FEKETE-SZEG PROBLEMS FOR SEVERAL QUASI-SUBORDINATION SUBCLASSES OF ANALYTIC AND BI-UNIVALENT FUNCTIONS ASSOCIATED WITH THE DZIOK-SRIVASTAVA OPERATOR
2020-08-13LONGPinhongLIXingWANGWenshuai
LONG Pin-hong,LI Xing,WANG Wen-shuai
(School of Mathematics and Statistics,Ningxia University,Yinchuan 750021,China)
Abstract:In the article we introduce two quasi-subordination subclasses of the function classof analytic and bi-univalent functions associated with the Dziok-Srivastava operator,and some problems for their coefficient estimation and Fekete-Szeg functional.By using differential quasisubordination and convolution operator theory,we obtain some results about the corresponding bound estimations of the coefficient a2 and a3 as well as Fekete-Szeg functional inequalities for theses subclasses,which generalize and improve some earlier known results.
Keywords:Fekete-Szeg problem;bi-univalent function;Gaussian hypergeometric function;Dziok-Srivastava operator;quasi-subordination
1 Introduction
In the article,our aim focuses on the certain quasi-subordination subclasses of analytic and bi-univalent functions associated with the Dziok-Srivastava operator.To state our results,at first we will recall some notations and basic properties for analytic and bi-univalent functions and Dziok-Srivastava operator.
LetAbe the class of normalized analytic functionf(z)by
in the open unit disk∆={z∈C:|z|<1}.
Let the subclassSofAbe the set of all univalent functions in∆.According to the Koebe one quarter theorem[1],the inversef−1of everyf∈Ssatisfies
If both the functionf∈Aand its inversef−1are univalent in∆,then it is bi-univalent.Denote by Σ the class of all bi-univalent functionsf∈Ain∆.
For givenf,g∈A,define the Hadamard product or convolutionf∗gby
wheref(z)is given by(1.1)andAssume that the Gaussian hypergeometric functionqFs(α1,···,αq;β1,···,βs;z)is defined by
for the complex parametersαkandβjwithβj60,−1,−2,−3,...(k=1,···,q;j=1,···,s),where(‘)ndenotes the Pochhammer symbol or shifted factorial by
Dziok and Srivastava[2,3]ever introduced the convolution operatorqIs(α1,···,αq;β1,···,βs)=qIslater named by themselves as follows
where
Note that
Here we remind some reduced versions of Dziok-Srivastava operatorqIs(α1,···,αq;β1,···,βs)for suitable parametersαk(k=1,···,q)andβj(j=1,···,s);refer to the generalized Bernardi operatorJη=2I1(1,1+η;2+η)(<(η)>−1)[4];Carlson-Shaffer operatorL(a,c)=2I1(a,1;c)[5];Choi-Saigo-Srivastava operatorIλ,µ=2I1(µ,1;λ+1)(λ>−1,µ≥0)[6];Hohlov operator;Noor integral operatorIn=2I1(2,1;n+1)[9];Owa-Srivastava fractional differential operator;Ruscheweyh derivative operatorDδ=2I1(1+δ,1;1)[12].
In 1967,Lewin[13]introduced the analytic and bi-univalent function and proved that|a2|<1.51.Moreover,Brannan and Clunie[14]conjectured thatand Netanyahu[15]obtained thatLater,Styer and Wright[17]showed that there exists functionf(z)so thatHowever,so far the upper bound estimate|a2|<1.485 of coefficient for functions inby Tan[18]is best.Unfortunately,as for the coefficient estimate problem for every Taylor-Maclaurin coefficient|an|(n∈N{1,2})it is probably still an open problem.Based on the works of Brannan and Taha[19]and Srivastava et al.[20],many subclasses of analytic and bi-univalent functions classwere introduced and investigated,and the non-sharp estimates of first two Taylor-Maclaurin coefficients|a2|and|a3|were given;refer to Deniz[21],Frasin and Aouf[22],Hayami and Owa[23],Patil and Naik[24,25],Srivastava et al.[26,27],Tang et al.[28]and Xu et al.[29,30]for more detailed information.Recently,Srivastava et al.[31,32]gave some new subclasses of the function classof analytic and bi-univalent functions to unify the works of Deniz[21],Frasin[33],Keerthi and Raja[34],Srivastava et al.[35],Murugusundaramoorthy et al.[36]and Xu et al.[29],etc.Besides,we also refer to Goyal et al.[37]for the subclasses of analytic and bi-univalent associated with quasi-subordination.Since Fekete-Szeg[38]studied the determination of the sharp upper bounds for the subclass ofS,Fekete-Szegfunctional problem was considered in many classes of functions;refer to Abdel-Gawad[39]for class of quasi-convex functions,Koepf[40]for class of close-to-convex functions,Orhan and Rducanu[16]for class of starlike functions,Magesh and Balaji[41]for class of convex and starlike functions,Orhan et al.[42]for the classes of bi-convex and bi-starlike type functions,Panigrahi and Raina[43]for class of quasi-subordination functions,Tang et al.[28]for classes of m-mold symmetric bi-univalent functions.In addition,Murugusundaramoorthy et al.[36,44,45]and Patil and Naik[46]ever introduced and investigated several new subclasses of the function classof analytic and bi-univalent functions involving the hohlov operator.Moreover,Al-Hawary et al.[47]studied the Fekete-Szegfunctional problem for the classes of analytic functions of complex order defined by the Dziok-Srivastava operator.Motivated by the statements above,in the article we are ready to introduce and investigate two new subclasses of the function classof analytic and bi-univalent functions associated with the Dziok-Srivastava operator and quasi-subordination,and consider the corresponding bound estimates of the coefficienta2anda3as well as the corresponding Fekete-Szegfunctional inequalities.Furthermore,the consequences and connections to some earlier known results would be pointed out.
