T.Panigrahiand R.N.Mohapatra

1Department of Mathematics,School of Applied Sciences,KIIT,Bhubaneswar-751024,Odisha,India

2Department of Mathematics,University of Central Florida,Orlando,FL 32816,USA

Abstract.In the present paper,the authors introduce a new subclass of p-valent analytic functions with complex order defined on the open unit disk U={z:z∈C and|z|<1}and obtain coefficient inequalities for the functions in these class.Application of these results for the functions defined by the convolution are also obtained.

Key Words:p-valent function,subordination,coefficient inequalities,convolution.

1 Introduction and definition

Let Ap(p∈N:={1,2,3,···})be the class of functions f(z)of the form

that are regular and p-valent in the open unit disk

In particular,for n=1,we write A1=A.

For the functions f(z)given by(1.1)and g(z)given by

their convolution(or Hadamard product)denoted by f?g,is defined by

For two analytic functions f and g,the function f is subordinate to g,written as f(z)?g(z)(z∈U),if there exists a Schwarz function w,which(by definition)is analytic in U with w(0)=0 and|w(z)|<1 such thatf(z)=g(w(z))(z∈U).It follows from this definition that

In particular,if the function g is univalent in U,then we have the following equivalence relation(see[9]).

Let φ(z)be analytic function in U with φ(0)=1,φ0(0)>0 and<{φ(z)}>0 which maps the open unit disk U onto a region starlike with respect to 1 and is symmetric with respect to the real axis.In[1]Ali et al.defined and introduced the classto be the class of function in f∈Apfor which

and the corresponding class Cb,p(φ)of all functions in f∈Apfor which

Further,they also defined and studied the following classes:

and

Further,Ramachandran et al.[5]introduced the class Rp,b,α,β(φ)to be the class of function in f∈Apfor which

Motivated by the aforementioned works,in this paper we introduce certain subclass of pvalent functions and obtain the sharp coefficient bounds for the functions in these class.Application of these results for the functions defined by the convolution are also obtained.

Definition 1.1.Let φ(z)be a univalent starlike function with respect to 1 which maps the open unit disk U onto a region in the right half plane and is symmetric with respect to the real axis,φ(0)=1 and φ0(0)>0.A function f∈Apis said to be in the classif

where 0≤α,λ≤1;0<β,γ≤1 and b is a non-zero complex number.

Note that,by specializing the parameters p,b,α,β,γ and λ,the classreduces to the following classes studied by various earlier researchers.

?defined and studied by Ravichandran et al.[7].

?introduced and studiedby Panigrahiand Murugusun daramoor-

?introduc edandstudied by Maand Minda[3].

2 Preliminaries

Let ? be the class of analytic functions of the form

in the unit disk U satisfying the condition|w(z)|<1.

In order to derive our main results,we need the following:

Lemma 2.1(see[1]).If w∈?,then

When t1,equality holds if and only if w(z)=z or one of its rotations.If?1

or one of its rotations.

Also the sharp upper bound above can be improved as follows.

When?1

and

Lemma 2.2(see[2]).If w∈?,then for any complex number t,

The result is sharp for the functions w(z)=z2or w(z)=z.

3 Coefficient bounds

Theorem 3.1.Let φ(z)=1+B1z+B2z2+···,where Bn’s are real with B1>0 and B2≥0.If f(z)given by(1.1)belongs to the class(φ)then for any complex number μ,

where

The result is sharp.

A computation shows that

Therefore,we have

and

In view of(3.4)and(3.5),we have

Since

therefore,we have

Using(3.6)and(3.7)in(3.3)and then comparing the coefficients of z and z2on both sides,we get

and

which on simplification give

and

where

For any complex numberμ,we have

where

An application of Lemma 2.2 to(3.11)gives

where σ1is defined in(3.2).This prove the inequality(3.1).

The result is sharp i-e

The proof of the Theorem 3.1 is completed.

Putting b=β=γ=1 and α=λ=0 in Theorem 3.1 we get the following result.

Corollary 3.1(see[1]).Let φ(z)=1+B1z+B2z2+···where Bn’s are real with B1>0 and B2≥0.If f(z)given by(1.1)belongs to the class(φ),then for any complex numberμ,

The result is sharp.

