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 ℋ(U) be the class of analytic functions in the open unit disc U and let ℋ[a,n] be the subclass of ℋ(U) consisting functions of the form f(z)=a+anzn+an+1zn+1+⋯, with ℋ0=ℋ[0,1] and ℋ=ℋ[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 f1≺f2 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 f1≺f2 is equivalent to f1(0)=f2(0) and f1(U)⊂f2(U).

Let f,h∈ℋ(U) and ψ:ℂ3×U→ℂ. 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 f≺F 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 [1]. In that direction, many differential subordination and differential superordination problems for analytic functions defined by means of linear operators were investigated. See [2–11] for related results.

Let 𝒜p denote the class of functions of the formf(z)=zp+∑n=1∞an+pzn+p(z∈U,p∈N=1,2,3,…),
which are analytic and p-valent in U. For f satisfying (1.2), let the generalized Srivastava-Attiya operator [12] be denoted byJs,bf(z)=Gs,b*f(z)(b∈C∖Z¯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 [13]. 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) [14],

J1,γf(z)=ℐγ(f)(z),(γ>-1) [15, 16],

Jσ,1f(z)=Iσ(f)(z),(σ>0) [17],

Jα,βf(z)=Pβα(f)(z),(α≥1,β>1) [18].

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)={ζ∈∂U:limz→ζ=∞}
and are such that q′(ζ)≠0 for ζ∈∂U∖E(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 ℂ,q∈Q, and let n be a positive integer. The class of admissible functions Ψn[Ω,q] consists of those functions ψ:ℂ3×U→ℂ 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},z∈U,ζ∈∂U∖E(q), and k≥n. 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 ℂ, q∈ℋ[a,n] with q′(z)≠0. The class of admissible functions Ψn′[Ω,q] consists of those functions ψ:ℂ3×U→ℂ 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},z∈U,ζ∈∂U, and m≥n≥1. 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 ψ∈Ψn[Ω,q] with q(0)=a. If the analytic function j(z)∈ℋ[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 ψ∈Ψn′[Ω,q] with q(0)=a. If j∈Q(a) and ψ(j(z),zj′(z),z2j′′(z);z) is univalent in U, then
Ω⊂{ψ(j(z),zj′(z),z2j′′(z);z):z∈U}
implies q(z)≺j(z).

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

Let Ω be a set in ℂ and q∈Q0∩ℋ[0,p]. The class of admissible functions ΦJ[Ω,q] consists of those functions ϕ:ℂ3×U→ℂ 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(b∈C∖Z¯0=0,-1,-2,…,p∈N),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},z∈U,ζ∈∂U∖E(q), and k≥p.

Theorem 2.2.

Let ϕ∈ΦJ[Ω,q]. If f∈𝒜p satisfies
{ϕ(Js+2,bf(z),Js+1,bf(z),Js,bf(z);z):z∈U}⊂Ω,
then
Js+2,bf(z)≺q(z)(z∈U).

Proof.

The following relation obtained in [13]
zJs+1,b′f(z)=[p-(1+b)]Js+1,bf(z)+(1+b)Js,bf(z)
is equivalent to
Js,bf(z)=zJs+1,b′f(z)-[p-(1+b)]Js+1,bf(z)1+b,
and hence
Js+1,bf(z)=zJs+2,b′f(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 ℂ3 to ℂ 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 ϕ∈ΦJ[Ω,q] is equivalent to the admissibility condition for ψ as given in Definition 1.2, hence ψ∈Ψp[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:ℂ3×U→ℂ that satisfy the admissibility condition:
v=kζq′(ζ)-[p-(1+b)]q(ζ)1+b∉Ω,z∈U,ζ∈∂U∖E(q), and k≥p and ϕ∈ΦJv[Ω,q]. If f∈𝒜p satisfies
Js+1,bf(z)⊂Ω,
then
Js+2,bf(z)≺q(z)(z∈U).

If Ω≠ℂ is a simply connected domain, then Ω∈h(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 ϕ∈ΦJ[Ω,q]. If f∈𝒜p 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⊂ℂ, q be univalent in Uand q(0)=0. Let ϕ∈ΦJ[Ω,qρ] for some ρ∈(0,1) where qρ(z)=q(ρz). If f∈𝒜p 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:ℂ3×U→ℂ satisfy one of the following conditions:

ϕ∈ΦJ[h,qρ], for some ρ∈(0,1), or

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

If f∈𝒜p satisfies (2.21), then
Js+2,bf(z)≺q(z). Proof.

