IJDEInternational Journal of Differential Equations1687-96511687-9643Hindawi Publishing Corporation90283010.1155/2011/902830902830Research ArticleDifferential Subordination and Superordination for Srivastava-Attiya OperatorHaji MohdMaisarah1DarusMaslina1EzzinbiKhalilSchool of Mathematical SciencesFaculty of Science and TechnologyUniversiti Kebangsaan Malaysia43600 Bangi, Selangor D. EhsanMalaysiaukm.my20111182011201128032011010620112011Copyright © 2011 Maisarah Haji Mohd and Maslina Darus.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Due to the well-known Srivastava-Attiya operator, we investigate here some results relating the p-valent of the operator with differential subordination and subordination. Further, we obtain some interesting results on sandwich-type theorem for the same.

1. Introduction and Motivation

Let (U) be the class of analytic functions in the open unit disc U and let [a,n] be the subclass of (U) consisting functions of the form f(z)=a+anzn+an+1zn+1+, with 0=[0,1] and =[1,1]. For two functions f1 and f2 analytic in U, the function f1 is subordinate to f2, or f2 superordinate to f1, written as f1f2 if there exists a function w(z), analytic in U with w(0)=0 and |w(z)|<1 such that f1(z)=f2(w(z)). In particular, if the function f2 is univalent in U, then f1f2 is equivalent to f1(0)=f2(0) and f1(U)f2(U).

Let f,h(U) and ψ:3×U. If f and ψ(f(z),zf(z),z2f′′(z);z) are univalent and f satisfies the second-order differential subordinationψ(f(z),zf(z),z2f′′(z);z)h(z), then f is called a solution of the differential subordination. The univalent function F is called a dominant if fF for all f satisfying (1.1). Miller and Mocanu discussed many interesting results containing the above mentioned subordination and also many applications of the field of differential subordination in . In that direction, many differential subordination and differential superordination problems for analytic functions defined by means of linear operators were investigated. See  for related results.

Let 𝒜p denote the class of functions of the formf(z)=zp+n=1an+pzn+p(zU,  pN=1,2,3,), which are analytic and p-valent in U. For f satisfying (1.2), let the generalized Srivastava-Attiya operator  be denoted byJs,bf(z)=Gs,b*f(z)(bCZ¯0=0,-1,-2,), whereGs,b=(1+b)s[φ(z,s,b)-b-s], withφ(z,s,b)=1bs+zp(1+b)s+z1+p(2+b)s+, and the symbol (*) denotes the usual Hadamard product (or convolution). From the equations, we can see thatJs,bf(z)=zp+n=1(1+bn+1+b)san+pzn+p. Note that for p=1 in (1.6), Js,bf(z) coincides with the Srivastava-Attiya operator . Further, observe that for proper choices of s and b, the operator Js,bf(z) coincides with the following:

J0,bf(z)=f(z),

J1,0f(z)=A(f)(z) ,

J1,γf(z)=γ(f)(z),  (γ>-1) [15, 16],

Jσ,1f(z)=Iσ(f)(z),  (σ>0) ,

Jα,βf(z)=Pβα(f)(z),  (α1,  β>1) .

Since the above mentioned operator, the generalized Srivastava-Attiya operator, Js,bf(z) reduces to the well-known operators introduced and studied in the literature by suitably specializing the values of s and b and also in view of the several interesting properties and characteristics of well-known differential subordination results, we aim to associate these two motivating findings and obtain certain other related results. Further, we consider the differential superordination problems associated with the same operator. In addition, we also obtain interesting sandwich-type theorems.

The following definitions and theorems were discussed and will be needed to prove our results.

Definition 1.1 (see [<xref ref-type="bibr" rid="B16">1</xref>], Definition 2.2b, page 21).

Denote by Q the set of all functions q that are analytic and injective on U¯E(q) where E(q)={ζU:limzζ=} and are such that q(ζ)0 for ζUE(q). Further let the subclass of Q for which q(0)=a be denoted by Q(a), Q(0)Q0, and Q(1)Q1.

