Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 813687, 12 pages
doi:10.1155/2009/813687
Research Article
On Bounded Boundary and
Bounded Radius Rotations
K. I. Noor, W. Ul-Haq, M. Arif, and S. Mustafa
Department of Mathematics, COMSATS Institute of Information Technology, 44000 Islamabad, Pakistan
Correspondence should be addressed to M. Arif, [email protected]
Received 6 January 2009; Revised 6 March 2009; Accepted 19 March 2009
Recommended by Narendra Kumar Govil
We establish a relation between the functions of bounded boundary and bounded radius rotations
by using three different techniques. A well-known result is observed as a special case from our
main result. An interesting application of our work is also being investigated.
Copyright q 2009 K. I. Noor 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.
1. Introduction
Let A be the class of functions f of the form
f
z
z
∞
n2
a
n
z
n
z ∈ E
, 1.2
the convolution Hadamard product of f and g is defined by
f ∗ g
z
∞
n0
a
n
b
n
z
n
z ∈ E
. 1.3
2 Journal of Inequalities and Applications
We denote by S
∗
α,Cα, 0 ≤ α<1, the classes of starlike and convex functions of order
z
∈ S
∗
α
,z∈ E
.
1.4
For α 0, we have the well-known classes of starlike and convex univalent functions denoted
by S
∗
and C, respectively.
Let P
k
α be the class of functions pz analytic in the unit disc E satisfying the
properties p01and
2π
0
Re
p
z
2π
2π
0
1
1 − 2α
ze
−it
1 − ze
−it
dμ
t
,z∈ E, 1.6
where μt is a function with bounded variation on 0, 2π such that
2π
0
dμ
t
2π,
2π
0
2π
0
1
1 − 2α
ze
−it
1 − ze
−it
dμ
1
t
−
k
4
−
1
2
1
2π
2π
0
1
z
−
k
4
−
1
2
p
2
z
,z∈ E, 1.9
where P α is the class of functions with real part greater than α and p
i
∈ Pα,fori 1, 2.
Journal of Inequalities and Applications 3
We define the following classes:
R
k
α
f: f ∈ A and
zf
f
z
∈ P
k
α
, 0 ≤ α<1
.
1.10
We note that
f ∈ V
k
α
⇐⇒ zf
∈ R
k
α
. 1.11
For α 0, we obtain the well-known classes R
k
2α1
,α
/
1
2
,
1
2ln2
,α
1
2
,
1.12
and this result is sharp.
In this paper, we prove the result of Goel 6 for the classes V
k
α and R
k
α by using
three different methods. The first one is the same as done by Goel 6, while the second and
third are the convolution and subordination techniques.
2. Preliminary Results
We need the following results to obtain our results.
Lemma 2.1. Let f ∈ V
k
α. Then there exist s
1
,s
2
z
g
z
1−α
,z∈ E, see
2
. 2.2
4 Journal of Inequalities and Applications
From Brannan 7 representation form for functions with bounded boundary rotations, we
have
g
z
g
1
z
−
⎛
⎝
1
2
⎞
⎠
,g
i
∈ S
∗
,i 1, 2. 2.3
Now, it is shown in 8 that for s
i
∈ S
∗
α, we can write
s
i
z
z
g
i
z
z
∈ D and v
1
≤−1/21 u
2
2
.
If hz1 c
1
z ··· is a function analytic in E such that hz,zh
z ∈ D and
Re Ψhz,zh
z > 0 for z ∈ E, then Re hz > 0 in E.
Lemma 2.3. Let β>0, β γ>0, and α ∈ α
0
, 1,with
α
0
max
β − γ − 1
2β
,
−γ
β
. 2.5
If
z
≺
1
1 − 2α
z
1 − z
, 2.7
where
Q
z
1
βG
z
−
γ
β
,
G
z
,
2.8
Journal of Inequalities and Applications 5
2
F
1
denotes Gauss hypergeometric function. From 2.7, one can deduce the sharp result that h ∈ P β,
with
β β
α, β, γ
min Re Q
z
Q
−1
. 2.9
This result is a special case of the one given in [10, page 113].
3. Main Results
By using the same method as that of Goel 6, we prove the following result. We include all
the details for the sake of completeness.
3.1. First Method
Theorem 3.1. Let f ∈ V
k
α.Thenf ∈ R
s
1
z
−
k
4
−
1
2
zs
2
z
s
2
z
k
4
f
2
z
,
3.1
where s
i
∈ S
∗
α and f
i
∈ Cα, i 1, 2.
