Volume 10 (2009), Issue 1, Article 17, 7 pp.
THE ALEXANDER TRANSFORMATION OF A SUBCLASS OF SPIRALLIKE
FUNCTIONS OF TYPE β
1
1
QINGHUA XU AND 1,2 SANYA LU
S CHOOL OF M ATHEMATICS AND I NFORMATION S CIENCE
J IANGXI N ORMAL U NIVERSITY
J IANGXI , 330022, C HINA
xuqhster@gmail.com
2
D EPARTMENT OF S CIENCE ,
NANCHANG I NSTITUTE OF T ECHNOLOGY
J IANGXI , 330099, C HINA
yasanlu@163.com
Received 13 August, 2008; accepted 27 December, 2008
Communicated by G. Kohr
A BSTRACT. In this paper, a subclass of spirallike function of type β denoted by Ŝαβ is introduced
in the unit disc of the complex plane. We show that the Alexander transformation of class of Ŝαβ
1
is univalent when cos β ≤ 2(1−α)
, which generalizes the related results of some authors.
Key words and phrases: Univalent functions, Starlike functions of order α, spirallike functions of type β, Integral transformations.
2000 Mathematics Subject Classification. 30C45.
1. I NTRODUCTION
Let A denote the class of analytic functions f on the unit disk D = {z ∈ C : |z| < 1}
normalized by f (0) = 0 and f 0 (0) = 1, S denote the subclass of A consisting of univalent
functions, and S ∗ denote starlike functions on D. Obviously, S ∗ ⊂ S ⊂ A holds.
In [1], Robertson introduced starlike functions of order α on D.
Definition 1.1. Let α ∈ [0, 1), f ∈ S and
0 zf (z)
<e
> α,
f (z)
z ∈ D.
We say that f is a starlike function of order α. Let S ∗ (α) denote the whole starlike functions of
order α on D.
This research has been supported by the Jiangxi Provincial Natural Science Foundation of China (Grant No. 2007GZS0177) and Specialized
Research Fund for the Doctoral Program of JiangXi Normal University.
225-08
2
Q INGHUA X U AND S ANYA L U
Spaček [2] extended the class of S ∗ ,and obtained the class of spirallike functions of type β.
In the same article, the author gave an analytical characterization of spirallikeness of type β on
D.
Theorem 1.1. Let f ∈ S and β ∈ (− π2 , π2 ). Then f (z) is a spirallike function of type β on D if
and only if
0
iβ zf (z)
<e e
> 0, z ∈ D.
f (z)
We denote the whole spirallike functions of type β on D by Ŝβ .
From Definition 1.1 and Theorem 1.1, it is easy to see that starlike functions of order α and
spirallike functions of type β have some relationships on geometry. Spirallike functions of type
0 (z)
β map D into the right half complex plane by the mapping eiβ zff (z)
, while starlike functions
of order α map D into the right half complex plane whose real part is greater than α by the
0 (z)
0 (z)
mapping zff (z)
. Since lim eiβ zff (z)
= eiβ , we can deduce that if we restrict the image of the
0 (z)
eiβ zff (z)
z→0
mapping
in the right complex plane whose real part is greater than a certain constant,
then the constant must be smaller than cos β. According to this, we introduce the functions class
Ŝαβ on D.
Definition 1.2. Let α ∈ [0, 1), β ∈ (− π2 , π2 ), f ∈ S, then f ∈ Ŝαβ if and only if
0
iβ zf (z)
<e e
> α cos β, z ∈ D.
f (z)
Obviously, when β = 0 , f ∈ S ∗ (α); while α = 0, f ∈ Ŝβ .
Example 1.1. Let f (z) =
such that
z
2(1−α)
(1−z) 1+i tan β
, z ∈ D. The branch of the power function is chosen
2(1−α)
[(1 − z)] 1+i tan β = 1.
z=0
It is easily proved that f ∈ Ŝαβ . We omit the proof.
S
For our applications, we set Ŝ = β Ŝαβ .
In this paper, we first establish the relationships among Ŝαβ and some important subclasses
of S, then investigate the Alexander transformation of Ŝαβ preserving univalence. Furthermore,
some other properties of the class of Ŝαβ are obtained. These results generalize the related works
of some authors.
2. I NTEGRAL T RANSFORMATIONS AND L EMMAS
Integral Transformation 1. The integral transformation
Z z
f (ζ)
J[f ](z) =
dζ
ζ
0
is called the Alexander Transformation and it was introduced by Alexander in [4]. Alexander
was the first to observe and prove that the Integral transformation J maps the class S ∗ of starlike
functions onto the class K of convex functions in a one-to-one fashion.
