INTRODUCTION LetAdenote the class of functions f :f(z) =z

http://jipam.vu.edu.au/

Volume 7, Issue 2, Article 49, 2006

ON CERTAIN CLASSES OF ANALYTIC FUNCTIONS

KHALIDA INAYAT NOOR MATHEMATICSDEPARTMENT

COMSATS INSTITUTE OFINFORMATIONTECHONOLGY

ISLAMABAD, PAKISTAN

khalidanoor@hotmail.com

Received 28 August, 2005; accepted 21 October, 2005 Communicated by Th.M. Rassias

ABSTRACT. LetAbe the class of functionsf : f(z) = z+P∞

n=2anzⁿ which are analytic in the unit diskE.We introduce the classBk(λ, α, ρ)⊂ Aand study some of their interesting properties such as inclusion results and covering theorem. We also consider an integral operator for these classes.

Key words and phrases: Analytic functions, Univalent, Functions with positive real part, Convex functions, Convolution, In- tegral operator.

2000 Mathematics Subject Classification. 30C45, 30C50.

1. INTRODUCTION

LetAdenote the class of functions

f :f(z) =z+

∞

X

n=2

a_nzⁿ

which are analytic in the unit disk E ={z : |z| < 1}and letS ⊂ A be the class of functions univalent inE.

LetP_k(ρ)be the class of functionsp(z)analytic inEsatisfying the propertiesp(0) = 1and (1.1)

Z 2π

0

Rep(z)−ρ 1−ρ

dθ ≤kπ,

wherez =reⁱθ, k ≥ 2and0 ≤ ρ <1.This class has been introduced in [7]. We note that, forρ = 0,we obtain the classPkdefined and studied in [8], and forρ= 0, k= 2,we have the well known classP of functions with positive real part. The casek = 2gives the classP(ρ)of functions with positive real part greater thanρ.

ISSN (electronic): 1443-5756

c 2006 Victoria University. All rights reserved.

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

054-06

From (1.1) we can easily deduce thatp∈P_k(ρ)if, and only if, there existp₁, p₂ ∈P(ρ)such that, forE,

(1.2) p(z) =

k 4 + 1

2

p₁(z)− k

4 −1 2

p₂(z).

Letf andg be analytic inE withf(z) = P∞

m=0a_mz^mandg(z) = P∞

m=0b_mz^m inE.Then the convolution?(or Hadamard Product) off andg is defined by

(f ? g)(z) =

∞

X

m=0

ambmz^m, m∈N⁰ ={0,1,2, . . .}.

Definition 1.1. Letf ∈ A.Thenf ∈Bk(λ, α, ρ)if and only if (1.3)

(1−λ)

f(z) z

α

+λzf⁰(z) f(z)

f(z) z

α

∈P_k(ρ), z ∈E,

whereα >0, λ >0, k ≥2and0≤ρ <1.The powers are understood as principal values.

Fork = 2and with different choices ofλ, αandρ,these classes have been studied in [2, 3, 4, 10]. In particularB₂(1, α, ρ)is the class of Bazilevic functions studied in [1].

We shall need the following results.

Lemma 1.1 ([9]). Ifp(z)is analytic inEwithp(0) = 1and ifλis a complex number satisfying Reλ≥0, (λ6= 0),then

Re[p(z) +λzp⁰(z)]> β (0≤β <1) implies

Rep(z)> β+ (1−β)(2γ−1), whereγis given by

γ =γ_Reλ = Z 1

0

(1 +t^Reλ)⁻¹dt.

Lemma 1.2 ([5]). Letc >0, λ > 0, ρ <1andp(z) = 1 +b₁z+b₂z² +· · · be analytic inE.

LetRe[p(z) +cλzp⁰(z)]> ρinE,then

Re[p(z) +czp⁰(z)]≥2ρ−1 + 2(1−ρ)

1− 1 λ

1 cλ

Z 1

0

u

1 cλ−1

1 +udu.

This result is sharp.

2. MAINRESULTS

Theorem 2.1. Letλ, α >0, 0≤ρ <1and letf ∈b_k(λ, α, ρ).Then

f(z) z

^α

∈P_k(ρ₁),where ρ1 is given by

(2.1) ρ₁ =ρ+ (1−ρ)(2γ−1),

and

γ = Z 1

0

1 +t^αλ

−1

dt.

Proof. Let

f(z) z

α

=p(z) = k

4 + 1 2

p₁(z)− k

4 − 1 2

p₂(z).

Thenp(z) = 1 +αa₂z+· · · is analytic inE,and

(2.2) (f(z))^α =z^αp(z).

Differentiation of (2.2) and some computation give us (1−λ)

f(z) z

α

+λzf⁰(z) f(z)

f(z) z

α

=p(z) + λ

αzp⁰(z).