Definition 1.2 (see [<xref ref-type="bibr" rid="B16">1</xref>], Definition 2.3a, page 27).

Let Ω be a set in ,  qQ, and let n be a positive integer. The class of admissible functions Ψn[Ω,q] consists of those functions ψ:3×U that satisfy the admissibility condition ψ(c,d,e;z)Ω whenever c=q(ζ),  d=kζq(ζ), and Re{ed+1}kRe{ζq′′(ζ)q(ζ)+1},zU,  ζUE(q), and kn. Let Ψ1[Ω,q]=Ψ[Ω,q].

Definition 1.3 (see [<xref ref-type="bibr" rid="B17">19</xref>], Definition 3, page 817).

Let Ω be a set in , q[a,n] with q(z)0. The class of admissible functions Ψn[Ω,q] consists of those functions ψ:3×U that satisfy the admissibility condition ψ(c,d,e;ζ)Ω whenever c=q(z),  d=zq(z)/m, and Re{ed+1}1mRe{ζq′′(ζ)q(ζ)+1},zU,  ζU, and mn1. Let Ψ1[Ω,q]=Ψ[Ω,q].

Theorem 1.4 (see [<xref ref-type="bibr" rid="B16">1</xref>], Theorem 2.3b, page 28).

Let ψΨn[Ω,q] with q(0)=a. If the analytic function j(z)[a,n] satisfies ψ(j(z),zj(z),z2j′′(z);z)Ω, then j(z)q(z).

Theorem 1.5 (see [<xref ref-type="bibr" rid="B17">19</xref>], Theorem 1, page 818).

Let ψΨn[Ω,q] with q(0)=a. If jQ(a) and ψ(j(z),zj(z),z2j′′(z);z) is univalent in U, then Ω{ψ(j(z),zj(z),z2j′′(z);z):zU} implies q(z)j(z).

2. Subordination Results Associated with Generalized Srivastava-Attiya OperatorDefinition 2.1.

Let Ω be a set in and qQ0[0,p]. The class of admissible functions ΦJ[Ω,q] consists of those functions ϕ:3×U that satisfy the admissibility condition: ϕ(u,v,w;z)Ω whenever u=q(ζ),v=kζq(ζ)-[p-(1+b)]q(ζ)1+b(bCZ¯0=0,-1,-2,,pN),Re{(1+b)2w-[p-(1+b)]2u(1+b)v+[p-(1+b)]u+2[p-(1+b)]}kRe{ζq′′(ζ)q(ζ)+1},zU,  ζUE(q), and kp.

Theorem 2.2.

Let ϕΦJ[Ω,q]. If f𝒜p satisfies {ϕ(Js+2,bf(z),Js+1,bf(z),Js,bf(z);z):zU}Ω, then Js+2,bf(z)q(z)(zU).

Proof.

The following relation obtained in  zJs+1,bf(z)=[p-(1+b)]Js+1,bf(z)+(1+b)Js,bf(z) is equivalent to Js,bf(z)=zJs+1,bf(z)-[p-(1+b)]Js+1,bf(z)1+b, and hence Js+1,bf(z)=zJs+2,bf(z)-[p-(1+b)]Js+2,bf(z)1+b. Define the analytic function j in U by j(z)=Js+2,bf(z), and then we get Js+1,bf(z)=zj(z)-[p-(1+b)]j(z)1+b,Js,bf(z)=z2j′′(z)+(1-2[p-(1+b)])zj(z)+[p-(1+b)]2j(z)(1+b)2. Further, let us define the transformations from 3 to by u=c,v=d-[p-(1+b)]c1+b,w=e+(1-2[p-(1+b)])d+[p-(1+b)]2c(1+b)2. Let ψ(c,d,e;z)=ϕ(u,v,w;z),ϕ(u,v,w;z)=ϕ(c,d-[p-(1+b)]c1+b,e+(1-2[p-(1+b)])d+[p-(1+b)]2c(1+b)2;z). The proof will make use of Theorem 1.4. Using (2.8) and (2.9), from (2.12) we obtain ψ(j(z),zj(z),z2j′′(z);z)=ϕ(Js+2,bf(z),Js+1,bf(z),Js,bf(z);z). Hence (2.3) becomes ψ(j(z),zj(z),z2j′′(z);z)Ω. Note that ed+1=(1+b)2w-[p-(1+b)]2u(1+b)v+[p-(1+b)]u+2[p-(1+b)], and since the admissibility condition for ϕΦJ[Ω,q] is equivalent to the admissibility condition for ψ as given in Definition 1.2, hence ψΨp[Ω,q], and by Theorem 1.4, j(z)q(z), or Js+2,bf(z)q(z).

