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,and of the form

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For two functions f andg,analytic in ∆,we say that the function f is subordinate to g in ∆and we write it as if there exists a Schwartz function ω,which is analytic in∆with such that

Indeed,it is known that and f(∆)⊂g(∆).

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

The functions in the class are starlike with respect to symmetric points.Further Sakaguchi has shown that the functions in are close-to-convex and hence are univalent.The concept of starlike functions with respect to symmetric points have been extended to starlike functions with respect to N-symmetric points by Ratanchand[24]and Prithvipal Singh[21],Ram Reddy[22]studied the class of closeto-convex functions with respect to N-symmetric points and proved that the class is closed under convolution with convex univalent functions.Das and Singh[3]introduced another class Cs namely convex functions with respect to symmetric points and satisfying the condition

From the definition of and Cs it is evident that f ∈Cs if and only if Ashwah and Thomas in[6]introduced another class namely the class consisting of functions starlike with respect to conjugate points.

见我对背诵如此深恶痛绝，同学们有些不解：“你的记忆力不是很好吗？怕啥？”其实，我可不是怕背诵，我是讨厌被强迫。我喜欢的文字自然是能记得住的，但别人强迫我记住的文字，我可能便无法再喜欢了。后来成了作家，我最担心的就是自己的文章被选入中小学课本，那该多招孩子们的痛恨啊。即使不用背诵我的文章，但对于相关问题的标准答案也仍是得牢记的呀。

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Let be the subclass of S consisting of functions given by(1.1)and satisfying the 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

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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 be the class of functions of the form(1.1)and satisfying the condition

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

Also let be the class of functions of the form(1.1)and satisfying the condition

In view of Koebe’stheorem,every function f ∈S has an inversedefined by f −1(f(z))=z,(z∈∆)andIn fact the inverse function is given by

A function f ∈S is said to be bi-univalent in ∆if both f and are univalent in ∆.

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Let denote the class of all bi-univalent functions defined in the unit disc ∆. We notice that is non empty.One of the best examples of bi-univalent functions is f(z)=which maps the unit disc univalent onto the strip|Imw|<π/2,which in turn contains the unit disc.Other examples are

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 class are

In 1967,Lewin[15]first introduced the classof bi-univalent functions and showed that|a2|≤1.51 for every Subsequently,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 of were 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].

Definition 1.1.The function maps the unit disc onto a shell shaped region in the right half plane,which is analytic and univalent on C{i,−i}.It is symmetric with respect to real axis.It is a function with positive real part with

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

Thus the function f ∈S∗(q)if l ies in a shell shaped region.Since this region is contained in right half plane,functions subordinate to q(z)be starlike,and in particular univalent.Here super ordinate functionmaps ∆onto a shell shaped region.

Let C(q) be the class of functions defined by C(q) =Solkol [28,29] have studied and obtained some coefficient inequalities for the class S∗(q).

Let S∗(q)be the class of functions defined by

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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 that A anabsolute constant)for areally mean univalent functions. One can easily observe that the Fekete-Szegö functionalFekete-Szegö[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 determinant for 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-Szegö inequalities for functions in these classes.

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

where g is the extension of to ∆.

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

where g is the extension of to ∆.

Definition 2.3.A function is said to be in the class if it satisfies the following conditions

where g is the extension of to ∆.

To prove our results we require the following Lemmas.

这个定理指出，闭合电流回路周围的磁场可以看成是由许多微小电流元(很短的直导线电流元)单独存在时产生的磁场经过矢量叠加产生的。式(1)中的μ0为真空磁导率，而I0dl表示的是为电流强度为I0dl、长度为dl的电流微元，其矢量的方向代表电流的流向，r为电流微元到空间某一个场点P的位置矢量。

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.

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

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(3.1)and

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

Then Eqs.(3.1)and(3.2)becomes

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

And

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

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And

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

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

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

According to Lemma 2.1,we get that

for some x,y,z and w withand|w|≤1.Using(3.10a)and(3.10b)

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

Thus for and 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

Thus we can easily obtain 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

Then

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

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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,T3 and T4 in the function K,then

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

式中：Rp表示节域物元；Cn(apn,bpn)表示节域物元关于特征值Cn的量值范围；p表示草原生态安全的全体等级。

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 class andThen

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

Where

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 class Then

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 ΣCs andThen

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 class Then

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

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,T3 and T4 in 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 class andThen

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

Where Then

This completes the proof.