Sincef ∈B_k(λ, α, ρ),so{p(z) + _α^λzp⁰(z)} ∈P_k(ρ)forz ∈E.This implies that Re

p_i(z) + λ αzp⁰_i(z)

> ρ, i= 1,2.

Using Lemma 1.1, we see that Re{p_i(z)} > ρ₁, where ρ₁ is given by (2.1). Consequently

p∈P_k(ρ₁)forz ∈E,and the proof is complete.

Corollary 2.2. Letf =zF₁⁰ andf ∈B₂(λ,1, ρ).ThenF₁is univalent inE.

Proceeding as in Theorem 2.1 and using Lemma 1.2, we have the following.

Theorem 2.3. Let α > 0, λ > 0, 0 ≤ ρ < 1and let f ∈ B_k(λ, α, ρ).Then ^zf_f(z)^0(z)(^f(z)_z )^α ∈ P_k(ρ₂),where

ρ2 = 2ρ−1 + 1−ρ

λ + 2(1−ρ)

1− 1 λ

α λ

Z 1

0

u

α λ−1

1 +udu.

This result is sharp.

Fork = 2,we note thatf is univalent, see [1].

Theorem 2.4. Let, forα >0, λ > 0, 0 ≤ρ < 1, f ∈ B_k(λ, α, ρ)and define I(f) : A −→ A as

(2.3) I(f) =F(z) = 1

λz^α−

1 λ

Z z

0

t

1 λ−1−α

(f(z))^αdt

α1

, z ∈E.

ThenF ∈B_k(αλ, α, ρ₁)forz ∈E,whereρ₁ is given by (2.1).

Proof. Differentiating (2.3), we have (1−αλ)

F(z) z

α

+αλzF⁰(z) F(z)

F(z) z

α

=

f(z) z

α

.

Now, using Theorem 2.1, we obtain the required result.

Theorem 2.5. Let

f :f(z) =z+

∞

X

n=2

a_nzⁿ ∈B_k(λ, α, ρ).

Then

|a_n| ≤ k(1−ρ) λ+α .

The functionf_λ,α,ρ(z)defined as f_λ,α,ρ(z)

z α

= α λ

Z 1

0

k 4 + 1

2

u^αλ−11 + (1−2ρ)uz 1−uz

− k

4 − 1 2

u^αλ−11−(1−2ρ)uz 1 +uz

du

shows that this inequality is sharp.

Proof. Sincef ∈B_k(λ, α, ρ),so (1−λ) 1 +

∞

X

n=2

a_nzⁿ⁻¹

!α

+λ 1 +

∞

X

n=2

na_nzⁿ⁻¹

! 1 +

∞

X

n=2

a_nzⁿ⁻¹

!α

=H(z) = 1 +

∞

X

n=1

c_nzⁿ

!

∈P_k(ρ).

It is known that |c_n| ≤ k(1− ρ) for all n and using this inequality, we prove the required

result.

Different choices ofk, λ, αandρyield several known results.

Theorem 2.6 (Covering Theorem). Letλ > 0and0 < ρ < 1.Letf = zF₁⁰ ∈ B₂(λ,1, ρ).If Dis the boundary of the image ofE underF1,then every point ofDhas a distance of at least

λ+1

(3+2λ−ρ) from the origin.

Proof. LetF₁(z)6=w₀, w₀ 6= 0.Thenf₁(z) = _w^w0F1(z)

0+F1(z) is univalent inEsinceF₁ is univalent.

Let

f(z) =z+

∞

X

n=2

anzⁿ, F1(z) =z+

∞

X

n=2

bnzⁿ. Thena₂ = 2b₂.Also

f1(z) = z+

b2+ 1 w₀

z²+· · · , and so|b₂+ _w¹

0| ≤2.Since, by Theorem 2.5,|b₂| ≤ ^1−ρ_1+λ,we obtain|w₀| ≥ _3+2λ−ρ^λ+1 . Theorem 2.7. For eachα >0, B_k(λ₁, α, ρ)⊂B_k(λ₂, α, ρ)for0≤λ₂ < λ₁.

Proof. For λ₂ = 0, the proof is immediate. Letλ₂ > 0and let f ∈ B_k(λ₁, α, ρ).Then there exist two functionsh1, h2 ∈Pk(ρ)such that, from Definition 1.1 and Theorem 2.1,

(1−λ)

f(z) z

α

+λ₁zf⁰(z) f(z)

f(z) z

α

=h₁(z), and

f(z) z

α

=h₂(z).

Hence

(2.4) (1−λ₂)

f(z) z

α

+λ₂zf⁰(z) f(z)

f(z) z

α

= λ₂

λ₁h₁(z) +

1−λ₂ λ₁

h₂(z).

Since the classP_k(ρ)is a convex set, see [6], it follows that the right hand side of (2.4) belongs

toP_k(ρ)and this proves the result.

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