In the case ϕ(u,v,w;z)=v, we have the following example.

Example 2.3.

Let the class of admissible functions ΦJv[Ω,q] consist of those functions ϕ:3×U that satisfy the admissibility condition: v=kζq(ζ)-[p-(1+b)]q(ζ)1+bΩ,zU,  ζUE(q), and kp and ϕΦJv[Ω,q]. If f𝒜p satisfies Js+1,bf(z)Ω, then Js+2,bf(z)q(z)(zU).

If Ω is a simply connected domain, then Ωh(U) for some conformal mapping h(z) of U onto Ω and the class is written as ΦJ[h,q]. The following result follows immediately from Theorem 2.2.

Theorem 2.4.

Let ϕΦJ[Ω,q]. If f𝒜p satisfies ϕ(Js+2,bf(z),Js+1,bf(z),Js,bf(z);z)h(z), then Js+2,bf(z)q(z).

The next result occurs when the behavior of q on U is not known.

Corollary 2.5.

Let Ω, q be univalent in Uand q(0)=0. Let ϕΦJ[Ω,qρ] for some ρ(0,1) where qρ(z)=q(ρz). If f𝒜p and ϕ(Js+2,bf(z),Js+1,bf(z),Js,bf(z);z)Ω, then Js+2,bf(z)q(z).

Proof.

From Theorem 2.2, we see that Js+2,bf(z)qρ(z) and the proof is complete.

Theorem 2.6.

Let h and q be univalent in U, with q(0)=0 and set qρ(z)=q(ρz) and hρ(z)=h(ρz). Let ϕ:3×U satisfy one of the following conditions:

ϕΦJ[h,qρ], for some ρ(0,1), or

there exists ρ0(0,1) such that ϕΦJ[hρ,qρ], for all ρ0(0,1).

If f𝒜p satisfies (2.21), then Js+2,bf(z)q(z).

Proof.

The proof is similar to the one in  and therefore is omitted.

The next results give the best dominant of the differential subordination (2.21).

Theorem 2.7.

Let h be univalent in U. Let ϕ:3×U. Suppose that the differential equation ϕ(q(z),zq(z),z2q′′(z);z)=h(z) has a solution q with q(0)=0 and satisfy one of the following conditions:

qQ0 and ϕΦJ[h,q],

q is univalent in U and ϕΦJ[h,qρ], for some ρ(0,1), or

q is univalent in U and there exists ρ0(0,1) such that ϕΦJ[hρ,qρ], for all ρ0(0,1).

If f𝒜p satisfies (2.21), then Js+2,bf(z)q(z), and q is the best dominant.

Proof.

Following the same arguments in , we deduce that q is a dominant from Theorem 2.4 and Theorem 2.6. Since q satisfies (2.26), it is also a solution of (2.21) and therefore q will be dominated by all dominants. Hence q is the best dominant.

Definition 2.8.

Let Ω be a set in and qQ00. The class of admissible functions ΦJ,1[Ω,q] consists of those functions ϕ:3×U that satisfy the admissibility condition: ϕ(u,v,w;z)Ω whenever u=q(ζ),v=kζq(ζ)-bq(ζ)1+b(bCZ¯0=0,-1,-2,,pN),Re{(1+b)2w-b2u(1+b)v+bu-2b}kRe{ζq′′(ζ)q(ζ)+1},zU,  ζUE(q), and k1.

