let f hzgz ∈ S H , and let ϕ z hz − gz satisfy the normalization 1.1 Then, for |z| < r, one has
f ≤ |1 − b1|
1− |b1|
r
0
1 |b1|ρ1 ρ
1− ρ3 dρ
Proof Since f hz gz, we have the following inequalities:
f hz gz r
0
h
ρe iθ
e iθ dρ r
0
g
ρe iθ e iθ dρ
r
0
f z
ρe iθ
e iθ dρ r
0
f z
ρe iθ
e −iθ dρ.
2.15
Hence,
f hz gz ≤ |hz| g z ≤ r
0
f z
ρe iθdρ r
0
f z
ρe iθdρ. 2.16
Applying2.9 to the above expression yields 2.14
Corollary 2.5 If one takes b1 0, then one obtains
f ≤ r
This growth was found by Schaubroeck [ 3 ].
Theorem 2.6 Let f hz gz ∈ S H , and let f be convex in the direction of the real axis If
ϕ z hz − gz satisfies the normalization 1.1, then
1 − |b1|1 |b1|r1 − r4|1 − b1|2
1 |b1|1 − |b1|r1 r41 r22 ≤ J f z ≤ 1 |b1|1 |b1|r1 r|1 − b1|2
1 − |b1|1 − |b1|r1 − r5 . 2.18
Proof Since J f z |hz|2 − |gz|2 |hz|21 − |wz|2, then using Lemma 2.1 and
Theorem 2.2and after straightforward calculations, we get2.18
Trang 6Remark 2.7 We note that the distortion and growth theorem in our study is sharp, because by
choosing the suitable analytic dilatation and ϕz, we can find the extremal function in the
following manner:
ϕ z hz − gz ⇒ ϕz hz − gz,
w z gz
hz ⇒ 0 wzhz − gz,
hz f z ϕz
1− wz ⇒ hz
z
0
ϕξ
1− wξ dξ,
gz f z ϕzwz
1− wz ⇒
g z z
0
ϕξwξ
1− wξ dξ
z
0
ϕξwξ − ϕξ ϕξ
1− wξ dξ
z
0
ϕξ
1− wξ − ϕξ dξ⇒
g z z
0
ϕξ
1− wξ dξ−
z
0
ϕξdξ z
0
ϕξ
1− wξ dξ − ϕz.
2.19 Therefore we have
f hz gz z
0
ϕξ
1− wξ dξ
z
0
ϕξ
1− wξ dξ − ϕz
z
0
ϕξ
1− wξ dξ
z
0
ϕξ
1− wξ dξ − ϕz ⇒
f z Re z
0
2ϕξ
1− wξ dξ
− ϕz.
2.20
References
1 W Hengartner and G Schober, “On Schlicht mappings to domains convex in one direction,”
Commentarii Mathematici Helvetici, vol 45, pp 303–314, 1970.
2 J Clunie and T Sheil-Small, “Harmonic univalent functions,” Annales Academiae Scientiarum Fennicae Series A I Mathematica, vol 9, pp 3–25, 1984.
3 L E Schaubroeck, “Growth, distortion and coefficient bounds for plane harmonic mappings convex
in one direction,” The Rocky Mountain Journal of Mathematics, vol 31, no 2, pp 625–639, 2001.
... Mathematica, vol 9, pp 3–25, 1984.3 L E Schaubroeck, “Growth, distortion and coefficient bounds for plane harmonic mappings convex
in one direction, ” The Rocky Mountain... “On Schlicht mappings to domains convex in one direction, ”
Commentarii Mathematici Helvetici, vol 45, pp 303–314, 1970.
2 J Clunie and T Sheil-Small, ? ?Harmonic univalent... bounds for plane harmonic mappings convex
in one direction, ” The Rocky Mountain Journal of Mathematics, vol 31, no 2, pp 625–639, 2001.