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volume 7, issue 2, article 49, 2006.

Received 28 August, 2005;

accepted 21 October, 2005.

Communicated by:Th.M. Rassias

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Journal of Inequalities in Pure and Applied Mathematics

ON CERTAIN CLASSES OF ANALYTIC FUNCTIONS

KHALIDA INAYAT NOOR

Mathematics Department

COMSATS Institute of Information Techonolgy Islamabad, Pakistan

EMail:khalidanoor@hotmail.com

c

2000Victoria University ISSN (electronic): 1443-5756 054-06

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On Certain Classes of Analytic Functions

Khalida Inayat Noor

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J. Ineq. Pure and Appl. Math. 7(2) Art. 49, 2006

http://jipam.vu.edu.au

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

n=2anzⁿ which are analytic in the unit diskE.We introduce the classB_k(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.

2000 Mathematics Subject Classification:30C45, 30C50.

Key words: Analytic functions, Univalent, Functions with positive real part, Convex functions, Convolution, Integral operator.

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

1. Introduction

LetAdenote the class of functions

f :f(z) = z+

∞

X

n=2

a_nzⁿ

which are analytic in the unit diskE ={z :|z|<1}and letS⊂ Abe the class of functions univalent inE.

LetP_k(ρ)be the class of functionsp(z)analytic inE satisfying the proper- tiesp(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 class Pk defined and studied in [8], and for ρ = 0, k = 2,we have the well known class P of functions with positive real part. The case k = 2 gives the classP(ρ) of functions with positive real part greater thanρ.

From (1.1) we can easily deduce thatp ∈ P_k(ρ)if, and only if, there exist p₁, p₂ ∈P(ρ)such that, forE,

(1.2) p(z) =

k 4 + 1

2

p₁(z)− k

4 − 1 2

p₂(z).

Letf andgbe analytic inEwithf(z) = P∞

m=0amz^mandg(z) = P∞

m=0bmz^m

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inE.Then the convolution?(or Hadamard Product) off andg is defined by (f ? g)(z) =

∞

X

m=0

a_mb_mz^m, m∈N0 ={0,1,2, . . .}.

Definition 1.1. Letf ∈ A.Thenf ∈B_k(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, where α > 0, λ > 0, k ≥ 2and 0 ≤ ρ < 1. The powers are understood as principal values.

Fork = 2 and with different choices ofλ, α andρ,these classes have been studied in [2, 3, 4,10]. In particular B₂(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 inE withp(0) = 1 and ifλis a complex number satisfyingReλ≥0, (λ6= 0),then

Re[p(z) +λzp⁰(z)]> β (0≤β <1) implies

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

γ =γ_Re_λ = Z 1

0

(1 +t^Re^λ)⁻¹dt.

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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.

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2. Main Results

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

f(z) z

^α

∈ Pk(ρ1),whereρ1 is given by

(2.1) ρ₁ =ρ+ (1−ρ)(2γ−1),

and

γ = Z 1

0

1 +t^λ^α

−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).

Since f ∈ B_k(λ, α, ρ),so{p(z) + _α^λzp⁰(z)} ∈ P_k(ρ)for z ∈ 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).

Consequentlyp∈Pk(ρ1)forz ∈E,and the proof is complete.

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Corollary 2.2. Letf =zF₁⁰ andf ∈B₂(λ,1, ρ).ThenF₁ is univalent inE.

Proceeding as in Theorem2.1and using Lemma1.2, we have the following.

Theorem 2.3. Let α > 0, λ > 0, 0 ≤ ρ < 1 and let f ∈ Bk(λ, α, ρ). Then

zf⁰(z)

f(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 −→ Aas

(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,whereρ₁is 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 Theorem2.1, we obtain the required result.

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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 functionfλ,α,ρ(z)defined as

fλ,α,ρ(z) z

α

= α λ

Z 1

0

k 4 +1

2

u^α^λ⁻¹1 + (1−2ρ)uz 1−uz

− k

4 − 1 2

u^α^λ⁻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 allnand using this inequality, we prove the required result.

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Different choices ofk, λ, αandρyield several known results.

Theorem 2.6 (Covering Theorem). Letλ > 0and0< ρ <1.Letf =zF₁⁰ ∈ B₂(λ,1, ρ).IfD is the boundary of the image ofE underF₁,then every point ofDhas a distance of at least _(3+2λ−ρ)^λ+1 from the origin.

Proof. LetF₁(z)6=w₀, w₀ 6= 0.Thenf₁(z) = _w^w⁰^F¹^(z)

0+F1(z) is univalent inE since F₁is 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| ≥

λ+1 3+2λ−ρ.

Theorem 2.7. For eachα >0, B_k(λ₁, α, ρ)⊂B_k(λ₂, α, ρ)for0≤λ₂ < λ₁. Proof. Forλ₂ = 0,the proof is immediate. Letλ₂ >0and letf ∈B_k(λ₁, α, ρ).

Then there exist two functionsh₁, h₂ ∈P_k(ρ)such that, from Definition1.1and Theorem2.1,

(1−λ)

f(z) z

α

+λ₁zf⁰(z) f(z)

f(z) z

α

=h₁(z),

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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 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 Re^f^(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. LINGANDG.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.

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[10] S. OWAANDM. OBRADOVIC, Certain subclasses of Bazilevic functions of typeα, Internat. J. Math. and Math. Sci., 9 (1986), 97–105.