Theorem 2.9.

Let ϕΦJ,1[Ω,q]. If f𝒜p satisfies {ϕ(Js+2,bf(z)zp-1,Js+1,bf(z)zp-1,Js,bf(z)zp-1;z):zU}Ω, then Js+2,bf(z)zp-1q(z)(zU).

Proof.

Define the analytic function j in U by j(z)=Js+2,bf(z)zp-1. Using the relations (2.5) and (2.32), we get Js+1,bf(z)zp-1=zj(z)-bj(z)1+b,Js,bf(z)zp-1=z2j′′(z)+(2b+1)zj(z)+b2j(z)(1+b)2. Further, let us define the transformations from 3 to by u=c,v=d+bc1+b,w=e+(2b+1)d+b2c(1+b)2. Let ψ(c,d,e;z)=ϕ(u,v,w;z)=ϕ(c,d+bc1+b,e+(2b+1)d+b2c(1+b)2;z). The proof will make use of Theorem 1.4. Using (2.32) and (2.33), from (2.35) we obtain ψ(j(z),zj(z),z2j′′(z);z)=ϕ(Js+2,bf(z)zp-1,Js+1,bf(z)zp-1,Js,bf(z)zp-1;z). Hence (2.30) becomes ψ(j(z),zj(z),z2j′′(z);z)Ω. Note that ed+1=(1+b)2w-b2u(1+b)v+bu-2b, and since the admissibility condition for ϕΦJ,1[Ω,q] is equivalent to the admissibility condition for ψ as given in Definition 1.2, hence ψΨ[Ω,q], and by Theorem 1.4, j(z)q(z), or Js+2,bf(z)zp-1q(z).

In the case ϕ(u,v,w;z)=v-u, we have the following example.

Example 2.10.

Let the class of admissible functions ΦJv,1[Ω,q] consist of those functions ϕ:3×U that satisfy the admissibility condition: v-u=kζq(ζ)-pq(ζ)1+bΩ,zU,    ζUE(q), and kp and ϕΦJv,1[Ω,q]. If f𝒜p satisfies Js+1,bf(z)zp-1-Js,bf(z)zp-1Ω(zU), then Js+2,bf(z)zp-1q(z)(zU).

If Ω is a simply connected domain, then Ωh(U) for some conformal mapping h(z) of U onto Ω and the class is written as ΦJ,1[h,q]. The following result follows immediately from Theorem 2.9.

Theorem 2.11.

Let ϕΦJ,1[Ω,q]. If f𝒜p satisfies ϕ(Js+2,bf(z)zp-1,Js+1,bf(z)zp-1,Js,bf(z)zp-1;z)h(z), then Js+2,bf(z)zp-1q(z).

Definition 2.12.

Let Ω be a set in and qQ1. The class of admissible functions ΦJ,2[Ω,q] consists of those functions ϕ:3×U that satisfy the admissibility condition: ϕ(u,v,w;z)Ω whenever u=q(ζ),v=q(ζ)+kζq(ζ)(1+b)q(ζ)(bCZ¯0=0,-1,-2,,pN),Re{(w-u)(1+b)uv-u+(1+b)(w-3u)}kRe{ζq′′(ζ)q(ζ)+1},zU,  ζUE(q), and k1.

Theorem 2.13.

Let ϕΦJ,2[Ω,q]. If f𝒜p satisfies {ϕ(Js+2,bf(z)Js+3,bf(z),Js+1,bf(z)Js+2,bf(z),Js,bf(z)Js+1,bf(z);z):zU}Ω, then Js+2,bf(z)Js+3,bf(z)q(z)(zU).

Proof.

