Rayaprolu Bharavi Sharma and Kalikota Rajya Laxmi

1 Department of Mathematics,Kakatiya University,Warangal,Telangana State,India 506009

2 SR International Institute of Technology,Hyderabad,Telangana State,India,501301

Abstract.In this paper,we have investigated second Hankel determinants and Fekete-Szeg? inequalities for some subclasses of Bi-univalent functions with respect to symmetric and Conjugate points which are subordinate to a shell shaped region in the open unit disc ?.

Key Words:Analytic functions,univalent functions,Bi-univalent functions,second Hankel determinants,Fekete-Szeg? inequalities,symmetric points,conjugate points.

1 Introduction

Let A be the class of all functions of the form

which are analytic in the open unit disc ?={z:|z|<1}.Let S be the class of all functions in A which are univalent in ?.

Let P denote the family of functions p(z)which are analytic in ?such that p(0)=1,andof the form

For two functions f andg,analytic in ?,we say that the function f is subordinate to g in ?and we write it asif there exists a Schwartz function ω,which is analytic in?withsuch that

In 1959,Sakaguchi[26]defined a subclassof S which satisfies following condition

In terms of subordination following Ma and Minda,Ravichandran[25]defined the classesand Cs(φ)as below.

A function f ∈A is in the classif

And in the class Cs(φ)if

In terms of subordination,Goel and Mehrol[8]in 1982 generalized above classes and they are denoted by

Let Cs(A,B)be the class of functions of the form(1.1)and satisfying the condition

A function f ∈S is said to be bi-univalent in ?if both f andare univalent in ?.

However the Koebe function is not a member of ∑because it maps the unit disc univalently onto the entire complex plane minus a slit along ?1/4 to ?∞. Hence the image domain does not contain the unit disc.Other examples of univalent function that are not in the classare

In 1967,Lewin[15]first introduced the classof bi-univalent functions and showed that|a2|≤1.51 for everySubsequently,in 1967,Branan and Clunie[2]conjectured thatfor bi-Star like functions and|a2|≤1 for bi-Convex functions.Only the last estimate is sharp;equality occurs only foror its rotation.

In 1985 Tan[?]obtained that|a2|<1.485,which is the best known estimate for biunivalent functions. Since then various subclasses of the bi-univalent function ofwere introduced and non-sharp estimates on the first two coefficients|a2|and|a3|in the Taylor-Maclaurin’s series expansion were found in several investigations.The coefficient estimate problem for each of|an|(n∈N{2,3})is still an open problem.So many authors have studied coefficient estimates of analytic starlike functions and convex functions with respect to symmetric points and starlike functions with respect to conjugate points[12,14,27].

In 1976,Noonan and Thomas[17]defined qthHankel determinant of f for q ≥1 and n≥1,which is stated by

The Hankel determinant plays an important role in the study of singularities;for instance,see[4]and Edrei[5].Hankel determinant plays an important role in the study of power series with integral coefficients. In 1966,Pommerenke[19]investigated the Hankel determinant of areally mean p-valent functions,univalent functions as well as of starlike functions,and in 1967[20]he proved that the Hankel determinants of univalent functions satisfy

where β>1/4000 and K depends only on q.

Later Hayman[11]proved thatA anabsolute constant)for areally mean univalent functions. One can easily observe that the Fekete-Szeg? functionalFekete-Szeg?[7]gave a sharp estimate offorμreal.It is a combination of the two coefficients which describes the area problems posed earlier by Gronwall[10]in 1914-1915.Recently S.K.Lee et al.[13]obtained the second Hankel determinantfor functions belonging to the subclasses of Ma-Minda starlike and convex functions.T.Ram Reddy et al.[23]obtained the Hankel determinant for starlike and convex functions with respect to symmetric points.In 2015,Second Hankel determinant for bi-univalent functions was obtained by Murugusundharmoorthy et al.[16].

2 Preliminaries

Many authors have established the second Hankel determinant for analytic functions.Motivated by the aforementioned work,in the present paper we introduced three subclasses of bi-univalent functions namely bi-starlike with respect to symmetric points,biconvex functions with respect to symmetric points and bi-starlike functions with respect to conjugate points which are subordinate to a shell shaped region,and obtain the second Hankel determinant and Fekete-Szeg? inequalities for functions in these classes.

Definition 2.1.A function f ∈Σ is said to be in the classif it satisfies the following conditions

where g is the extension ofto ?.

Definition 2.2.A function f ∈Σ is said to be in the class ΣCsif it satisfies the following conditions

Definition 2.3.A functionis said to be in the classif it satisfies the following conditions

where g is the extension ofto ?.

To prove our results we require the following Lemmas.

Lemma 2.1(see[18]).Let the function be given by the following series

Then the sharp estimate is given by|pn|≤2,(n∈N).

Lemma 2.2(see[9]).If the function is given by the series

then

for some x,z with|x|≤1 and|z|≤1.

Another result that will be required is the optimal value of quadratic expression.Standard computations show that

3 Main results

Theorem 3.1.Let the function given by(1.1)be in the classThen

Proof.Since

then there exists two Schwarz functions u(z),v(w)with and|u(z)|≤1,|v(w)|≤1,such that

(3.1)and

Define two functions p(z),q(w)such that

Then Eqs.(3.1)and(3.2)becomes

Now equating the coefficients in(3.3a)and(3.3b),we have

And

Now from(3.4a)and(3.5a),we get that

And

Now from(3.4b)and(3.5b),we get that

Also from(3.4c)and(3.5c),we get that

Thus we can easily obtain that

According to Lemma 2.1,we get that

Since p∈P,so|p1|≤2.Letting p1=p,we may assume without any restriction that p∈[0,2].