In 1960, Biernacki conjectured that J(S) ⊂ S , but Krzyz and Lewandowski disproved it in
1963 by giving the example f (z) = z(1 − iz)i−1 , which is a spirallike function of type π4 but is
transformed into a non-univalent function by J [4]. In 1969, Robertson studied the Alexander
Integral Transformation of spirallike functions of type β. The author showed that J(Ŝβ ) ⊂ S
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 17, 7 pp.
http://jipam.vu.edu.au/
A LEXANDER T RANSFORMATION
3
holds when β satisfies a certain condition, that is cos β ≤ x0 (a constant). Robertson also
noticed that x0 cannot be replaced by any number greater than 21 and asked about the best value
for this [3]. In 2007, Y.C. Kim and T. Sugawa proved that J(Ŝβ ) ⊂ S holds precisely when
cos β ≤ 21 or β = 0 [4].
Integral Transformation 2. Let γ ∈ C, f (z) ∈ A be locally univalent, and the Integral
transformation Iγ [5] be defined by
Z z
Z 1
0
γ
Iγ [f ](z) =
[f (ζ)] dζ = z
[f 0 (tz)]γ dt.
0
0
Based on the definition of Iγ , we may easily show that Iγ ◦ Iγ 0 = Iγγ 0 .
Let A(F ) = {γ ∈ C : Iγ (F ) ⊂ S}, F ⊂ A be locally univalent. According to the definition
of the A(F ), J(Ŝαβ ) ⊂ S is equivalent to 1 ∈ A(J(Ŝαβ )).
For the proof of the theorems in this paper, we need the following lemma, which establishes
the relationships among Ŝαβ and some important subclasses of S.
Lemma 2.1. For α ∈ [0, 1), β ∈ (− π2 , π2 ), c = e−iβ cos β, the following assertions hold:
(i) ([6, 7]) f ∈ S ∗ (α) if and only if
1−α
u(z)
f (z)
=
, z ∈ D,
z
z
h i1−α ∗
where u(z) ∈ S . The branch of the power function is chosen such that u(z)
z
= 1.
z=0
(ii) f ∈ Ŝαβ if and only if
c
g(z)
f (z)
=
,
z
z
z ∈ D,
∗
where g(z) ∈ S (α). The branch of the power function is chosen such that
h
ic g(z)
z
=1.
z=0
(iii) f ∈ Ŝαβ if and only if
(1−α)c
f (z)
s(z)
=
,
z
z
z ∈ D,
∗
where s(z)∈S . The branch of the power function is chosen such that
h
i(1−α)c s(z)
z
=1.
z=0
Now we give the proof of (ii) and (iii).
Proof. (ii). First, assume that f (z) ∈
tions we may obtain the equality
Ŝαβ .
Setting g(z) = z
h
f (z)
z
eiβ
i cos
β
, through simple calcula-
zg 0 (z)
zf 0 (z)
= (1 + i tan β)
− i tan β.
g(z)
f (z)
Therefore the following inequality holds,
0 0
zg (z)
1
α cos β
iβ zf (z)
<e
=
<e e
>
= α,
g(z)
cos β
f (z)
cos β
namely g(z) ∈ S ∗ (α).
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 17, 7 pp.
http://jipam.vu.edu.au/
4
Q INGHUA X U AND S ANYA L U
Conversely, suppose g(z) ∈ S ∗ (α), then according to the above calculation, we have the
inequality
0 0
zg (z)
1
iβ zf (z)
<e e
= <e
> α.
cos β
f (z)
g(z)
This implies
0
iβ zf (z)
<e e
> α cos β,
f (z)
i.e., f (z) ∈ Ŝαβ .
(iii). It is easy to see from (ii) that f ∈ Ŝαβ if and only if g ∈ S ∗ (α) such that
c = e−iβ cos β. Noting that g(z) ∈ S ∗ (α) if and only if s(z) ∈ S ∗ such that
g(z)
z
f (z)
z
=
h
=
h
s(z)
z
ic
g(z)
z
i1−α
, here
which
holds in (i), we may obtain an important relationship between the class of Ŝαβ and the class of
h i(1−α)c
s(z)
S ∗ : f ∈ Ŝαβ if and only if there exists s(z) ∈ S ∗ such that f (z)
=
. Here, c =
z
z
h i(1−α)c e−iβ cos β and the branch of the power function is chosen such that s(z)
= 1.
z
z=0
Lemma 2.1 expresses the relations of the Ŝαβ and S ∗ classes, which play a key role in this
paper.
Lemma 2.2 ([5], [8]). A(K) = |γ| ≤ 12 ∪ 12 , 32 .
Lemma 2.3. For α ∈ [0, 1), β ∈ − π2 , π2 , J(Ŝαβ ) = I(1−α)e−iβ cos β (K).
