Hindawi Publishing Corporation
International Journal of Differential Equations
Volume 2011, Article ID 902830, 19 pages
doi:10.1155/2011/902830
Research Article
Differential Subordination and Superordination for
Srivastava-Attiya Operator
Maisarah Haji Mohd and Maslina Darus
School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia,
43600 Bangi, Selangor D. Ehsan, Malaysia
Correspondence should be addressed to Maslina Darus, maslina@ukm.my
Received 28 March 2011; Accepted 1 June 2011
Academic Editor: Khalil Ezzinbi
Copyright q 2011 M. Haji Mohd and M. 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 pvalent 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 HU be the class of analytic functions in the open unit disc U and let Ha, n be the
subclass of HU consisting functions of the form fz a an zn an1 zn1 · · · , with
H0 H0, 1 and H H1, 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 wz,
analytic in U with w0 0 and |wz| < 1 such that f1 z f2 wz. 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 ∈ HU and ψ : C3 × U → C. If f and ψfz, zf z, z2 f z; z are univalent
and f satisfies the second-order differential subordination
ψ fz, zf z, z2 f z; z ≺ hz,
1.1
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
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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 form
fz zp ∞
z ∈ U, p ∈ N 1, 2, 3, . . . ,
anp znp
1.2
n1
which are analytic and p-valent in U. For f satisfying 1.2, let the generalized SrivastavaAttiya operator 12 be denoted by
Js,b fz Gs,b ∗ fz
b ∈ C \ Z0 0, −1, −2, . . . ,
1.3
where
Gs,b 1 bs ϕz, s, b − b−s ,
1.4
with
ϕz, s, b zp
z1p
1
··· ,
s s
b
1 b
2 bs
1.5
and the symbol ∗ denotes the usual Hadamard product or convolution. From the
equations, we can see that
s
∞ 1b
anp znp .
Js,b fz z n1b
n1
p
1.6
Note that for p 1 in 1.6, Js,b fz coincides with the Srivastava-Attiya operator 13.
Further, observe that for proper choices of s and b, the operator Js,b fz coincides with the
following:
i J0,b fz fz,
ii J1,0 fz Afz 14,
iii J1,γ fz Iγ fz, γ > −1 15, 16,
iv Jσ,1 fz I σ fz, σ > 0 17,
v Jα,β fz Pβα fz, α ≥ 1, β > 1 18.
Since the above mentioned operator, the generalized Srivastava-Attiya operator,
Js,b fz 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.
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The following definitions and theorems were discussed and will be needed to prove
our results.
Definition 1.1 see 1, Definition 2.2b, page 21. Denote by Q the set of all functions q that are
analytic and injective on U \ Eq where
E q ζ ∈ ∂U : lim ∞
z→ζ
1.7
0 for ζ ∈ ∂U \ Eq. Further let the subclass of Q for which q0 a
and are such that q ζ /
be denoted by Qa, Q0 ≡ Q0 , and Q1 ≡ Q1 .
Definition 1.2 see 1, 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 :
/ Ω whenever c qζ, d C3 × U → C that satisfy the admissibility condition ψc, d, e; z ∈
kζq ζ, and
e
ζq ζ
1 ≥ k Re
1
,
Re
d
q ζ
1.8
z ∈ U, ζ ∈ ∂U \ Eq, and k ≥ n. Let Ψ1 Ω, q ΨΩ, q.
Definition 1.3 see 19, Definition 3, page 817. Let Ω be a set in C, q ∈ Ha, 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 ψc, d, e; ζ ∈
/ Ω whenever c qz, d zq z/m, and
Re
1
ζq ζ
1 ≥
Re
1
,
d
m
q ζ
e
1.9
z ∈ U, ζ ∈ ∂U, and m ≥ n ≥ 1. Let Ψ1 Ω, q Ψ Ω, q.
Theorem 1.4 see 1, Theorem 2.3b, page 28. Let ψ ∈ Ψn Ω, q with q0 a. If the analytic
function jz ∈ Ha, n satisfies
ψ jz, zj z, z2 j z; z ∈ Ω,
1.10
then jz ≺ qz.
Theorem 1.5 see 19, Theorem 1, page 818. Let ψ ∈ Ψn Ω, q with q0 a. If j ∈ Qa and
ψjz, zj z, z2 j z; z is univalent in U, then
Ω ⊂ ψ jz, zj z, z2 j z; z : z ∈ U
implies qz ≺ jz.
