In this paper, a subclass of spirallike function of typeβdenoted bySˆαβis introduced in the unit disc of the complex plane

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THE ALEXANDER TRANSFORMATION OF A SUBCLASS OF SPIRALLIKE FUNCTIONS OF TYPE β

1QINGHUA XU AND^1,2SANYA LU

1SCHOOL OFMATHEMATICS ANDINFORMATIONSCIENCE

JIANGXINORMALUNIVERSITY

JIANGXI, 330022, CHINA

xuqhster@gmail.com

2DEPARTMENT OFSCIENCE, NANCHANGINSTITUTE OFTECHNOLOGY

JIANGXI, 330099, CHINA

yasanlu@163.com

Received 13 August, 2008; accepted 27 December, 2008 Communicated by G. Kohr

ABSTRACT. In this paper, a subclass of spirallike function of typeβdenoted bySˆ_α^βis introduced in the unit disc of the complex plane. We show that the Alexander transformation of class ofSˆ^β_α is univalent whencosβ≤ _2(1−α)¹ , which generalizes the related results of some authors.

Key words and phrases: Univalent functions, Starlike functions of orderα, spirallike functions of typeβ, Integral transformations.

2000 Mathematics Subject Classification. 30C45.

1. INTRODUCTION

Let A denote the class of analytic functions f on the unit disk D = {z ∈ C : |z| < 1}

normalized by f(0) = 0 and f⁰(0) = 1, S denote the subclass of A consisting of univalent functions, andS^∗ denote starlike functions onD. Obviously,S^∗ ⊂S ⊂Aholds.

In [1], Robertson introduced starlike functions of orderαonD.

Definition 1.1. Letα∈[0,1),f ∈Sand

<e

zf⁰(z) f(z)

> α, z∈D.

We say thatf is a starlike function of orderα. LetS^∗(α)denote the whole starlike functions of orderαonD.

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.

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Spaˇcek [2] extended the class ofS^∗,and obtained the class of spirallike functions of typeβ.

In the same article, the author gave an analytical characterization of spirallikeness of typeβon D.

Theorem 1.1. Letf ∈S andβ ∈(−^π₂,^π₂). Thenf(z)is a spirallike function of typeβonDif and only if

<e

e^iβzf⁰(z) f(z)

>0, z ∈D.

We denote the whole spirallike functions of typeβonDbySˆ_β.

From Definition 1.1 and Theorem 1.1, it is easy to see that starlike functions of orderαand spirallike functions of typeβhave some relationships on geometry. Spirallike functions of type β mapD into the right half complex plane by the mappinge^{iβ zf}_f_(z)⁰^(z), while starlike functions of order α map D into the right half complex plane whose real part is greater than α by the mapping ^zf_f(z)⁰^(z). Since lim

z→0e^{iβ zf}_f(z)⁰^(z) = e^iβ, we can deduce that if we restrict the image of the mappinge^{iβ zf}_f(z)⁰^(z) in the right complex plane whose real part is greater than a certain constant, then the constant must be smaller thancosβ. According to this, we introduce the functions class Sˆ_α^β onD.

Definition 1.2. Letα∈[0,1),β ∈(−^π₂,^π₂),f ∈S, thenf ∈Sˆ_α^β if and only if

<e

e^iβzf⁰(z) f(z)

> αcosβ, z ∈D.

Obviously, whenβ= 0,f ∈S^∗(α); whileα= 0, f ∈Sˆβ. Example 1.1. Letf(z) = ^z

(1−z)

2(1−α) 1+itanβ

, z ∈ D. The branch of the power function is chosen such that

[(1−z)]¹⁺ⁱ^2(1−α)^tanβ z=0

= 1.

It is easily proved thatf ∈Sˆ_α^β. We omit the proof.

For our applications, we setSˆ=S

βSˆ_α^β.

In this paper, we first establish the relationships among Sˆ_α^β and some important subclasses ofS, then investigate the Alexander transformation ofSˆ_α^β preserving univalence. Furthermore, some other properties of the class ofSˆ_α^βare obtained. These results generalize the related works of some authors.