For two analytic functionsfandg,if there exist two analytic functionsϕandhwith|ϕ(z)|≤1,h(0)=0 and|h(z)|<1 forz∈∆ so thatf(z)=ϕ(z)g(h(z)),thenfis quasi-subordinate tog,i.e.,f≺quasig.Note that ifϕ≡1,thenfis subordinate togin∆,i.e.,f≺g.Further,ifh(z)=z,thenfis majorized bygin∆,i.e.f≤g.For the related work on quasi-subordination,refer to Robertson[48],and Frasin and Aouf[22].Write
First we will introduce the following general subclasses of analytic and bi-univalent functions associated with the Dziok-Srivastava operator.
Definition1.1A functionf(z)∈given by(1.1),belongs to the classif the following quasi-subordinations are satisfied
and
forz,w∈∆,whereη ≥0 and the functiongis the inverse offgiven by(1.2).
Definition 1.2A functiongiven by(1.1),belongs to the classif the following quasi-subordinations are satisfied:
and
forz,w∈∆,whereτ∈C{0},0≤µ≤1,0≤λ≤1,0≤γ≤1 and the functiongis the inverse offgiven by(1.2).
Lemma 1.3(see[1,49])LetPbe the class of all analytic functionsq(z)of the following form
satisfying
2 Coefficient Bounds for the Function Class
Denote the functionssandtinPby
Equivalently,from(2.1)we know that
and
Givenφ∈Pwithφ0(0)>0,letφ(∆)be symmetric with respect to the real axis.When the series expansion form ofφis denoted by
by(2.2)–(2.3)and(2.4)it follows that
and
In the section we study the estimates for the class.Now,we establish the next theorem.
Theorem 2.1Iff(z)given by(1.1)belongs to the class,then
and
where
ProofIfthen by Definition 1.1 and Lemma 1.3,there exist two analytic functionsu(z)andv(w)∈Pso that
and
Expanding the left half parts of(2.11)and(2.12),we obtain that
and
In addition,we know that
and
Therefore,from(2.10)–(2.15)we have that
and
From(2.16)and(2.18),it infers that
Then,we show that
and
By(2.17)and(2.19),we have that
Therefore,by(2.22)–(2.23)we obtain that
We follow from Lemma 1.3 and(2.22)–(2.24)that
and
then(2.7)holds.Similarly,from(2.17),(2.19)and(2.21),it also implies that
Hence,from(2.22)and(2.25),we obtain that
Therefore,from Lemma 1.3 it shows that
On the other hand,by(2.23)and(2.25),we infer that
Thus,from Lemma 1.3,we see that
Next,we consider Fekete-Szegproblems for the class.
Theorem 2.2Iff(z)given by(1.1)belongs to the classandδ∈R,then
ProofFrom(2.25),it follows that
By(2.24)we easily obtain that
Hence,from Lemma 1.3,we imply that
Corollary 2.3Iff(z)given by(1.1)belongs to the classandδ∈R,then
Remark 2.4Without quasi-subordination(i.e.,ϕ(z)≡1),if we choose some suitable parametersαk(k=1,···,q),βj(j=1,···,s)andη,we obtain the following reduced versions forin Theorem 2.1.
Remark 2.5Without Dziok-Srivastava operator,we can collect the following reduced versions forin Theorem 2.1.
3 Coefficient Bound Estimates for the Function Class
Now,we study the coefficients for the classand establish the next theorem.
Theorem 3.1Iff(z)given by(1.1)belongs to the class,then
for
and
and
for
and
where
and
ProofHere,we follow the method of Theorem 2.1.Ifthen by Definition 1.2 there exist two analytic functionsu(z),v(z):∆→∆withu(0)=0 andv(0)=0 such that
and
Expanding the left half parts of(3.5)and(3.6),we have that
and
Therefore,From(2.14)–(2.15)and(3.5)–(3.8),we get that
and
From(3.9)and(3.11),we know that
Then,it infers that
and
By(3.10)and(3.12),we have that
Therefore,by(3.15)–(3.16)we know that
Therefore,from(3.15)–(3.17)and Lemma 1.3,we obtain that
and
Similarly,from(3.10)and(3.12),it implies that
Hence,by(3.15)and(3.18),it follows that
So,we obtain from Lemma 1.3 that
On the other hand,by(3.16)and(3.18),we infer that
Thus,from Lemma 1.3 we see that
Next,we consider Fekete-Szegproblems for the class.
Theorem 3.2Letf(z)given by(1.1)belong to the classandδ∈R.Then
if
or
if
where Φ = Φ(µ,λ,γ,p2(q,s),p3(q,s)),Θ = Θ(µ,λ,γ)and Ψ = Ψ(µ,λ,γ)are the same as in Theorem 3.1.
ProofFrom(3.18),it follows that