1.11
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2. Subordination Results Associated with
Generalized Srivastava-Attiya Operator
Definition 2.1. Let Ω be a set in C and q ∈ Q0 ∩ H0, p. The class of admissible functions
ΦJ Ω, q consists of those functions φ : C3 × U → C that satisfy the admissibility condition:
φu, v, w; z ∈
/Ω
2.1
kζq ζ − p − 1 b qζ b ∈ C \ Z0 0, −1, −2, . . . , p ∈ N ,
1b
2
1 b2 w − p − 1 b u
ζq ζ
1 ,
Re
2 p − 1 b ≥ k Re
q ζ
1 bv p − 1 b u
2.2
whenever
u qζ,
v
z ∈ U, ζ ∈ ∂U \ Eq, and k ≥ p.
Theorem 2.2. Let φ ∈ ΦJ Ω, q. If f ∈ Ap satisfies
φ Js2,b fz, Js1,b fz, Js,b fz; z : z ∈ U ⊂ Ω,
2.3
then
Js2,b fz ≺ qz
z ∈ U.
2.4
Proof. The following relation obtained in 13
fz p − 1 b Js1,b fz 1 bJs,b fz
zJs1,b
2.5
is equivalent to
Js,b fz fz − p − 1 b Js1,b fz
zJs1,b
1b
2.6
,
and hence
Js1,b fz fz − p − 1 b Js2,b fz
zJs2,b
1b
.
2.7
Define the analytic function j in U by
jz Js2,b fz,
2.8
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and then we get
Js,b fz zj z − p − 1 b jz
,
Js1,b fz 1b
2
z2 j z 1 − 2 p − 1 b zj z p − 1 b jz
1 b2
2.9
.
Further, let us define the transformations from C3 to C by
w
d − p − 1 b c
,
u c, v 1b
2
e 1 − 2 p − 1 b d p − 1 b c
1 b2
2.10
.
Let
ψc, d, e; z φu, v, w; z,
2.11
2
d − p − 1 b c e 1 − 2 p − 1 b d p − 1 b c
,
;z .
φu, v, w; z φ c,
1b
1 b2
2.12
The proof will make use of Theorem 1.4. Using 2.8 and 2.9, from 2.12 we obtain
ψ jz, zj z, z2 j z; z φ Js2,b fz, Js1,b fz, Js,b fz; z .
2.13
Hence 2.3 becomes
ψ jz, zj z, z2 j z; z ∈ Ω.
2.14
Note that
2
1 b2 w − p − 1 b u
e
1
2 p − 1 b ,
d
1 bv p − 1 b u
2.15
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,
jz ≺ qz,
2.16
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or
Js2,b fz ≺ qz.
2.17
In the case φu, 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:
kζq ζ − p − 1 b qζ
∈
/ Ω,
v
1b
2.18
z ∈ U, ζ ∈ ∂U \ Eq, and k ≥ p and φ ∈ ΦJv Ω, q. If f ∈ Ap satisfies
Js1,b fz ⊂ Ω,
2.19
then
Js2,b fz ≺ qz
z ∈ U.
2.20
If Ω /
C is a simply connected domain, then Ω ∈ hU for some conformal mapping
hz 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 Js2,b fz, Js1,b fz, Js,b fz; z ≺ hz,
2.21
Js2,b fz ≺ qz.
2.22
then
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 q0 0. Let φ ∈ ΦJ Ω, qρ for some ρ ∈ 0, 1
where qρ z qρz. If f ∈ Ap and
φ Js2,b fz, Js1,b fz, Js,b fz; z ∈ Ω,
2.23
Js2,b fz ≺ qz.
2.24
then
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Proof. From Theorem 2.2, we see that Js2,b fz ≺ qρ z and the proof is complete.
Theorem 2.6. Let h and q be univalent in U, with q0 0 and set qρ z qρz and hρ z hρz. Let φ : C3 × U → C satisfy one of the following conditions:
1 φ ∈ ΦJ h, qρ , for some ρ ∈ 0, 1, or
2 there exists ρ0 ∈ 0, 1 such that φ ∈ ΦJ hρ , qρ , for all ρ0 ∈ 0, 1.
If f ∈ Ap satisfies 2.21, then
Js2,b fz ≺ qz.