Define the analytic function j in U by j(z)=Js+2,bf(z)Js+3,bf(z). Differentiating (2.50) yields zj(z)j(z)=zJs+2,bf(z)Js+2,bf(z)-Js+3,bf(z)Js+3,bf(z). From the relation (2.5) we get zJs+2,bf(z)Js+2,bf(z)=[p-(1+b)]+(1+b)j+zj(z)j(z), and hence Js+1,bf(z)Js+2,bf(z)=j(z)+zj(z)(1+b)j(z). Further computations show that Js,bf(z)Js+1,bf(z)=j(z)+[2(1+b)j(z)+1]zj(z)+z2j′′(z)(1+b)2j(z)2+(1+b)zj(z). Let us define the transformations from 3 to by u=c,v=c+d(1+b)c,w=c+[2(b+1)c+1]d+e(1+b)2c2+(1+b)d. Let ψ(c,d,e;z)=ϕ(u,v,w;z)=ϕ(c,c+d(1+b)c,c+[2(b+1)c+1]d+e(1+b)2c2+(1+b)d;z). The proof will make use of Theorem 1.4. Using (2.50), (2.53) and (2.54), from (2.56) we obtain ψ(j(z),zj(z),z2j′′(z);z)=ϕ(Js+2,bf(z)Js+3,bf(z),Js+1,bf(z)Js+2,bf(z),Js,bf(z)Js+1,bf(z);z). Hence (2.48) becomes ψ(j(z),zj(z),z2j′′(z);z)Ω. Note that ed+1=(w-u)(1+b)uv-u+(1+b)(w-3u), and since the admissibility condition for ϕΦJ,2[Ω,q] is equivalent to the admissibility condition for ψ as given in Definition 1.2, hence ψΨ[Ω,q] and by Theorem 1.4, j(z)q(z), or Js+2,bf(z)Js+3,bf(z)q(z).

If Ω is a simply connected domain, then Ωh(U) for some conformal mapping h(z) of U onto Ω and the class is written as ΦJ,2[h,q]. The following result follows immediately from Theorem 2.13.

Theorem 2.14.

Let ϕΦJ,2[Ω,q]. If f𝒜p satisfies ϕ(Js+2,bf(z)Js+3,bf(z),Js+1,bf(z)Js+2,bf(z),Js,bf(z)Js+1,bf(z);z)h(z), then Js+2,bf(z)Js+3,bf(z)q(z).

3. Superordination Results Associated with Generalized Srivastava-Attiya OperatorDefinition 3.1.

Let Ω be a set in and q[0,p] with zq(z)0. The class of admissible functions ΦJ[Ω,q] consists of those functions ϕ:3×U¯ that satisfy the admissibility condition: ϕ(u,v,w;ζ)Ω whenever u=q(z),v=zq(z)-m[p-(1+b)]q(z)m(1+b)(bCZ¯0=0,-1,-2,,pN),Re{(1+b)2w-[p-(1+b)]2u(1+b)v+[p-(1+b)]u+2[p-(1+b)]}1mRe{zq′′(z)q(z)+1},zU,  ζU, and mp.

Theorem 3.2.

Let ϕΦJ[Ω,q]. If f𝒜p, Js+2,bfQ0 and ϕ(Js+2,bf(z),Js+1,bf(z),Js,bf(z);z) is univalent in U, then Ω{ϕ(Js+2,bf(z),Js+1,bf(z),Js,bf(z);z):zU} implies that q(z)Js+2,bf(z).

Proof.

From (2.13) and (3.4), we have Ω{ψ(j(z),zj(z),z2j′′(z);z):zU}. From (2.10), we see that the admissibility condition for ϕΦJ[Ω,q] is equivalent to the admissibility condition for ψ as given in Definition 1.3. Hence ψΨp[Ω,q], and by Theorem 1.5, q(z)j(z) or q(z)Js+2,bf(z).

If Ω is a simply connected domain, then Ωh(U) for some conformal mapping h(z) of U onto Ω and the class is written as ΦJ[h,q]. The next result follows immediately from Theorem 3.2.

Theorem 3.3.

Let h be analytic in U and ϕΦJ[Ω,q]. If f𝒜p, Js+2,bf(z)Q0 and ϕ(Js+2,bf(z),Js+1,bf(z),Js,bf(z);z) is univalent in U, then h(z)ϕ(Js+2,bf(z),Js+1,bf(z),Js,bf(z);z), and then q(z)Js+2,bf(z).

