LI Shu-hai,MA Li-na,AO-en
(School of Mathematics and Statistics,Chifeng University,Chifeng 024001,China)
Convexity for New Integral Operator on k-uniformly p-valentα-convex Functions of Complex Order
LI Shu-hai,MA Li-na,AO-en
(School of Mathematics and Statistics,Chifeng University,Chifeng 024001,China)
In this paper,we obtain the convexity of new general integral operator on some classes of k-uniformly p-valentα-convex functions of complex order.These results extend some known theorems.
analytic functions;p-valently;k-uniformly;starlike;convex;integral operator
Let Apdenote the class of functions of the form
which are analytic in the open unit disk U={z∈C:|z|<1}.The class A1will be denoted by A.
Furthermore,a function f∈Cp(b,β)is p-valently convex of complex order b(b∈C−{0}) and typeβ(0≤β<p),if it satisfies the following inequality
For p=1,these classes were considered by[1-4].
A function f∈Jp(b,α;β)is p-valentlyα-convex of complex order b(b∈C−{0})and type β(0≤β<p),if it satisfies the following inequality
Similarly,forβ>p,the class Mp(b,α;β)reduce respectively to the equivalent class
Clearly,we have
and
The classes J1(1,α;β)and M1(1,α;β)were studied by Fukui[5].
For functions f and g,analytic in U,we say that f is subordinate to g in U and write
if there exists a Schwarz functionω,which(by definition)is analytic in U withω(0)=0 and |ω(z)|<1(z∈U),such that f(z)=g(ω(z))(z∈U).Indeed it is known that
Furthermore,if the function g is univalent in U,then we have the following equivalence[6, P4]:
As usual,we denote by S the subclass of A consisting offunctions which are also univalent in U.Furthermore,we denote by k−U C V and k−S T two interesting subclasses of S consisting, respectively,of functions which are k-uniformly convex and k-uniformly starlike in U.Thus, we have
and
The class k−U C V was introduced by Kanas and Wisniowska[7],where its geometric definition and connections with the conic domains were considered.The class k−S T was investigated in[8].In fact,it is related to the class k−U C V by means ofthe well-known Alexander equivalence between the usualclasses ofconvex and starlike functions(see also the work of Kanas and Srivastava[9]for further developments involving each of the classes k−U C V and k−S T).In particular,when k=1,we obtain
where U C V and S P are the familiar classes ofuniformly convex functions and parabolic starlike functions in U,respectively(see,for details,[10-13]).In fact,by making use of a certain fractionalcalculus operator,Srivastava and Mishra[14]presented a systematic and unified study of the classes U C V and S P.
For a function f∈Ap,we define the following operator
where j∈N0=N∪{0}.The differential operator Djwas introduced by Shenan et al[15].
When p=1,we get Sˇalˇagean diff erential operator(see[16]).
By using the operator Dj,we introduce the new class of p-valent analytic functions.
Defi nition 1.1 Letϕ(z)be an analytic univalent function in U withϕ(0)=1.The class U Jp,j(b,α,k;ϕ)consists of functions f(z)∈Apsatisfying
whereα≥0,k≥0,b∈C−{0}.
Letϕ(z)=ϕp,β(z):U→C be the function defined by
Whenβ<p,ϕp,β(z)is the half-plane defined by Reϕp,β(z)>,while in the caseβ> p,ϕp,β(z)is the half-plane defined by Reϕp,β(z)<
Thus forβ<p,the class U Jp,j(b,α,k;ϕp,β)reduce to the familiar class of k-uniformly p-valentα-convex functions of complex order b(b∈C−{0})and typeβ
Similarly,forβ>p,the class U Jp,j(b,α,k;ϕp,β)reduce to the equivalent class
Setting
Clearly,we have
For suitable choices of the functionϕand parametersα,j,k and p involved in the class U Jp,j(b,α,k;ϕ),we also obtain the following subclasses which were studied in many earlier works
(1)U Jp,j(b,0,0;β)(0≤β<p)and U Jp,j(b,1,0;β)(0≤β<p)(see[17]).
(2)U Jp,0(1,0,k;β)(0<k<1,−k p≤β<p)(see[18-19]).
(3)U M1(b,0,k;β)(β>1)and U M1(b,1,k;β)(β>1)(see[20]).
(4)U J1,0(1,0,k;ϕ)and U J1,0(1,1,k;ϕ)(see[21]).
(5)U Jp,0(1,0,0;β)(0≤β<p)and U Jp,0(1,1,0;β)(0≤β<p)(see[22-23]).
(6)U J1(1,1,k;β)and U J1,0(1,0,k;β)(k≥0,−1≤β<1)(see[24]).
The second family of integral operators was introduced by D Breaz and N Breaz[25]and it has the following form(see also a recent investigation on this subject by D Breaz et al[26]
whereα∈C,Reα>0;gj∈A;j=1,2,···,n.
