ON A GENERALIZATION OF ALPHA CONVEXITY

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Alpha Convexity Khalida Inayat Noor vol. 8, iss. 1, art. 16, 2007

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ON A GENERALIZATION OF ALPHA CONVEXITY

KHALIDA INAYAT NOOR

Mathematics Department,

COMSATS Institute of Information Technology, Islamabad, Pakistan.

EMail:khalidanoor@hotmail.com

Received: 28 November, 2006

Accepted: 06 February, 2007

Communicated by: N.E. Cho 2000 AMS Sub. Class.: 30C45, 30C50.

Key words: Starlike, Convex, Strongly alpha-convex, Bounded boundary rotation.

Abstract: In this paper, we introduce and study a classM˜k(α, β, γ), k ≥ 2of analytic functions defined in the unit disc. This class generalizes the concept of alpha- convexity and include several other known classes of analytic functions. Inclu- sion results, an integral representation and a radius problem is discussed for this class.

Acknowledgements: This research is supported by the Higher Education Commission, Pakistan, through research grant No: 1-28/HEC/HRD/2005/90.

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1. Introduction

LetP˜denote the class of functions of the form

(1.1) p(z) = 1 +c₁z+c₂z²+· · · ,

which are analytic in the unit discE ={z :|z|<1}.LetP˜(γ)be the subclass ofP˜ consisting of functionspwhich satisfy the condition

(1.2) |argp(z)| ≤ πγ

2 , for some γ(γ >0), z ∈E.

We note thatP˜(1) = P is the class of analytic functions with positive real part. We introduce the classP˜_k(γ)as follows:

An analytic functionpgiven by (1.1) belongs toP˜_k(γ),forz ∈E,if and only if there existp₁, p₂ ∈P˜(γ)such that

(1.3) p(z) =

k 4 + 1

2

p₁(z)− k

4 −1 2

p₂(z), k ≥2.

We now define the classM˜_k(α, β, γ)as follows:

Definition 1.1. Let α ≥ 0, β ≥ 0 (α+β 6= 0) and let f be analytic in E with f(0) = 0, f⁰(0) = 1and ^f⁰^(z)f_z ^(z) 6= 0.Then f ∈ M˜_k(α, β, γ)if and only if , for z ∈E,

α α+β

zf⁰(z)

f(z) + β α+β

(zf⁰(z))⁰ f⁰(z)

∈P˜_k(γ).

We note that, fork = 2, β = (1−α),we have the classM˜₂(α,1−α, γ) = ˜M_α(γ) of strongly alpha-convex functions introduced and studied in [4].

We also have the following special cases.

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(i) M˜₂(α,0,1) = S^?, M˜₂(0, β,1) = C, where S^? and C are respectively the well-known classes of starlike and convex functions. It is known [3] that M˜_α(γ)⊂S^?andM˜₂(α,0, γ)coincides with the class of strongly starlike functions of orderγ,see [1,7,8].

(ii) M˜k(α,0,1) = Rk, M˜k(0, β,1) = Vk,whereRk is the class of functions of bounded radius rotation and V_k is the class of functions of bounded boundary rotation.

AlsoM˜_k(0, β, γ) = ˜V_k(γ)⊂V_kandM˜_k(α,0, γ) = ˜R_k(γ)⊂R_k.

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2. Main Results

Theorem 2.1. A functionf ∈ M˜_k(α, β, γ), α, β >0,if and only if, there exists a functionF ∈R˜_k(γ)such that

(2.1) f(z) =

"

α+β α

Z z

0

(F(t))^α+β^β t dt

#_α+β^β .

Proof. A simple calculation yields α

α+β

zf⁰(z)

f(z) + β α+β

(zf⁰(z))⁰

f⁰(z) = zF⁰(z) F(z) .

If the right hand side belongs toP˜_k(γ)so does the left and conversely, and the result follows.

Theorem 2.2. Letf ∈M˜k(α, β, γ).Then the function

(2.2) g(z) =f(z)

zf⁰(z) f(z)

_α+β^β

belongs toR˜k(γ)forz ∈E.

Proof. Differentiating (2.2) logarithmically, we have zg⁰(z)

g(z) = α α+β

zf⁰(z)

f(z) + β α+β

(zf⁰(z))⁰ f⁰(z) , and, sincef ∈M˜_k(α, β, γ),we obtain the required result.

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Theorem 2.3. Let f ∈ M˜_k(α, β, γ), β > 0,0 < γ ≤ 1. Then f ∈ R˜_k(γ) for z ∈E.

Proof. Let ^zf_f(z)⁰^(z) =p(z).Then (zf⁰(z))⁰

f⁰(z) =p(z) + zp⁰(z) p(z) . Therefore, forz ∈E,

(2.3) α

α+β

zf⁰(z)

f(z) + β α+β

(zf⁰(z))⁰ f⁰(z) =

p(z) + β α+β

zp⁰(z) p(z)

∈P˜_k(γ).