2. INTEGRALTRANSFORMATIONS AND LEMMAS

Integral Transformation 1. The integral transformation J[f](z) =

Z z

0

f(ζ) ζ dζ

is called the Alexander Transformation and it was introduced by Alexander in [4]. Alexander was the first to observe and prove that the Integral transformationJmaps the classS^∗of starlike functions onto the classK of convex functions in a one-to-one fashion.

In 1960, Biernacki conjectured thatJ(S)⊂ S, but Krzyz and Lewandowski disproved it in 1963 by giving the examplef(z) = z(1−iz)ⁱ⁻¹, which is a spirallike function of type ^π₄ but is transformed into a non-univalent function byJ [4]. In 1969, Robertson studied the Alexander Integral Transformation of spirallike functions of type β. The author showed thatJ( ˆS_β) ⊂ S

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holds when β satisfies a certain condition, that is cosβ ≤ x₀ (a constant). Robertson also noticed thatx₀ cannot be replaced by any number greater than ¹₂ and asked about the best value for this [3]. In 2007, Y.C. Kim and T. Sugawa proved that J( ˆS_β) ⊂ S holds precisely when cosβ ≤ ¹₂ orβ = 0[4].

Integral Transformation 2. Let γ ∈ C, f(z) ∈ A be locally univalent, and the Integral transformationI_γ[5] be defined by

I_γ[f](z) = Z z

0

[f⁰(ζ)]^γdζ =z Z 1

0

[f⁰(tz)]^γdt.

Based on the definition ofI_γ, we may easily show thatI_γ◦I_γ⁰ =I_γγ⁰.

LetA(F) ={γ ∈C:I_γ(F)⊂S}, F ⊂Abe locally univalent. According to the definition of theA(F),J( ˆS_α^β)⊂Sis equivalent to1∈A(J( ˆS_α^β)).

For the proof of the theorems in this paper, we need the following lemma, which establishes the relationships amongSˆ_α^β and some important subclasses ofS.

Lemma 2.1. Forα∈[0,1),β ∈(−^π₂,^π₂),c=e^−iβcosβ, the following assertions hold:

(i) ([6, 7])f ∈S^∗(α)if and only if f(z)

z = u(z)

z 1−α

, z ∈D,

whereu(z)∈S^∗. The branch of the power function is chosen such thath_u(z)

z

i^1−α z=0

= 1.

(ii) f ∈Sˆ_α^β if and only if

f(z) z =

g(z) z

c

, z∈D,

whereg(z)∈S^∗(α). The branch of the power function is chosen such that h

g(z) z

ic

_z=0=1.

(iii) f ∈Sˆ_α^β if and only if

f(z) z =

s(z) z

(1−α)c

, z ∈D,

wheres(z)∈S^∗. The branch of the power function is chosen such that hs(z)

z

i(1−α)c z=0

=1.

Now we give the proof of (ii) and (iii).

Proof. (ii). First, assume thatf(z)∈Sˆ_α^β. Settingg(z) =zh_f(z)

z

i_cosβ^eiβ

, through simple calcula- tions we may obtain the equality

zg⁰(z)

g(z) = (1 +itanβ)zf⁰(z)

f(z) −itanβ.

Therefore the following inequality holds,

<e

zg⁰(z) g(z)

= 1

cosβ<e

e^iβzf⁰(z) f(z)

> αcosβ cosβ =α, namelyg(z)∈S^∗(α).

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Conversely, suppose g(z) ∈ S^∗(α), then according to the above calculation, we have the inequality

1 cosβ<e

e^iβzf⁰(z) f(z)

=<e

zg⁰(z) g(z)

> α.

This implies

<e

e^iβzf⁰(z) f(z)

> αcosβ,

i.e.,f(z)∈Sˆ_α^β.