Therefore, from 2.4, we have
zf
z
f
z
k
4
4
−
1
2
z
g
2
z
/z
1−α
z
0
g
2
φ
/φ
1−α
dφ
, 3.2
that is,
φ
g
1
z
1−α
dφ
z
⎤
⎦
−1
−
k
4
−
1
2
⎡
⎣
z
0
z
f
z
k
4
1
2
p
1
z
−
k
4
1
2
p
2
z
1−α
dφ
z
⎤
⎦
−1
, 3.5
where p
i
01 and hence by 11 we have
p
i
z
−
1 r
2
1 − r
2
min
f
i
∈C
α
min
|
z
|
r
p
i
z
. 3.7
Let z re
iθ
and φ Re
iθ
,0<R<r<1. For fixed z and φ, we have from 2.4
z
0
z
φ
1−α
g
i
φ
g
i
z
1−α
dφ
z
2
1−α
dR
r
, 3.10
with R rt,0<t<1, we have
T
r
1
0
1 r
1 rt
2
1−α
dt. 3.11
Journal of Inequalities and Applications 7
By differentiating we note that
T
r
T
1
2
21−α
1
0
dt
1 t
2
1−α
⎧
⎪
⎪
⎨
⎪
⎪
⎩
2 − 4
z
0
z
φ
1−α
g
i
φ
g
i
z
1−α
dφ
z
⎤
1
2
1 −
1 − 2α
z
1 z
−
k
4
−
1
2
1
1 − 2α
z
1 − z
. 3.15
It is easy to check that f
0
∈ R
z
1 − β
p
z
β
1 − β
k
4
1
2
p
1
z
−
k
p
z
β
1 − β
zp
z
1 − β
p
z
β
, 3.18
that is,
1
1 − α
zf
z
1 − β
p
z
β
β − α
1 − α
1 − β
1 − α
p
z
1/
1 − β
zp
p
z
1/
1 − β
zp
z
p
z
β/
1 − β
∈ P
k
,z∈ E. 3.20
We define
k
4
1
2
ϕ
a,b
z
z
∗ p
1
z
−
k
4
−
1
2
ϕ
a,b
4
1
2
p
1
z
azp
1
z
p
1
z
b
−
k
4
−
1 − β
1 − α
p
i
z
azp
i
z
p
i
z
b
∈ P, i 1, 2. 3.24
Journal of Inequalities and Applications 9
We now form t he functional Ψu, v by choosing u p
i
z,v zp
1 − β
1
1 − α
β − α
v
1
β/
1 − β
u
2
2
β/1 − β
2
≤
1
u
2
2
β/1 − β
2
−
1 u
2
2
β/
1 − β
2
u
2
2
β/1 − β
β/
1 − β
u
2
2
2
u
2
2
β/1 − β
2
1 − α
A Bu
2
2
2C
, 2C>0,
3.25
where
u
2
2
β
1 − β
2
> 0.
3.26
The right-hand side of 3.25 is negative if A ≤ 0andB ≤ 0. From A ≤ 0, we have
β β
α
1
4
2α − 1
4α
2
⎩
2α − 1
2 − 2
21−α
, if α
/
1
2
,
1
2ln2
, if α
1
2
.
3.28
Proof. Let
zf
z
f
z
p
z
2
z
s
2
z
, 3.29
and let
zs
i
z
s
i
z
p
i
z
,i 1, 2. 3.30
Then p, p
i
k
4
1
2
zs
1
z
s
1
z
−
k
4
−
1
2
zs
z
p
1
z
−
k
4
−
1
2
p
2
z
zp
2
z
p
z
p
i
z
∈ P
α
, 3.32
where zs
i
z/s
i
zp
i
z, i 1, 2. We use Lemma 2.3 with γ 0,β 1 > 0,α∈ 0, 1, and
h p
i
in 3.32, to have p
i
∈ Pβ, where β is given in 3.28 and this estimate is best possible,
extremal function Q is given by
Q
z − 1
log
1 − z
, if α
1
2
,
3.33
see 10. MacGregor 13 conjectured the exact value given by 3.28.Thuss
i
∈ S
∗
β and
consequently f ∈ R
k
β, where the exact value of β is given by 3.28.
Journal of Inequalities and Applications 11
3.4. Application of Theorem 3.3
Theorem 3.4. Let g and h belong to V
k
α.ThenFz, defined by
F
z
z
g
z
η
zh
z
h
z
1 −
μ η
. 3.35
Since g and h belong to V
k
α, then, by Theorem 3.3, zg
z/gz and zh
z/hz belong to
h
z
1 − β
q
2
z
β, q
2
∈ P
k
,
3.36
in 3.35, we have
1
1 − δ
zF
z
F
F
z
z
0
f
t
t
dt
Alexander’s integral operator
, 3.38
is in the class V
k
1/2.
Acknowledgments
The authors are grateful to Dr. S. M. Junaid Zaidi, Rector, CIIT, for providing excellent
research facilities and the referee for his/her useful suggestions on the earlier version of
this paper. W. Ul-Haq and M. Arif greatly acknowledge the financial assistance by the HEC,
Packistan, in the form of scholarship under indigenous Ph.D fellowship.
12 Journal of Inequalities and Applications
References
1 B. Pinchuk, “Functions of bounded boundary rotation,” Israel Journal of Mathematics, vol. 10, no. 1, pp.
6–16, 1971.
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