Let

(2.4) φ(α, β) = α

α+β z

1−z + β α+β

z (1−z)². Then, using (1.3) and (2.4), we have

p ? φ(α, β) z

= k

4 +1

2 p₁? φ(α, β) z

− k

4 − 1

2 p₂?φ(α, β) z

,

where?denotes the convolution (Hadamard product). This gives us p(z) + β

α+β zp⁰(z)

p(z) = k

4 + 1

2 p₁(z) + β α+β

zp⁰₁(z) p1(z)

− k

4 − 1

2 p₂(z) + β α+β

zp⁰₂(z) p2(z)

.

From (2.3), it follows that

p_i+ β α+β

zp⁰_i p_i

∈P˜(γ), i= 1,2,

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and, using a result due to Nunokawa and Owa [6], we conclude thatp_i ∈ P˜(γ) in E, i= 1,2.Consequentlyp∈P˜_k(γ)and hencef ∈R˜_k(γ)forz ∈E.

Theorem 2.4. Let, for(α₁ +β₁)6= 0, α₁

α₁+β₁ < α

α+β, β₁

α₁+β₁ < β

α+β and 0≤γ <1.

Then

M˜_k(α, β, γ)⊂M˜_K(α₁, β₁, γ), z ∈E.

Proof. We can write α₁ α₁ +β₁

zf⁰(z)

f(z) + β₁ α₁+β₁

(zf⁰(z))⁰ f⁰(z)

=

1− β₁(α+β) β(α₁+β₁)

zf⁰(z) f(z) +

β₁(α+β) β(α₁+β₁)

α α+β

zf⁰(z)

f(z) + β α+β

(zf⁰(z))⁰ f⁰(z)

=

1− β₁(α+β) β(α₁+β₁)

H₁(z) + β₁(α+β)

β(α₁+β₁)H₂(z),

whereH₁, H₂ ∈P˜_k(γ)by using Definition1.1and Theorem2.3. Since0< γ ≤1, the classP˜(γ)is a convex set and consequently, by (1.3), the classP˜_k(γ)is a convex set. This impliesH ∈ P˜_k(γ)and thereforef ∈ M˜_k(α₁, β₁, γ). This completes the proof.

Theorem 2.5. Letf ∈M˜_k(α, β, γ).Then

(2.5) h(z) = Z z

0

(f⁰(t))

β α+β

f(t) t

_α+β^α

dt belongs to V˜k(γ) for z ∈E.

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Proof. From (2.5), we have

h⁰(z) = (f⁰(z))

β α+β

f(z) z

_α+β^α .

Now the proof is immediate when we differentiate both sides logarithmically and use the fact thatf ∈M˜_k(α, β, γ).

In the following we study the converse case of Theorem2.3withγ = 1.

Theorem 2.6. Let f ∈ R˜_k(1). Then f ∈ M˜_k(α, β,1), β > 0 for |z| < r(α, β), where

(2.6) r(α, β) = 1−ρ²¹₂

−ρ, with ρ= β α+β. This result is best possible.

Proof. Sincef ∈R˜_k(1), ^zf_f_(z)⁰^(z) ∈P˜_k(1) =P_k,and α

α+β

zf⁰(z)

f(z) + β α+β

(zf⁰(z))⁰

f⁰(z) =p(z) + β α+β

zp⁰(z) p(z) .

Letφ(α, β) be as given by (2.4). Now using (1.3) and convolution techniques, we have

p(z) + β α+β

zp⁰(z)

p(z) =p(z)? φ(α, β) z

= k

4 +1

2 p₁(z)? φ(α, β) z

− k

4 −1

2 p₂(z)?φ(α, β) z

.

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Sincep_i ∈P˜₂(1) =P and it is known [2] thatRen

φ(α,β) z

o

> ¹₂ for|z|< r(α, β),it follows from a well known result, see [5] thath

p_i? ^φ(α,β)_z i

∈P for|z|< r(α, β), i= 1,2.withr(α, β)given by (2.6). The functionφ(α, β)given by (2.4) shows that the radiusr(α, β)is best possible.

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References

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[2] J.L. LIU AND K. INAYAT NOOR, On subordination for certain analytic func- tions associated with Noor integral operator, Appl. Math. Computation, (2006), in press.

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[4] K. INAYAT NOOR, On strongly alpha-convex and alpha-quasi-convex func- tions, J. Natural Geometry, 10(1996), 111–118.

[5] K. INAYAT NOOR, Some properties of certain analytic functions, J. Natural Geometry, 7 (1995), 11–20.

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Math. J., 3 (1993), 35–38.

[7] J. STANSKIEWICS, Some remarks concerning starlike functions, Bull. Acad.

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[8] J. STANSKIEWICS, On a family of starlike functions, Ann. Univ. Mariae-Curie- Skl. Sect. A., 22(24) (1968/70), 175–181.