GUO DONGAND LI ZONG-TAO

(1.Foundation Department,Chuzhou Vocational and Technical College,Chuzhou,Anhui,239000)

(2.Foundation Department,Guangzhou Civil Aviation College,Guangzhou,510403)

Communicated by Ji You-qing

1 Introduction

LetAdenote the class of functions of the form

thenfis in the class of starlike functionsS?(0)forαbe a real number and is in the class of convex functionsK(0)forα≥1.

Further,We say thatf(z)∈Aisα-starlike inUiff(z)satisfies

For suchα-starlike functions,Lewandowskiet al.[3]proved that allα-starlike functions are univalent and starlike for allα(α∈R).

In[4],it was shown that if

For 0≤α＜1 andλ≥0,we letQλ(h,α)be the subclass ofAconsisting of functionsf(z)of the form(1.1)and functionsh(z)given by

and satisfying the analytic criterion:

It is easy to see thatQλ1(h,α)?Qλ2(h,α)forλ1＞λ2≥0.Thus,forλ≥1,0≤α＜1,Qλ(h,α)?Q1(h,α)={f,h∈A:Re(f?h)′(z)＞α,0≤α＜1}and henceQλ(h,α)is univalent class(see[5]–[7]).

It is well known that every functionf∈Shas an inversef?1,defined by

A functionf∈Ais said to be bi-univalent inUif bothf(z)andf?1(z)are univalent inU.

The object of the present paper is to introduce several subclasses of the function classΣand find estimates on the coefficients|a2|and|a3|for functions in these new subclasses of the function classΣemploying the techniques used earlier by Penget al.[16]

2 Coefficient Estimates

In the sequel,it is assumed thatφis an analytic function with positive real part in the unit diskU,satisfyingφ(0)=1,φ′(0)＞0,andφ(U)is symmetric with respect to the real axis.Such a function has a Taylor series of the form

Suppose thatu(z)andv(z)are analytic in the unit diskUwithu(0)=v(0)=0,|u(z)|＜1,|v(z)|＜1,and

It is well known that(see[18],P.172)

By a simple calculation,we have

Definition 2.1A function f∈Σ given by(1.1)is said to be in the class MΣ(h,α,φ),α≥0,if the following conditions are satisfied:

where the function h(z)is given by(1.3)and(f?h)?1(ω)is defined by:

Theorem 2.1Let f given by(1.1)be in the class MΣ(h,α,φ),α≥0.Then

Proof.Letf∈MΣ(h,α,φ),α≥0.Then there are analytic functionsu,v:U?→Ugiven by(2.2)such that

Now,equating the coefficients in(2.9)and(2.10),we get

From(2.11)and(2.13)we get

Adding(2.12)and(2.13),we have

Substituting(2.15)and(2.16)into(2.17),we get

Substituting(2.15)and(2.18)into(2.16),we get

Then,in view of(2.3),we have

From(2.11)and(2.20)we get

Next,from(2.12)and(2.14)we have

Then,in view of(2.3),we have

Notice that

This completes the proof of Theorem 2.1.

Example 2.1(1)For

this operator contains in turn many interesting operator(see[19]).Theorem 2.1 becomes

Definition 2.2A function f∈Σ given by(1.1)is said to be in the class BΣ(h,λ,φ),λ≥0,if the following conditions are satisfied:

where the function h(z)is given by(1.3)and(f?h)?1(ω)is given by(2.6).

Theorem 2.2Let f given by(1.1)be in the class BΣ(h,λ,φ),λ≥0.Then

Proof.Letf(z)∈BΣ(h,λ,φ),λ≥0.Then there are analytic functionsu,v:U?→Ugiven by(2.2)such that

it follows from(2.4),(2.5),(2.26)and(2.27)that

From(2.28)and(2.30)we get

By adding(2.29)to(2.31),we have

Substituting(2.32)and(2.33)into(2.34),we get

Substituting(2.32)and(2.35)into(2.33),we get

Then,in view of(2.3)and(2.32),we have

From(2.28)and(2.37)we get

By subtracting(2.29)from(2.31)and a computation using(2.32)finally leads to

Then,in view of(2.3)and(2.32),we have

It follows from(2.28)that

Notice that(2.38),we have

This completes the proof of Theorem 2.2.

The bounds on|a2|and|a3|given in(2.40)and(2.41)are more accurate than that given by Theorem 1 in[17].

The bounds on|a2|and|a3|given in(2.42)and(2.43)are more accurate than that given by Theorem 2 in[17].

Definition 2.3A function f∈Σ given by(1.1)is said to be in the class CΣ(h,λ,φ),λ≥0,if the following conditions are satisfied:

where the function h(z)is given by(1.3)and(f?h)?1(ω)is given by(2.6).

By applying the method of the proof of Theorem 2.2,we can prove the following result.

Theorem 2.3Let f given by(1.1)be in the class CΣ(h,λ,φ),λ≥0.Then

Definition 2.4A function f∈Σ given by(1.1)is said to be in the class LΣ(h,α,φ),α≥0,if the following conditions are satisfied:

By applying the method of the proof of Theorem 2.1,we can prove the following result.

Theorem 2.4Let f given by(1.1)be in the class LΣ(h,α,φ),α≥0.Then

Definition 2.5A function f∈Σ given by(1.1)is said to be in the class STΣ(h,α,φ),α≥0,if the following conditions are satisfied:

By applying the method of the proof of Theorem 2.1,we can prove the following result.

Theorem 2.5Let f given by(1.1)be in the class STΣ(h,α,φ),α≥0.Then

Definition 2.6A function f∈Σ given by(1.1)is said to be in the class BΣ(h,λ,k),λ≥0,0＜k≤1,if the following conditions are satisfied:

where the function h(z)is given by(1.3)and(f?h)?1(ω)is given by(2.6).

Theorem 2.6Let f given by(1.1)be in the class BΣ(h,λ,k),λ≥0,0＜k≤1.Then

Proof.Letf(z)∈BΣ(h,λ,k),λ≥0,0＜k≤1.Then there are analytic functionsu,v:U?→Ugiven by(2.2)such that

Now,equating the coefficients in(2.44)and(2.45),we get

From(2.46)and(2.48)we get

By adding(2.47)to(2.49),we have

From(2.3)and(2.51)we have

Subtracting(2.47)from(2.49)we have

Then,in view of(2.3)and(2.53),we have

It follows from(2.46)that

Notice that(2.52),we have

This completes the proof of Theorem 2.6.

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