The proof is similar to the one in [1] 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:ℂ3×U→ℂ. 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:

q∈Q0 and ϕ∈ΦJ[h,q],

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

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

If f∈𝒜p satisfies (2.21), then
Js+2,bf(z)≺q(z),
and q is the best dominant. Proof.

Following the same arguments in [1], 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 ℂ and q∈Q0∩ℋ0. The class of admissible functions ΦJ,1[Ω,q] consists of those functions ϕ:ℂ3×U→ℂ that satisfy the admissibility condition:
ϕ(u,v,w;z)∉Ω
whenever
u=q(ζ),v=kζq′(ζ)-bq(ζ)1+b(b∈C∖Z¯0=0,-1,-2,…,p∈N),Re{(1+b)2w-b2u(1+b)v+bu-2b}≥kRe{ζq′′(ζ)q′(ζ)+1},z∈U,ζ∈∂U∖E(q), and k≥1.

Theorem 2.9.

Let ϕ∈ΦJ,1[Ω,q]. If f∈𝒜p satisfies
{ϕ(Js+2,bf(z)zp-1,Js+1,bf(z)zp-1,Js,bf(z)zp-1;z):z∈U}⊂Ω,
then
Js+2,bf(z)zp-1≺q(z)(z∈U).

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 ℂ3 to ℂ 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 ϕ∈ΦJ,1[Ω,q] is equivalent to the admissibility condition for ψ as given in Definition 1.2, hence ψ∈Ψ[Ω,q], and by Theorem 1.4,
j(z)≺q(z),
or
Js+2,bf(z)zp-1≺q(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:ℂ3×U→ℂ that satisfy the admissibility condition:
v-u=kζq′(ζ)-pq(ζ)1+b∉Ω,z∈U,ζ∈∂U∖E(q), and k≥p and ϕ∈ΦJv,1[Ω,q]. If f∈𝒜p satisfies
Js+1,bf(z)zp-1-Js,bf(z)zp-1⊂Ω(z∈U),
then
Js+2,bf(z)zp-1≺q(z)(z∈U).

If Ω≠ℂ is a simply connected domain, then Ω∈h(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 ϕ∈ΦJ,1[Ω,q]. If f∈𝒜p 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-1≺q(z).

Definition 2.12.

Let Ω be a set in ℂ and q∈Q1∩ℋ. The class of admissible functions ΦJ,2[Ω,q] consists of those functions ϕ:ℂ3×U→ℂ that satisfy the admissibility condition:
ϕ(u,v,w;z)∉Ω
whenever
u=q(ζ),v=q(ζ)+kζq′(ζ)(1+b)q(ζ)(b∈C∖Z¯0=0,-1,-2,…,p∈N),Re{(w-u)(1+b)uv-u+(1+b)(w-3u)}≥kRe{ζq′′(ζ)q′(ζ)+1},z∈U,ζ∈∂U∖E(q), and k≥1.

Theorem 2.13.

Let ϕ∈ΦJ,2[Ω,q]. If f∈𝒜p 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):z∈U}⊂Ω,
then
Js+2,bf(z)Js+3,bf(z)≺q(z)(z∈U).

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,b′f(z)Js+2,bf(z)-Js+3,b′f(z)Js+3,bf(z).
From the relation (2.5) we get
zJs+2,b′f(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 ℂ3 to ℂ 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 ϕ∈ΦJ,2[Ω,q] is equivalent to the admissibility condition for ψ as given in Definition 1.2, hence ψ∈Ψ[Ω,q] and by Theorem 1.4,
j(z)≺q(z),
or
Js+2,bf(z)Js+3,bf(z)≺q(z).

If Ω≠ℂ is a simply connected domain, then Ω∈h(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 ϕ∈ΦJ,2[Ω,q]. If f∈𝒜p 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 ℂ and q∈ℋ[0,p] with zq′(z)≠0. The class of admissible functions ΦJ′[Ω,q] consists of those functions ϕ:ℂ3×U¯→ℂ 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)(b∈C∖Z¯0=0,-1,-2,…,p∈N),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},z∈U,ζ∈∂U, and m≥p.

Theorem 3.2.