2.25
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 qz, zq z, z2 q z; z hz
2.26
has a solution q with q0 0 and satisfy one of the following conditions:
1 q ∈ Q0 and φ ∈ ΦJ h, q,
2 q is univalent in U and φ ∈ ΦJ h, qρ , for some ρ ∈ 0, 1, or
3 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
Js2,b fz ≺ qz,
2.27
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:
φu, v, w; z ∈
/Ω
2.28
whenever
u qζ,
kζq ζ − bqζ v
b ∈ C \ Z0 0, −1, −2, . . . , p ∈ N ,
1b
ζq ζ
1 b2 w − b2 u
Re
− 2b ≥ k Re
1
,
q ζ
1 bv bu
z ∈ U, ζ ∈ ∂U \ Eq, and k ≥ 1.
2.29
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Theorem 2.9. Let φ ∈ ΦJ,1 Ω, q. If f ∈ Ap satisfies
Js2,b fz Js1,b fz Js,b fz
φ
,
,
; z : z ∈ U ⊂ Ω,
zp−1
zp−1
zp−1
2.30
then
Js2,b fz
≺ qz
zp−1
z ∈ U.
2.31
Proof. Define the analytic function j in U by
jz Js2,b fz
.
zp−1
2.32
Using the relations 2.5 and 2.32, we get
Js1,b fz zj z − bjz
,
1b
zp−1
Js,b fz z2 j z 2b 1zj z b2 jz
.
zp−1
1 b2
2.33
Further, let us define the transformations from C3 to C by
u c,
w
v
d bc
,
1b
e 2b 1d b2 c
1 b2
2.34
.
Let
d bc e 2b 1d b2 c
,
;z .
ψc, d, e; z φu, v, w; z φ c,
1b
1 b2
2.35
The proof will make use of Theorem 1.4. Using 2.32 and 2.33, from 2.35 we obtain
Js2,b fz Js1,b fz Js,b fz
ψ jz, zj z, z2 j z; z φ
,
,
;
z
.
zp−1
zp−1
zp−1
2.36
Hence 2.30 becomes
ψ jz, zj z, z2 j z; z ∈ Ω.
2.37
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Note that
e
1 b2 w − b2 u
1
− 2b,
d
1 bv bu
2.38
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 ∈ ΨΩ, q, and by Theorem 1.4,
jz ≺ qz,
2.39
Js2,b fz
≺ qz.
zp−1
2.40
or
In the case φu, 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 − pqζ
∈
/ Ω,
1b
2.41
z ∈ U, ζ ∈ ∂U \ Eq, and k ≥ p and φ ∈ ΦJv ,1 Ω, q. If f ∈ Ap satisfies
Js1,b fz Js,b fz
−
⊂ Ω z ∈ U,
zp−1
zp−1
2.42
then
Js2,b fz
≺ qz
zp−1
z ∈ U.
2.43
If Ω / C is a simply connected domain, then Ω ∈ hU for some conformal mapping
hz 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
Js2,b fz Js1,b fz Js,b fz
φ
,
,
; z ≺ hz,
zp−1
zp−1
zp−1
2.44
Js2,b fz
≺ qz.
zp−1
2.45
then
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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:
φu, v, w; z ∈
/Ω
2.46
whenever
u qζ,
kζq ζ
1 bqζ
v qζ b ∈ C \ Z 0 0, −1, −2, . . . , p ∈ N ,
ζq ζ
w − u1 bu
1 bw − 3u ≥ k Re
Re
1
,
v−u
q ζ
2.47
z ∈ U, ζ ∈ ∂U \ Eq, and k ≥ 1.
Theorem 2.13. Let φ ∈ ΦJ,2 Ω, q. If f ∈ Ap satisfies
Js2,b fz Js1,b fz Js,b fz
,
,
; z : z ∈ U ⊂ Ω,
φ
Js3,b fz Js2,b fz Js1,b fz
2.48
then
Js2,b fz
≺ qz
Js3,b fz
z ∈ U.
2.49
Proof. Define the analytic function j in U by
jz Js2,b fz
.
Js3,b fz
2.50
Differentiating 2.50 yields
zj z zJs2,b fz Js3,b fz
−
.
jz
Js2,b fz
Js3,b fz
2.51
From the relation 2.5 we get
zJs2,b
fz
zj z
p − 1 b 1 bj ,
Js2,b fz
jz
2.52
Js1,b fz
zj z
jz .
Js2,b fz
1 bjz
2.53
and hence
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Further computations show that
21 bjz 1 zj z z2 j z
Js,b fz
jz .