Proof. Let f ∈ J(Ŝαβ ), then there exists g(z) ∈ Ŝαβ such that f (z) =
(iii) of Lemma 2.1 there is s(z) ∈ S ∗ such that
s(z)
g(z) = z
z
Rz
0
g(ζ)
dζ.
ζ
According to
(1−α)e−iβ cos β
,
therefore
Z
z
f (z) =
0
s(ζ)
ζ
(1−α)e−iβ cos β
dζ.
By the relationship of the S ∗ class and the K class, there exists u(z) ∈ K such that s(z) =
zu0 (z), thus
Z z
−iβ
f (z) =
[u0 (ζ)](1−α)e cos β dζ,
0
i.e., f (z) ∈ I(1−α)e−iβ cos β (K). As a result, J(Ŝαβ ) ⊂ I(1−α)e−iβ cos β (K) holds.
Conversely, when f (z) ∈ I(1−α)e−iβ cos β (K), we can trace back the above procedure to get
f ∈ J(Ŝαβ ), so I(1−α)e−iβ cos β (K) ⊂ J(Ŝαβ ).
From the above proof, we obtain the assertion.
Remark 1. If, in the hypothesis of Lemma 2.3, we set α = 0, we arrive at Lemma 4 of [4].
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 17, 7 pp.
http://jipam.vu.edu.au/
A LEXANDER T RANSFORMATION
5
3. T HE M AIN R ESULTS AND T HEIR P ROOFS
In this section, we let [z, w] denote the closed line segment with endpoints z and w for
z, w ∈ C.
Now we give the main results and their proofs.
Theorem 3.1. For α ∈ [0, 1), β ∈ (− π2 , π2 ),
[
eiβ
1
3eiβ
β
A(J(Ŝα )) = |γ| ≤
,
.
2(1 − α) cos β
2(1 − α) cos β 2(1 − α) cos β
Proof. By Lemma 2.3, we have
Iγ (J(Ŝαβ )) = Iγ (I(1−α)e−iβ cos β (K)) = Iγ(1−α)e−iβ cos β (K).
Therefore, γ ∈ A(J(Ŝαβ )) if and only if γ(1 − α)e−iβ cos β ∈ A(K), and by Lemma 2.2 we
may easily get the result.
Remark 2. In this theorem, if we set α = 0, we obtain Theorem 3 of [4].
Theorem 3.2. For α ∈ [0, 1), β ∈ (− π2 , π2 ), the inclusion relation J(Ŝαβ ) ⊂ S holds precisely if
1
or α = β = 0.
either cos β ≤ 2(1−α)
Proof. As α = β = 0, the result holds evidently by Integral transformation 1; while for α = 0
and β 6= 0, the result is Theorem 1 of [4] and was proved by Y.C. Kim and T. Sugawa [4].
If α 6= 0 and β = 0, then f (z) ∈ S ∗ (α). By Lemma 2.1(i), there exists u(z) ∈ S ∗ such that
1
1−α
1 f (z) 1−α u(z) = z f (z)
.
The
branch
of
the
power
function
is
chosen
such
that
= 1.
z
z
z=0
From Integral transformation 1, we can easily see that there exists g(z) ∈ J(Ŝαβ ) such that
1
Z z
f (ζ) 1−α
g(z) =
dζ.
ζ
0
For
zg 00 (z)
1 zf 0 (z)
<e 1 + 0
= <e
g (z)
1 − α f (z)
h 0 i
h
i
zf (z)
zg 00 (z)
and <e f (z) > α, we can deduce that <e 1 + g0 (z) > 0. This implies g(z) ∈ K and
J(S ∗ (α)) ⊂ S.
let α 6= 0 andi β 6= 0. Since J(Ŝαβ ) ⊂ S is equivalent to 1 ∈ A(J(Ŝαβ )) and 1 ∈
/
h Now
3eiβ
1
1
eiβ
,
, by Theorem 3.1, we deduce that 1 ≤ 2(1−α) cos β , i.e., cos β ≤ 2(1−α) .
2(1−α) cos β 2(1−α) cos β
Summarizing the above procedure, for α ∈ [0, 1), β ∈ (− π2 , π2 ), J(Ŝαβ ) ⊂ S holds when
1
or α = β = 0.This completes the proof.
cos β ≤ 2(1−α)
Remark 3. This theorem is an extension of Theorem 1 of [4]. Indeed, if we set α = 0, we will
obtain the result of [4].
Theorem 3.3. For α ∈ [0, 1), β ∈ (− π2 , π2 ),
A(J(Ŝ)) = |γ| ≤
1
2(1 − α) cos β
.
S
Proof. In view of Ŝ = β Ŝαβ and A(F ) = {γ ∈ C : Iγ (F ) ⊂ S}, we deduce A(J(Ŝ)) =
T
(J(Ŝαβ )). With the aid of Theorem 3.1, a simple observation gives A(J(Ŝ)) =
o
nβ
1
|γ| ≤ 2(1−α) cos β . Thus the proof is now complete.