Forλi>0,α≥0 and fi(z)∈Ap,we define the following general integral operators
Remark 1.1 Takingα=0 andα=1,we obtain the generalintegraloperators Fn,p(j,0;z) and Fn,p(j,1;z)introduced and studied by G¨ulsah Saltik et al[17].Forα=0,j=0 andα= 1,j=0,we obtain the general integral operators Fn,p(0,0;z)and Fn,p(0,1;z)introduced and studied by B A Frasin[18].And for p=1,we obtain the general integral operators Fn,1(0,0;z) and Fn,1(0,1;z)introduced and studied by Breaz et al[1,2730].
In[31],Li and Yang investigated a class of analytic functions of complex order.
In this paper,we obtain the order of convexity of the operator Fn,p(j,α;z)on the class U Jp,j(b,α,k;β),continue and extend the considerations of the paper[17-18].
First,we prove the following inclusion relation.
Theorem 2.1 For k≥0,β<p,b∈C−{0},let f(z)∈U Jp,j(b,α,k;β),then
Proof Let f(z)∈U Jp,j(b,α,k;β).Then the quantity
satisfies
The inequality
yields
Thus
Theorem 2.2 Letλi>0,αi≥0,ki≥0,−1≤βi<p for all i=1,2,···,n and fi(z)∈U Jp,j(b,αi,ki;βi)for all i=1,2,···,n.If
then the integral operator Fn,p(j,αi;z)is p-valently convex of order b and type
Proof From the definition(1.10),we observe that Fn,p(j,αi;z)∈Ap.On the other hand, it is easy to see that
Now we diff erentiate(2.2)logarithmically and multiply by z,we obtain
Then multiplying the relation(2.3)with
The relation(2.4)is equivalent to
Taking the real part of both sides of(2.5),we have
Since fi(z)∈U Jp,j(b,αi,ki;βi)for all i=1,2,···,n,from(2.6),we have
Because
for all i=1,2,···,n,we obtain
Therefore Fn,p(j,αi;z)is p-valently convex of complex order of b and type
This evidently completes the proof of Theorem 2.2.
Remark 2.1 Lettingαi=0,ki=0 and 0≤βi<p for all i={1,2,···,n}in Theorem 2.2,we improve Theorem 2.1 in[17].
Lettingαi=1,ki=0 and 0≤βi<p for all i={1,2,···,n}in Theorem 2.2,we improve Theorem 3.1 in[17].
Lettingαi=0,j=0,b=1 and−1≤βi<p for all i={1,2,···,n}in Theorem 2.2,we obtain Theorem 2.1 in[18].
Lettingαi=1,j=0,b=1 and−1≤βi<p for all i={1,2,···,n}in Theorem 2.2,we obtain Theorem 3.1 in[18].
Corollary 2.1 Letλi>0,ki≥0,−1≤βi<p and fi(z)∈U Jp,j(b,0,ki;βi)for all i=1,2,···,n.If
then the integraloperator Fn,p(j,0;z)is p-valently convex of order b and type
Theorem 2.3 Letλi>0,αi≥0,ki≥0,−1≤βi<p,fi(z)∈U Jp,j(b,αi,ki;βi)for all i=1,2,···,n and
where
for all i=1,2,···,n.Then the integral operator Fn,p(j,αi;z)is p-valently convex of order b in U.
Proof From(2.7)and(2.8)we easily get Fn,p(j,αi;z)∈Cp(b,0).
Remark 2.2 Lettingαi=0,j=0 and b=1 in Theorem 2.3,we obtain Theorem 2.4 in [18].
Theorem 2.4 Letλi>0,αi≥0,ki≥0,βi>p and fi(z)∈U Mp,j(b,αi,ki;βi)for all i=1,2,···,n.If
where
for all i=1,2,···,n,then the integral operator Fn,p(j,αi;z)∈Mp(b,0).
Proof From(2.7)and(2.9)we easily get Fn,p(j,αi;z)∈Mp(b,0).
We remark in conclusion that,by suitably specializing the parameters involved in the results presented in this paper,we can deduce numerous further corollaries and consequences of each of these results.
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tion:30C45,30C50,26D15
1002-0462(2014)01–0088–09
Chin.Quart.J.of Math. 2014,29(1):88—96
date:2013-01-21
Supported by the Natural Science Foundation of Inner Mongolia(2009MS0113);Supported by the Higher School Research Foundation of Inner Mongolia(NJzy08150)
Biographies:LI Shu-hai(1966-),male(mongolian),native of Chifeng,Inner Mongolia,a professor of Chifeng University,engages in geometric function theory and applications;MA Li-na(1982-),female(mongolian),native of Chifeng,Inner Mongolia,a lecturer of Chifeng University,M.S.D.,engagesin harmonic analysis;AO-en(1980-), female(mongolian),native of Chifeng,Inner Mongolia,a lecturer of Chifeng University,M.S.D.,engages in complex analysis.
CLC number:O174.51 Document code:A
Chinese Quarterly Journal of Mathematics2014年1期