Let ϕ∈ΦJ′[Ω,q]. If f∈𝒜p, Js+2,bf∈Q0 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):z∈U}
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):z∈U}.
From (2.10), we see that the admissibility condition for ϕ∈ΦJ′[Ω,q] is equivalent to the admissibility condition for ψ as given in Definition 1.3. Hence ψ∈Ψp′[Ω,q], and by Theorem 1.5, q(z)≺j(z) or
q(z)≺Js+2,bf(z).

If Ω≠ℂ is a simply connected domain, then Ω∈h(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 ϕ∈ΦJ′[Ω,q]. If f∈𝒜p, 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:ℂ3×U¯→ℂ. Suppose that the differential equation
ϕ(q(z),zq′(z),z2q′′(z);z)=h(z)
has a solution q∈Q0. If ϕ∈ΦJ′[Ω,q], f∈𝒜p, 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, q2∈Q0 with q1(0)=q2(0)=0 and ϕ∈ΦJ[h2,q2]∩ΦJ′[h1,q1]. If f∈𝒜p, Js+2,bf(z)∈ℋ[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 ℂ and q∈ℋ0 with zq′(z)≠0. The class of admissible functions ΦJ,1′[Ω,q] consists of those functions ϕ:ℂ3×U→ℂ that satisfy the admissibility condition:
ϕ(u,v,w;ζ)∈Ω
whenever
u=q(z),v=zq′(z)-mbq(z)m(1+b)(b∈C∖Z¯0=0,-1,-2,…,p∈N),Re{(1+b)2w-b2u(1+b)v+bu-2b}≥1mRe{zq′′(z)q′(z)+1},z∈U,ζ∈∂U, and m≥1.

The following result is associated with Theorem 2.9.

Theorem 3.7.

Let ϕ∈ΦH,1′[Ω,q]. If f∈𝒜p, Js+2,bf(z)/zp-1∈Q0, 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):z∈U}
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):z∈U}.
From (2.34), we see that the admissibility condition for ϕ∈ΦJ,1′[Ω,q] is equivalent to the admissibility condition for ψ as in Definition 1.3. Hence ψ∈Ψ′[Ω,q], and by Theorem 1.5, q(z)≺j(z) or
q(z)≺Js+2,bf(z)zp-1.

If Ω≠ℂ is a simply connected domain, then Ω∈h(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 q∈ℋ0, and let h be analytic on U, and let ϕ∈ΦJ,1′[Ω,q]. If f∈𝒜p, Js+2,bf(z)/zp-1∈Q0, 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, q2∈Q0 with q1(0)=q2(0)=0, and ϕ∈ΦJ,1[h2,q2]∩ΦJ,1′[h1,q1]. If f∈𝒜p, Js+2,bf(z)/zp-1∈ℋ0∩Q0, 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-1≺q2(z).

Definition 3.10.

Let Ω be a set in ℂ and q(z)≠0,zq′(z)≠0, and q∈ℋ. The class of admissible functions ΦJ,2′[Ω,q] consists of those functions ϕ:ℂ3×U→ℂ 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)(b∈C∖Z¯0=0,-1,-2,…,p∈N),Re{(w-u)(1+b)uv-u+(1+b)(w-3u)}≥1mRe{zq′′(z)q′(z)+1},z∈U,ζ∈∂U, and m≥1.

The following result is associated with Theorem 2.13.

Theorem 3.11.

Let ϕ∈ΦJ,2′[Ω,q]. If f∈𝒜p,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):z∈U}
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):z∈U}.
From (2.55), we see that the admissibility condition for ϕ∈ΦJ,2′[Ω,q] is equivalent to the admissibility condition for ψ as in Definition 1.3. Hence ψ∈Ψ′[Ω,q], and by Theorem 1.5, q(z)≺j(z) or
q(z)≺Js+2,bf(z)Js+3,bf(z).

If Ω≠ℂ is a simply connected domain, then Ω∈h(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∈ℋ, let h be analytic in U, and let ϕ∈ΦJ,2′[Ω,q]. If f∈𝒜p,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, q2∈Q0 with q1(0)=q2(0)=0, and ϕ∈ΦJ,2[h2,q2]∩ΦJ,2′[h1,q1]. If f∈𝒜p,Js+2,bf(z)/Js+3,bf(z)∈ℋ∩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 [20–25].

Acknowledgment

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

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