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

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

References

[1]Ajab Bai Akbarally and Nurul Atikah Mohd Isa,On new subclasses of analytic functions with respect to conjugate and symmetric conjugate points,Global J.Pure Appl.Math.,12(3)(2016),2849–2865.

[2]D.A.Brannan and J.G.Clunie,Aspects of Contemporary Complex Analysis,Proceedings of the NATO Advanced Study Institute(University of Durham,Durham;July(120,1979),Academic Press,New York and London,1980.

[3]R.N Das and P.Singh,On subclasses of Schlicht mapping,Indian J.Pure Appl.Math.,8(1977),864–872.

[4]P.Dienes,The Taylor Series,Dover,New York,1957.

[5]A.Edrei,Sur les determinants recurrent set les singularities dune function done por son development de Taylor,Compos.Math.,7(1940),20–88.

[6]R.M.El-Ashwah and D.K.Thomas,Some subclasses of close to-convex functions,Journal of Ramanujan Mathematical Society,2(1987),86–100.

[7]M.Feketo and G.Szegö,Eine Bemerkung uber ungerade Schlichte function,J.Landon Math Soc.,8(1933),85–89.

[8]R.M.Goel,B.C.Mehrok,A subclass of starlike functions with respect to symmetric points,Tamkang J.Math.,13(1)(1982),11–24.

[9]U.Grenander and G.Szegö,Toeplitz Forms and Their Applications,University of California Press,Berkeley,(1958).

[10]T.Gronwall,Some remarks on conformal representation,Aenn.Off.Math.,16(1914-15),72–76.

[11]W.K.Hayman,On the second Hankel determinant of mean univalent functions.Proc.Lond.Math.Soc.,3(1968),77–94.

[12]A.Janteng and S.A.F.M.Dahhar,A subclass of starlike functions with respect to conjugate points,Int.Math.Forum,4(2009),1373–1377.

[13]S.K.Lee,V.Ravichandran and S.Supramaniam,Bounds for the second Hankel determinant of certain univalent functions,J.Ineq.Appl.,(2013),281.

[14]M.Lewin,On a Coefficient problem for Bi-univalent functions,Proc.Amer.Math.Soc.,18(1967),63–68.

[15]G.Murugusundharamoorthy and K.Vijaya,Second Hankel determinant for bi-univalent analytic functions associated with Hohlov operator,Int.J.Anal.Appl.,8(1)(2015),22–29.

[16]J.W.Noonan and D.K.Thomas,On the second Hankel determinant of areally mean p-valent functions,Trans.Amer.Math.Soc.,223(2)(1976),337–346.

[17]Ch.Pommerenke,Univalent Functions,Vandenhoeck and Rupercht,Gotingen,1975.

[18]Ch.Pommerenke,On the coefficients and Hankel determinants of univalent functions,J.Lond.Math.Soc.,41(1966),111–122.

[19]Ch.Pommerenke,On the Hankel determinants of univalent functions,Mathematika,14(1967),108–112.

[20]Prithvipal Singh,A Study of Some Subclasses of Analytic Functions in the Unit Disc,Ph.D.Thesis,(1979),I.I.T.Kanpur.

[21]T.Ram Reddy,A Study of Certain Subclasses of Univalent Analytic Functions,Ph.D.Thesis,(1983),I.I.T.Kanpur.

[22]T.Ram Reddy and D.Vamshee Krishna,Hankel determinant for starlike and convex functions with respect to symmetric points,J.Indian Math.Soc.,79(2012),161–171.

[23]Ratanchand,Some Aspects of Functions Analytic in the Unit Disc,Ph.D.Thesis,(1978),I.I.T.Kanpur.

[24]V.Ravichandran,Starlike and convex functions with respect to conjugate points,Acta Mathematica:Academiae Paedagogicae Nyregyhaziensis,20(1)(2004),31–37.

[25]K.Sakaguchi,On certain univalent mappings,J.Math.Soc.Japan,11(1959),72–75.

[26]C.Selvaraj and N.Vasanthi,Subclasses of analytic functions with respect to symmetric and conjugate points,Tamkang J.Math.,42(2011),87–94.

[27]J.Sokol,On starlike functions connected with Fibonacci numbers,Folia Scient.Univ.Tech.Resoviensis,175(1999),111–116.

[28]J.Sokol,A certain class of starlike functions,Comput.Math.Appl.,62(2)(2011),611–619.

[29]D.L.Tan,Coefficient estimates for bi-univalent functions,Chinese Ann.Math.Ser.,A5(5)(1984),559–568.