(iii). It is easy to see from (ii) thatf∈Sˆ_α^β if and only ifg∈S^∗(α)such that ^f^(z)_z =h

g(z) z

ic

, here c=e^−iβcosβ. Noting thatg(z)∈S^∗(α)if and only ifs(z)∈S^∗ such that^g(z)_z =h

s(z) z

i1−α

which holds in (i), we may obtain an important relationship between the class of Sˆ_α^β and the class of S^∗ : f ∈ Sˆ_α^β if and only if there exists s(z) ∈ S^∗ such that ^f(z)_z = h_s(z)

z

i^(1−α)c

. Here, c = e^−iβcosβ and the branch of the power function is chosen such that h

s(z) z

i(1−α)c z=0

= 1.

Lemma 2.1 expresses the relations of the Sˆ_α^β and S^∗ classes, which play a key role in this paper.

Lemma 2.2 ([5], [8]). A(K) =

|γ| ≤ ¹₂ ∪₁

2,³₂ . Lemma 2.3. Forα∈[0,1),β ∈ −^π₂,^π₂

,J( ˆS_α^β) =I_(1−α)e^−iβ^cos^β(K).

Proof. Letf ∈ J( ˆS_α^β), then there existsg(z) ∈ Sˆ_α^β such thatf(z) = Rz 0

g(ζ)

ζ dζ. According to (iii) of Lemma 2.1 there iss(z)∈S^∗ such that

g(z) =z s(z)

z

^(1−α)e^−iβcosβ

,

therefore

f(z) = Z z

0

s(ζ) ζ

(1−α)e^−iβcosβ

dζ.

By the relationship of the S^∗ class and the K class, there exists u(z) ∈ K such thats(z) = zu⁰(z), thus

f(z) = Z z

0

[u⁰(ζ)]^(1−α)e^−iβ^cos^βdζ,

i.e.,f(z)∈I_(1−α)e^−iβ_cos_β(K). As a result,J( ˆS_α^β)⊂I_(1−α)e^−iβ_cos_β(K)holds.

Conversely, whenf(z) ∈ I_(1−α)e^−iβ_cos_β(K), we can trace back the above procedure to get f ∈J( ˆS_α^β), soI_(1−α)e^−iβ_cos_β(K)⊂J( ˆS_α^β).

From the above proof, we obtain the assertion.

Remark 1. If, in the hypothesis of Lemma 2.3, we setα= 0, we arrive at Lemma 4 of [4].

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3. THEMAINRESULTS AND THEIRPROOFS

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. Forα∈[0,1),β ∈(−^π₂,^π₂), A(J( ˆS_α^β)) =

|γ| ≤ 1

2(1−α) cosβ

[

e^iβ

2(1−α) cosβ, 3e^iβ 2(1−α) cosβ

. Proof. By Lemma 2.3, we have

I_γ(J( ˆS_α^β)) =I_γ(I_(1−α)e^−iβ_cos_β(K)) =I_γ(1−α)e^−iβ_cos_β(K).

Therefore, γ ∈ A(J( ˆS_α^β))if and only ifγ(1−α)e^−iβcosβ ∈ A(K), and by Lemma 2.2 we

may easily get the result.

Remark 2. In this theorem, if we setα= 0, we obtain Theorem 3 of [4].

Theorem 3.2. Forα ∈[0,1),β∈(−^π₂,^π₂), the inclusion relationJ( ˆS_α^β)⊂S holds precisely if eithercosβ ≤ _2(1−α)¹ orα=β = 0.

Proof. Asα = β = 0, the result holds evidently by Integral transformation 1; while forα = 0 andβ 6= 0, the result is Theorem 1 of [4] and was proved by Y.C. Kim and T. Sugawa [4].

Ifα 6= 0andβ = 0, thenf(z) ∈S^∗(α). By Lemma 2.1(i), there existsu(z)∈ S^∗ such that u(z) =z

f(z) z

_1−α¹

. The branch of the power function is chosen such that f(z)

z

_1−α¹ _z=0

= 1.

From Integral transformation 1, we can easily see that there existsg(z)∈J( ˆS_α^β)such that g(z) =

Z z

0

f(ζ) ζ

_1−α¹ dζ.