Taking b=β=γ=1 and α=0 in Theorem 3.1 we obtain the result for the class(λ,φ)as follows:

Corollary 3.2(see[1]).Let φ(z)=1+B1z+B2z2+···.If the function f(z)given by(1.1)belongs to the class(λ,φ),then for any complex numberμ,we have

The result is sharp.

Letting b=λ=γ=1 and β=1?α in Theorem 3.1 we obtain the result for the class Mp(α,φ)as follows:

Corollary 3.3(see[1]).Let φ(z)=1+B1z+B2z2+···.If f∈Mp(α,φ),then for any complex numberμ,we have

where

The result is sharp.

Taking b=1 and consideringμas a real number,we prove the following results for the function f in the class(φ).We denote the class

Theorem 3.2.Let φ(z)=1+B1z+B2z2+···,where Bn’s are real with B1>0 and B2≥ 0.Let 0≤α,λ≤1,0<β,γ≤1 and

where

If f(z)given by(1.1)belongs to(φ),then

Further,if δ1≤μ≤δ3,then

and if δ3≤μ≤δ2,then

where

These results are sharp.

where

where s1,s2,s3,s4and δ are defined in the statement of Theorem 3.2.

By application of Lemma 2.1 to the right hand side of relation(3.15)gives the following cases:

Case-1:Ifμ≤δ1,then

After simplification,we get v≤?1.Therefore,

Case-II:If δ1≤μ≤δ2,then

which on simplification reduces to

which implies

Case-III:Ifμ≥δ2,then

which implies

and

Thus,the results of(3.12)is established.

Case-IV:Furthermore,if δ1≤μ≤δ3,then

which implies that?1

Case-V:If δ3≤μ≤δ2then

which implies 0

Now

Sharpness:To prove that the bounds are sharp,we define the function Sφn(n=2,3,···),Tλand Uλ(0≤λ≤1)as follows.

Clearly,the functions Sφn,Tλ,Uλ∈(φ).Ifμ<δ1or μ>δ2,then the equality holds if and only if f is Sφ2or one of its rotations.When δ1<μ<δ2,then the equality holds if and only if f is Sφ3or one of its rotations.Ifμ=δ1,then the equality holds if and only if f is Tλor one of its rotations.Furthermore,ifμ=δ2,the equality holds if and only if f is Uλor one of its rotations.

Taking β=γ=1 and α=λ=0 in Theorem 3.2 we get the following:

Corollary 3.4(see[1]).Let φ(z)=1+B1z+B2z2+···and

If f(z)given by(1.1)belongs to,then

Further,if δ1≤μ≤δ3,then

If δ3≤ μ ≤ δ2,then

These results are sharp.

Putting β=γ=1 and α=0 in Theorem 3.2,we obtain the result for the class(λ,φ).

Corollary 3.5.Let φ(z)=1+B1z+B2z2+···and let

where

If f(z)given by(1.1)belongs to(λ,φ),then

Further if δ1≤ μ ≤ δ2,then

and if δ3≤μ≤δ2,then

where

These results are sharp.

Taking λ=γ=1 and β=1?α in Theorem 3.2 we get:

Corollary 3.6.Let φ(z)=1+B1z+B2z2+···and let

where

If f(z)given by(1.1)belongs to Mp(α,φ),then

Further if δ1≤μ≤δ2,then

and if δ3≤ μ ≤ δ2,then

where

These results are sharp.

4 Applications to functions defined by convolution

we get the following results.

where σ1is given by(3.2).The result is sharp.

Putting α= λ =0 and β=γ=1 in Theorem 4.1 we obtain the result due to Ali et al.[1,Theorem 2].

Taking b=1 andμto be a real number,we obtain the following result for the classWe denote the class

Theorem 4.2.Let φ(z)=1+B1z+B2z2+···.Let 0≤α,λ≤1,0<β,γ≤1 and

where

If f(z)given by(1.1)belongs to(φ),then

Further,if δ1≤μ≤δ3,then

and if δ3≤μ≤δ2,then

where

These results are sharp.

Taking α=λ=0 and β=γ=1 in Theorem 4.2 we get the result of Ali et al.(Corollary 1,[1]).

Acknowledgements

The authors thank the reviewer for many useful suggestions for revision which improved the content of the manuscript.Further,the present investigation of the first-named author is supported by CSIR research project scheme No.25(0278)/17/EMR-II,New Delhi,India.