Now we need to maximize F(γ1,γ2) in the closed square S :={(γ1,γ2):0≤γ1≤1,0≤γ2≤1}for p ∈[0,2].We must investigate the maximum of F(γ1,γ2)according to p ∈(0,2), p=0 and p=2 taking into account the sign of

First,let p∈(0,2).Since T3<0 and T3+2T4>0 for p∈(0,2),we conclude that

Thus the function F cannot have a local maximum in the interior of the square S.Now we investigate the maximum of F on the boundary of the square S.

For γ1=0 and 0≤γ2≤1(Similarly γ2=0 and 0≤γ1≤1),we obtain

Since T3+T4≥0 and 0≤γ2≤1 and for any fixed p with 0

that is G(γ2)is an increasing function.Hence for fixed p∈(0,2),the maximum of G(γ2)occurs at γ2=1 and

For γ1=1 and 0≤γ2≤1(similarly γ2=1 and 0≤γ1≤1),we obtain

Then

Since G(1)≤H(1)for p∈(0,2),maxF(γ1,γ2)=F(1,1)on the boundary of the square S.

Thus the maximum of F occurs at γ1=1 and γ2=1 on the boundary of the closed square S.Let K:(0,2)→R,

Substituting the values of T1,T2,T3and T4in the function K,then

Then

where t=p2.Then by using standard result of solving quadratic equation,

Thus,we complete the proof.

Theorem 3.2.Let the function given by(1.1)be in the classandThen

Proof.Subtracting(3.5b)from(3.4b)and applying(3.6),we get

Now summing(3.5b)and(3.4b)leads to

This equality and(3.4a),(3.5a)result in

From(3.11)and(3.12),it follows that

Then

This completes the proof.

Corollary 3.1.Let the function given by(1.1)be in the classThen

Corollary 3.2.Let the function given by(1.1)be in the class.Then

Theorem 3.3.Let the function given by(1.1)be in the classThen

Proof of this theorem is similar to that of above Theorem 3.1 and hence the details are omitted here.

Theorem 3.4.Let the function given by(1.1)be in the class ΣCsandThen

Proof of this theorem is similar to that of above Theorem 3.2.

Corollary 3.3.Let the function given by(1.1)be in the class ΣCs.Then

Corollary 3.4.Let the function given by(1.1)is in the class ΣCs.Then

Theorem 3.5.Let the function given by(1.1)be in the classThen

Proof.Since

then there exists two Schwarz’s functions u(z),v(w)with u(0)=0,v(0)=0 and|u(z)|≤1,|v(w)|≤1 such that

(3.19)And

Define two functions p(z),q(w)such that

Then the Eqs.(3.13)and(3.14)becomes

Now equating the coefficients in(3.15a)and(3.15b),we have

And

Now from(3.16a)and(3.17a)we get that

And

Now from(3.16b)and(3.17b),we get that

Also from(3.16c)and(3.17c),we get that

Thus we can easily obtain that

According to Lemma 2.1,and(3.10a)and(3.10b),the above equation becomes

so|p1|≤2.Letting p1=p,we may assume without any restriction that p∈[0,2].Thus for γ1=|x|≤1 and γ2=|y|≤1,we obtain

Now we need to maximize F(γ1,γ2) in the closed square S :={(γ1,γ2):0≤γ1≤1,0≤γ2≤1}for p ∈[0,2]. We must investigate the maximum of F(γ1,γ2)according to p∈(0,2),p=0 and p=2 taking into account the sign of

First let p∈(0,2).Since T3<0 and T3+2T4>0 for p∈(0,2),we conclude that Fγ1γ1Fγ2γ2?(Fγ1γ2)2<0.Thus the function Fcannot have a local maximum in the interior of the square S.Now we investigate the maximum of Fon the boundary of the square S.For γ1=0and 0≤γ2≤1(Similarly γ2=0and 0≤γ1≤1),we obtain

Case 1:If T3+T4≥0:In this case 0≤γ2≤1 and for any fixed p with 0

that is 0

Case 2:If T3+T4<0:since 2(T3+T4)γ2+T2≥0 for 0 ≤γ2≤1 and for any fixed p with 0

and so G'(γ2)>0.Hence for fixed p ∈(0,2)the maximum of G(γ2)occurs at γ2=1.By considering above two cases,for 0≤γ2≤1 and any fixed p with 0

For γ1=1 and 0≤γ2≤1(similarly γ2=1 and 0≤γ1≤1),we obtain

Similar to the above case we get that

Since G(1)≤H(1)for p∈(0,2),maxF(γ1,γ2)=F(1,1)on the boundary of the square S.Thus the maximum of F occurs at γ1=1 and γ2=1in the closed squareS.

Let K:(0,2)→R

Substituting the values of T1,T2,T3and T4in the function K,then

Then by using standard result of solving quadratic equation,

Thus,we complete the proof.

Theorem 3.6.Let the function f given by(1.1)be in the classandThen

Proof.Subtracting(3.17b)from(3.16b)and applying(3.18),we get

Now summing(3.17b)and(3.16b)leads to

This equality and(3.23),(3.26)result in

From(3.24)and(3.25)it follows that

This completes the proof.

Corollary 3.5.Let the function f given by(1.1)be in the classThen

Corollary 3.6.Let the function f given by(1.1)be in the classThen