For

<e

1 + zg⁰⁰(z) g⁰(z)

=<e 1

1−α

zf⁰(z) f(z)

and <eh

zf⁰(z) f(z)

i

> α, we can deduce that <eh

1 + ^zg_g0⁰⁰(z)^(z)

i

> 0. This impliesg(z) ∈ K and J(S^∗(α))⊂S.

Now let α 6= 0 and β 6= 0. Since J( ˆS_α^β) ⊂ S is equivalent to 1 ∈ A(J( ˆS_α^β)) and 1 ∈/ h e^iβ

2(1−α) cosβ,_{2(1−α) cos}^3e^iβ _βi

, by Theorem 3.1, we deduce that1≤ _{2(1−α) cos}¹ _β , i.e.,cosβ ≤ _2(1−α)¹ . Summarizing the above procedure, for α ∈ [0,1), β ∈ (−^π₂,^π₂), J( ˆS_α^β) ⊂ S holds when

cosβ ≤ _2(1−α)¹ orα =β = 0.This completes the proof.

Remark 3. This theorem is an extension of Theorem 1 of [4]. Indeed, if we setα = 0, we will obtain the result of [4].

Theorem 3.3. Forα∈[0,1),β ∈(−^π₂,^π₂), A(J( ˆS)) =

|γ| ≤ 1 2(1−α) cosβ

. Proof. In view of Sˆ = S

βSˆ_α^β and A(F) = {γ ∈ C : I_γ(F) ⊂ S}, we deduce A(J( ˆS)) = T

β(J( ˆS_α^β)). With the aid of Theorem 3.1, a simple observation gives A(J( ˆS)) = n|γ| ≤ _{2(1−α) cos}¹ _βo

. Thus the proof is now complete.

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Remark 4. Forα =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 derivativeTf =f⁰⁰/f⁰ off ∈Ais defined to be

kfk= sup

|z|<1

(1− |z|²)|Tf(z)|.

It is known thatf is bounded ifkfk<2and the bound depends only on the value ofkfk([9]).

Since

kIγ[f]k= sup

|z|<1

(1− |z|²)

Rz

0[f⁰(ζ)]^γdζ00

Rz

0[f⁰(ζ)]^γ0

= sup

|z|<1

(1− |z|²)

([f⁰(z)]^γ)⁰ f⁰(z)]^γ

= sup

|z|<1

(1− |z|²)

γf⁰⁰(z) f⁰(z)

=|γ|kfk.

We obtain the following assertion.

Proposition 3.4. For eachα∈[0,1), β ∈(−^π₂,^π₂), the sharp inequalitykfk ≤4(1−α) cosβ holds for f ∈ J( ˆS_α^β). Moreover, ifcosβ < _2(1−α)¹ , then a function in J( ˆS_α^β)is bounded by a constant depending onαandβ.

Proof. For each f ∈ J( ˆS_α^β), by Lemma 2.3, there is a function k ∈ K such that f = I_γ(k), whereγ = (1−α)e^−iβcosβ. Noting thatkkk ≤4[10], we obtain the following inequality

kfk=|γ|kkk ≤4|γ|= 4(1−α) cosβ.

Since the inequalitykkk ≤ 4is sharp, the above inequality is also sharp. Ifcosβ < _2(1−α)¹ , the above inequality implieskfk ≤ 4(1−α) cosβ < 2, sof is bounded by a constant depending

onαandβ.

Remark 5. If, in the statement of Proposition 3.4, we setα= 0, we arrive at the result of [4].

In the above proposition, the bound¹₂cannot be replaced by any number greater than√ ¹

2(1−α). Indeed, by the Alexander transformation, if the function

g(z) =z(1−z)^{−2(1−α)e}^−iβ^cos^β ∈Sˆ_α^β, then the function

f(z) = (1−z)^{1−2(1−α)e}^−iβ^cos^β−1

2(1−α)e^−iβcosβ−1 ∈J( ˆS_α^β), and we may verify that the latter is unbounded whencosβ > √ ¹

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