Js1,b fz
1 b2 jz2 1 bzj z
2.54
Let us define the transformations from C3 to C by
u c,
wc
vc
d
,
1 bc
2b 1c 1d e
1 b2 c2 1 bd
2.55
.
Let
d
2b 1c 1d e
ψc, d, e; z φu, v, w; z φ c, c ;z .
,c 1 bc
1 b2 c2 1 bd
2.56
The proof will make use of Theorem 1.4. Using 2.50, 2.53 and 2.54, from 2.56 we obtain
Js2,b fz Js1,b fz Js,b fz
,
,
;z .
ψ jz, zj z, z2 j z; z φ
Js3,b fz Js2,b fz Js1,b fz
2.57
Hence 2.48 becomes
ψ jz, zj z, z2 j z; z ∈ Ω.
2.58
Note that
e
w − u1 bu
1
1 bw − 3u,
d
v−u
2.59
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 ∈ ΨΩ, q and by Theorem 1.4,
jz ≺ qz,
2.60
Js2,b fz
≺ qz.
Js3,b fz
2.61
or
If Ω /
C is a simply connected domain, then Ω ∈ hU for some conformal mapping
hz of U onto Ω and the class is written as ΦJ,2 h, q. The following result follows
immediately from Theorem 2.13.
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Theorem 2.14. Let φ ∈ ΦJ,2 Ω, q. If f ∈ Ap satisfies
φ
Js2,b fz Js1,b fz Js,b fz
,
,
; z ≺ hz,
Js3,b fz Js2,b fz Js1,b fz
2.62
then
Js2,b fz
≺ qz.
Js3,b fz
2.63
3. Superordination Results Associated with
Generalized Srivastava-Attiya Operator
Definition 3.1. Let Ω be a set in C and q ∈ H0, 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:
φu, v, w; ζ ∈
/Ω
3.1
whenever
zq z − m p − 1 b qz b ∈ C \ Z 0 0, −1, −2, . . . , p ∈ N ,
u qz, v m1 b
2
1 b2 w − p − 1 b u
zq z
1
Re
1
,
Re
2 p − 1 b ≥
m
q z
1 bv p − 1 b u
3.2
z ∈ U, ζ ∈ ∂U, and m ≥ p.
Theorem 3.2. Let φ ∈ ΦJ Ω, q. If f ∈ Ap , Js2,b f ∈ Q0 and
φ Js2,b fz, Js1,b fz, Js,b fz; z
3.3
Ω ⊂ φ Js2,b fz, Js1,b fz, Js,b fz; z : z ∈ U
3.4
qz ≺ Js2,b fz.
3.5
Ω ⊂ ψ jz, zj z, z2 j z; z : z ∈ U .
3.6
is univalent in U, then
implies that
Proof. From 2.13 and 3.4, we have
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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, qz ≺ jz or
qz ≺ Js2,b fz.
3.7
If Ω /
C is a simply connected domain, then Ω ∈ hU for some conformal mapping
hz 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 , Js2,b fz ∈ Q0 and
φ Js2,b fz, Js1,b fz, Js,b fz; z
3.8
hz ≺ φ Js2,b fz, Js1,b fz, Js,b fz; z ,
3.9
qz ≺ Js2,b fz.
3.10
is univalent in U, then
and then
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 qz, zq z, z2 q z; z hz
3.11
has a solution q ∈ Q0 . If φ ∈ ΦJ Ω, q, f ∈ Ap , Js2,b fz ∈ Q0 , and
φ Js2,b fz, Js1,b fz, Js,b fz; z
3.12
hz ≺ φ Js2,b fz, Js1,b fz, Js,b fz; z
3.13
qz ≺ Js2,b fz,
3.14
is univalent in U, then
implies that
and qz is the best subordinant.
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International Journal of Differential Equations
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 , Js2,b fz ∈ H0, p ∩ Q0 , and
φ Js2,b fz, Js1,b fz, Js,b fz; z
3.15
h1 z ≺ φ Js2,b fz, Js1,b fz, Js,b fz; z ≺ h2 z
3.16
q1 z ≺ Js2,b fz ≺ q2 z.