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 17, 7 pp.
http://jipam.vu.edu.au/
6
Q INGHUA X U AND S ANYA L U
Remark 4. For α = β = 0, Theorem 3.3 implies the Theorem 2 of [4].
At the end of this paper, we mention the norm estimate of pre-Schwarzian derivatives. The
hyperbolic norm of the pre-Schwarzian derivative Tf = f 00 /f 0 of f ∈ A is defined to be
kf k = sup (1 − |z|2 )|Tf (z)|.
|z|<1
It is known that f is bounded if kf k < 2 and the bound depends only on the value of kf k ([9]).
Since
Rz
00 0
γ
[f
(ζ)]
dζ
kIγ [f ]k = sup (1 − |z|2 ) 0R z
0 [f 0 (ζ)]γ |z|<1
00 γ 0 ([f (z)] ) = sup (1 − |z|2 ) 0
γ f
(z)]
|z|<1
00 2 γf (z) = sup (1 − |z| ) 0
= |γ|kf k.
f (z) |z|<1
We obtain the following assertion.
Proposition 3.4. For each α ∈ [0, 1), β ∈ (− π2 , π2 ), the sharp inequality kf k ≤ 4(1 − α) cos β
1
holds for f ∈ J(Ŝαβ ). Moreover, if cos β < 2(1−α)
, then a function in J(Ŝαβ ) is bounded by a
constant depending on α and β.
Proof. For each f ∈ J(Ŝαβ ), by Lemma 2.3, there is a function k ∈ K such that f = Iγ (k),
where γ = (1 − α)e−iβ cos β. Noting that kkk ≤ 4 [10], we obtain the following inequality
kf k = |γ|kkk ≤ 4|γ| = 4(1 − α) cos β.
1
Since the inequality kkk ≤ 4 is sharp, the above inequality is also sharp. If cos β < 2(1−α)
, the
above inequality implies kf k ≤ 4(1 − α) cos β < 2, so f is bounded by a constant depending
on α and β.
Remark 5. If, in the statement of Proposition 3.4, we set α = 0, we arrive at the result of [4].
In the above proposition, the bound 21 cannot be replaced by any number greater than √
1
.
2(1−α)
Indeed, by the Alexander transformation, if the function
−iβ
g(z) = z(1 − z)−2(1−α)e
then the function
cos β
∈ Ŝαβ ,
−iβ
(1 − z)1−2(1−α)e cos β − 1
f (z) =
∈ J(Ŝαβ ),
2(1 − α)e−iβ cos β − 1
and we may verify that the latter is unbounded when cos β > √ 1 .
2(1−α)
R EFERENCES
[1] M.S. ROBERTSON, On the theory of univalent functions, Ann. Math., 37 (1936), 374–408.
[2] L. SPAČEK, Contribution à la théorie des fonctions univalentes, Casopis Pěst Math., 62 (1932),
12–19, (in Russian).
[3] M.S. ROBERTSON, Univalent functions f (z) for which zf 0 (z) is spirallike, Michigan Math. J.,
16 (1969), 97–101.
[4] Y.C. KIM AND T. SUGAWA, The Alexander transform of a spirallike function, J. Math. Anal.
Appl., 325(1) (2007), 608–611.
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 17, 7 pp.
http://jipam.vu.edu.au/
A LEXANDER T RANSFORMATION
7
[5] Y.C. KIM, S. PONNUSAMY AND T. SUGAWA, Mapping properties of nonlinear integral operators and pre-Schwarzian derivatives, J. Math. Anal. Appl., 299 (2004), 433–447.
[6] A.W. GOODMAN, Univalent functions, I-II, Mariner Publ.Co.,Tampa Florida,1983.
[7] I. GRAHAM AND G. KOHR, Geometric function theory in one and higher dimensions, Marcel
Dekker, New York ,2003.
[8] L.A. AKSENT’EV AND I.R. NEZHMETDINOV, Sufficient conditions for univalence of certain
integral transforms, Tr. Semin. Kraev. Zadacham. Kazan, 18 (1982), 3–11 (in Russian); English
translation in: Amer. Math. Soc. Transl., 136(2) (1987), 1–9.
[9] Y.C. KIM AND T. SUGAWA, Growth and coefficient estimates for uniformly locally univalent
functions on the unit disk, Rocky Mountain J. Math., 32 (2002), 179–200.
[10] S. YAMASHITA, Norm estimates for function starlike or convex of order alpha, Hokkaido Math.
J., 28 (1999), 217–230.
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 17, 7 pp.
http://jipam.vu.edu.au/