Theorems 3.2 and 3.3 can only be used to obtain subordinants for differential superordination of the form (3.4) and (3.9). The following theorems prove the existence of the best subordinant of (3.9) for certain ϕ.

Theorem 3.4.

Let h be analytic in U and ϕ:3×U¯. Suppose that the differential equation ϕ(q(z),zq(z),z2q′′(z);z)=h(z) has a solution qQ0. If ϕΦJ[Ω,q], f𝒜p, Js+2,bf(z)Q0, and ϕ(Js+2,bf(z),Js+1,bf(z),Js,bf(z);z) is univalent in U, then h(z)ϕ(Js+2,bf(z),Js+1,bf(z),Js,bf(z);z) implies that q(z)Js+2,bf(z), and q(z) is the best subordinant.

Proof.

The result can be obtained by similar proof of Theorem 2.7.

The next result, the sandwich-type theorem follows from Theorems 2.4 and 3.3.

Corollary 3.5.

Let h1  and q1  be analytic in U, and let h2  be univalent function in U, q2Q0  with q1(0)=q2(0)=0 and ϕΦJ[h2,q2]ΦJ[h1,q1]. If f𝒜p, Js+2,bf(z)[0,p]Q0, and ϕ(Js+2,bf(z),Js+1,bf(z),Js,bf(z);z) is univalent in U, then h1(z)ϕ(Js+2,bf(z),Js+1,bf(z),Js,bf(z);z)h2(z) implies that q1(z)Js+2,bf(z)q2(z).

Definition 3.6.

Let Ω be a set in and q0 with zq(z)0. The class of admissible functions ΦJ,1[Ω,q] consists of those functions ϕ:3×U that satisfy the admissibility condition: ϕ(u,v,w;ζ)Ω whenever u=q(z),v=zq(z)-mbq(z)m(1+b)(bCZ¯0=0,-1,-2,,pN),Re{(1+b)2w-b2u(1+b)v+bu-2b}1mRe{zq′′(z)q(z)+1},zU,  ζU, and m1.

The following result is associated with Theorem 2.9.

Theorem 3.7.

Let ϕΦH,1[Ω,q]. If f𝒜p, Js+2,bf(z)/zp-1Q0, and ϕ(Js+2,bf(z)zp-1,Js+1,bf(z)zp-1,Js,bf(z)zp-1;z) is univalent in U, then Ω{ϕ(Js+2,bf(z)zp-1,Js+1,bf(z)zp-1,Js,bf(z)zp-1;z):zU} implies that q(z)Js+2,bf(z)zp-1.

Proof.

From (2.36) and (3.21), we have Ω{ϕ(j(z),zj(z),z2j′′(z);z):zU}. From (2.34), we see that the admissibility condition for ϕΦJ,1[Ω,q] is equivalent to the admissibility condition for ψ as in Definition 1.3. Hence ψΨ[Ω,q], and by Theorem 1.5, q(z)j(z) or q(z)Js+2,bf(z)zp-1.

If Ω is a simply connected domain, then Ωh(U) for some conformal mapping h(z) of U onto Ω and the class is written as ΦJ,1[h,q]. The next result follows immediately from Theorem 3.7.

Theorem 3.8.

Let q0, and let h be analytic on U, and let ϕΦJ,1[Ω,q]. If f𝒜p, Js+2,bf(z)/zp-1Q0, and ϕ(Js+2,bf(z)zp-1,Js+1,bf(z)zp-1,Js,bf(z)zp-1;z) is univalent in U, then h(z)ϕ(Js+2,bf(z)zp-1,Js+1,bf(z)zp-1,Js,bf(z)zp-1;z) implies that q(z)Js+2,bf(z)zp-1.

Combining Theorems 2.11 and 3.8, we obtain the following sandwich-type theorem.

Corollary 3.9.