3.17
is univalent in U, then
implies that
Definition 3.6. Let Ω be a set in C and q ∈ H0 with zqz / 0. The class of admissible functions
ΦJ,1 Ω, q consists of those functions φ : C3 × U → C that satisfy the admissibility condition:
φu, v, w; ζ ∈ Ω
3.18
whenever
u qz,
v
zq z − mbqz
m1 b
1 b2 w − b2 u
Re
− 2b
1 bv bu
b ∈ C \ Z 0 0, −1, −2, . . . , p ∈ N ,
zq z
1
Re
≥
1
,
m
q z
3.19
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 , Js2,b fz/zp−1 ∈ Q0 , and
φ
Js2,b fz Js1,b fz Js,b fz
,
,
;
z
zp−1
zp−1
zp−1
3.20
is univalent in U, then
Js2,b fz Js1,b fz Js,b fz
,
,
;z : z ∈ U
Ω⊂ φ
zp−1
zp−1
zp−1
3.21
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15
implies that
qz ≺
Js2,b fz
.
zp−1
3.22
Proof. From 2.36 and 3.21, we have
Ω ⊂ φ jz, zj z, z2 j z; z : z ∈ U .
3.23
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,
qz ≺ jz or
qz ≺
Js2,b fz
.
zp−1
3.24
If Ω /
C is a simply connected domain, then Ω ∈ hU for some conformal mapping
hz 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 ,
Js2,b fz/zp−1 ∈ Q0 , and
φ
Js2,b fz Js1,b fz Js,b fz
,
,
;
z
zp−1
zp−1
zp−1
3.25
is univalent in U, then
hz ≺ φ
Js2,b fz Js1,b fz Js,b fz
,
,
;
z
zp−1
zp−1
zp−1
3.26
implies that
qz ≺
Js2,b fz
.
zp−1
3.27
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 , Js2,b fz/zp−1 ∈ H0 ∩ Q0 , and
Js2,b fz Js1,b fz Js,b fz
φ
,
,
;z
zp−1
zp−1
zp−1
3.28
16
International Journal of Differential Equations
is univalent in U, then
Js2,b fz Js1,b fz Js,b fz
h1 z ≺ φ
,
,
; z ≺ h2 z
zp−1
zp−1
zp−1
3.29
implies that
q1 z ≺
Js2,b fz
≺ q2 z.
zp−1
3.30
0, and q ∈ H. The class of admissible
Definition 3.10. Let Ω be a set in C and qz /
0, zq z /
functions ΦJ,2 Ω, q consists of those functions φ : C3 × U → C that satisfy the admissibility
condition:
φu, v, w; ζ ∈
/Ω
3.31
whenever
u qz,
v qz zq z
m1 bqz
b ∈ C \ Z 0 0, −1, −2, . . . , p ∈ N ,
w − u1 bu
1 bw − 3u
Re
v−u
zq z
1
≥
Re
1 ,
m
q z
3.32
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 , Js2,b fz/Js3,b fz ∈ Q1 , and
φ
Js2,b fz Js1,b fz Js,b fz
,
,
;z
Js3,b fz Js2,b fz Js1,b fz
3.33
is univalent in U, then
Js2,b fz Js1,b fz Js,b fz
Ω⊂ φ
,
,
;z : z ∈ U
Js3,b fz Js2,b fz Js1,b fz
3.34
implies that
qz ≺
Js2,b fz
.
Js3,b fz
3.35
Proof. From 2.57 and 3.34, we have
Ω ⊂ φ jz, zj z, z2 j z; z : z ∈ U .
3.36
International Journal of Differential Equations
17
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,
qz ≺ jz or
qz ≺
Js2,b fz
.
Js3,b fz
3.37
If Ω /
C is a simply connected domain, then Ω ∈ hU for some conformal mapping
hz 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 , Js2,b fz/
Js3,b fz ∈ Q1 and
φ
Js2,b fz Js1,b fz Js,b fz
,
,
;z
Js3,b fz Js2,b fz Js1,b fz
3.38
is univalent in U, then
hz ≺ φ
Js2,b fz Js1,b fz Js,b fz
,
,
;z
Js3,b fz Js2,b fz Js1,b fz
3.39
implies that
qz ≺
Js2,b fz
.
Js3,b fz
3.40
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 , Js2,b fz/Js3,b fz ∈ H ∩ Q1 ,
and
φ
Js2,b fz Js1,b fz Js,b fz
,
,
;z
Js3,b fz Js2,b fz Js1,b fz
3.41
Js2,b fz Js1,b fz Js,b fz
,
,
; z ≺ h2 z
Js3,b fz Js2,b fz Js1,b fz
3.42
is univalent in U, then
h1 z ≺ φ
implies that
q1 z ≺
Js2,b fz
≺ qz.
Js3,b fz
3.43
18
International Journal of Differential Equations
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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