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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

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

f :f(z) =z+

X

n=2

anzn

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

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

Z

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

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

054-06

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From (1.1) we can easily deduce thatp∈Pk(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=0amzmandg(z) = P

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

(f ? g)(z) =

X

m=0

ambmzm, 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

∈Pk(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 =γReλ = Z 1

0

(1 +tReλ)−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 −1

1 +udu.

This result is sharp.

2. MAINRESULTS

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

f(z) z

α

∈Pk1),where ρ1 is given by

(2.1) ρ1 =ρ+ (1−ρ)(2γ−1),

and

γ = Z 1

0

1 +tαλ

−1

dt.

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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 =zαp(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 ∈Bk(λ, α, ρ),so{p(z) + αλzp0(z)} ∈Pk(ρ)forz ∈E.This implies that Re

pi(z) + λ αzp0i(z)

> ρ, i= 1,2.

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

p∈Pk1)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 ∈ Bk(λ, α, ρ).Then zff(z)0(z)(f(z)z )α ∈ Pk2),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 ∈ Bk(λ, α, ρ)and define I(f) : A −→ A as

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

λzα−

1 λ

Z z

0

t

1 λ−1−α

(f(z))αdt

α1

, z ∈E.

ThenF ∈Bk(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

anzn ∈Bk(λ, α, ρ).

Then

|an| ≤ k(1−ρ) λ+α .

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The functionfλ,α,ρ(z)defined as fλ,α,ρ(z)

z α

= α λ

Z 1

0

k 4 + 1

2

uαλ−11 + (1−2ρ)uz 1−uz

− k

4 − 1 2

uαλ−11−(1−2ρ)uz 1 +uz

du

shows that this inequality is sharp.

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

X

n=2

anzn−1

!α

+λ 1 +

X

n=2

nanzn−1

! 1 +

X

n=2

anzn−1

!α

=H(z) = 1 +

X

n=1

cnzn

!

∈Pk(ρ).

It is known that |cn| ≤ 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) = ww0F1(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−ρ1+λ,we obtain|w0| ≥ 3+2λ−ρλ+1 . Theorem 2.7. For eachα >0, Bk1, α, ρ)⊂Bk2, α, ρ)for0≤λ2 < λ1.

Proof. For λ2 = 0, the proof is immediate. Letλ2 > 0and let f ∈ Bk1, 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

1zf0(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

2zf0(z) f(z)

f(z) z

α

= λ2

λ1h1(z) +

1−λ2 λ1

h2(z).

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

toPk(ρ)and this proves the result.

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REFERENCES

[1] I.E. BAZILEVIC, On a class of integrability in quadratures of the Loewner-Kuarev equation, Math.

Sb., 37 (1955), 471–476.

[2] M.P. CHEN, On the regular functions satisfying Ref(z)z > α, Bull. Inst. Math. Acad. Sinica, 3 (1975), 65–70.

[3] P.N. CHICHRA, New Subclass of the class of close-to-convex functions, Proc. Amer. Math. Soc., 62 (1977), 37–43.

[4] S.S. DING, Y. LING AND G.J. BAO, Some properties of a class of analytic functions, J. Math.

Anal. Appl., 195 (1995), 71–81.

[5] L. MING SHENG, Properties for some subclasses of analytic functions, Bull. Inst. Math. Acad.

Sinica, 30 (2002), 9–26

[6] K. INAYAT NOOR, On subclasses of close-to-convex functions of higher order, Internat. J. Math.

and Math. Sci., 15 (1992), 279–290.

[7] K. PADMANABHAN AND R. PARVATHAM, Properties of a class of functions with bounded boundary rotation, Ann. Polon. Math., 31 (1975), 311–323.

[8] B. PINCHUCK, Functions with bounded boundary rotation, Isr. J. Math., 10 (1971), 7–16.

[9] S. PONNUSAMY, Differential subordination and Bazilevic functions, Preprint.

[10] S. OWAANDM. OBRADOVIC, Certain subclasses of Bazilevic functions of typeα, Internat. J.

Math. and Math. Sci., 9 (1986), 97–105.

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