Let h1 and q1 be analytic in U, let h2 be univalent function in U, q2Q0 with q1(0)=q2(0)=0, and ϕΦJ,1[h2,q2]ΦJ,1[h1,q1]. If f𝒜p, Js+2,bf(z)/zp-10Q0, and ϕ(Js+2,bf(z)zp-1,Js+1,bf(z)zp-1,Js,bf(z)zp-1;z) is univalent in U, then h1(z)ϕ(Js+2,bf(z)zp-1,Js+1,bf(z)zp-1,Js,bf(z)zp-1;z)h2(z) implies that q1(z)Js+2,bf(z)zp-1q2(z).

Definition 3.10.

Let Ω be a set in and q(z)0,    zq(z)0, and q. The class of admissible functions ΦJ,2[Ω,q] consists of those functions ϕ:3×U that satisfy the admissibility condition: ϕ(u,v,w;ζ)Ω whenever u=q(z),v=q(z)+zq(z)m(1+b)q(z)(bCZ¯0=0,-1,-2,,pN),Re{(w-u)(1+b)uv-u+(1+b)(w-3u)}1mRe{zq′′(z)q(z)+1},zU,  ζU, and m1.

The following result is associated with Theorem 2.13.

Theorem 3.11.

Let ϕΦJ,2[Ω,q]. If f𝒜p,Js+2,bf(z)/Js+3,bf(z)Q1, and ϕ(Js+2,bf(z)Js+3,bf(z),Js+1,bf(z)Js+2,bf(z),Js,bf(z)Js+1,bf(z);z) is univalent in U, then Ω{ϕ(Js+2,bf(z)Js+3,bf(z),Js+1,bf(z)Js+2,bf(z),Js,bf(z)Js+1,bf(z);z):zU} implies that q(z)Js+2,bf(z)Js+3,bf(z).

Proof.

From (2.57) and (3.34), we have Ω{ϕ(j(z),zj(z),z2j′′(z);z):zU}. From (2.55), we see that the admissibility condition for ϕΦJ,2[Ω,q] is equivalent to the admissibility condition for ψ as in Definition 1.3. Hence ψΨ[Ω,q], and by Theorem 1.5, q(z)j(z) or q(z)Js+2,bf(z)Js+3,bf(z).

If Ω is a simply connected domain, then Ωh(U) for some conformal mapping h(z) of U onto Ω and the class is written as ΦJ,2[h,q]. The next result follows immediately from Theorem 3.11 as in the previous section.

Theorem 3.12.

Let q, let h be analytic in U, and let ϕΦJ,2[Ω,q]. If f𝒜p,  Js+2,bf(z)/Js+3,bf(z)Q1 and ϕ(Js+2,bf(z)Js+3,bf(z),Js+1,bf(z)Js+2,bf(z),Js,bf(z)Js+1,bf(z);z) is univalent in U, then h(z)ϕ(Js+2,bf(z)Js+3,bf(z),Js+1,bf(z)Js+2,bf(z),Js,bf(z)Js+1,bf(z);z) implies that q(z)Js+2,bf(z)Js+3,bf(z).

Combining Theorems 2.14 and 3.12, we obtain the following sandwich-type theorem.

Corollary 3.13.

Let h1  and q1 be analytic in U, let h2 be univalent function in U, q2Q0 with q1(0)=q2(0)=0, and ϕΦJ,2[h2,q2]ΦJ,2[h1,q1]. If f𝒜p,  Js+2,bf(z)/Js+3,bf(z)Q1, and ϕ(Js+2,bf(z)Js+3,bf(z),Js+1,bf(z)Js+2,bf(z),Js,bf(z)Js+1,bf(z);z) is univalent in U, then h1(z)ϕ(Js+2,bf(z)Js+3,bf(z),Js+1,bf(z)Js+2,bf(z),Js,bf(z)Js+1,bf(z);z)h2(z) implies that q1(z)Js+2,bf(z)Js+3,bf(z)q(z).

Other work related to certain operators concerning the subordination and superordination can be found in .

Acknowledgment

The work presented here was partially supported by UKM-ST-FRGS-0244-2010.

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